The accident happened on the Edinburgh bypass between Baberton and Dreghorn at about 17:00 on Monday.
Police said a woman on the bus was taken to hospital with a "serious leg injury". Her injury is not thought to be life-threatening.
Four other people who suffered minor injuries were also taken to hospital but were later released.
The eastbound lane of the road was closed following the accident causing lengthy tailbacks. It was re-opened at 18:45.

Kulbhushan Jadhav was arrested in the restive Balochistan province and accused of "espionage and sabotage activities against Pakistan".
Last month a top UN court told Pakistan not to execute Mr Jadhav until it had time to hear a case brought by India.
India dismissed the mercy petition and a "confessional video" by Mr Jadhav.
Pakistan on Thursday released a statement that said Mr Jadhav had sought "forgiveness for his actions" and had "requested the Chief of Army Staff to spare his life on compassionate grounds".
India, which denies Mr Jadhav is a spy, has dismissed Pakistan's statement and a second "confessional" video by Mr Jadhav saying, "India expects Pakistan to desist from attempting to influence the proceedings at the International Court of Justice (ICJ) in the case through false propaganda".
"The developments bring out once again the lack of transparency and farcical nature of proceedings against Jadhav on concocted charges, continued violation of his legal and consular rights and an attempt to introduce prejudice in the proceedings in the ICJ," external affairs ministry spokesperson Gopal Baglay told reporters on Thursday.
Last month India argued in the ICJ that Pakistan violated international laws by not allowing it access to Mr Jadhav since he was arrested in Balochistan in March 2016.
Meanwhile, there are differing accounts of how and where Mr Jadhav was detained.
Pakistan says he was detained on 3 March 2016 in restive Balochistan province, which has been hit by a separatist insurgency that Islamabad accuses India of backing. India says he was kidnapped by Pakistan while he was in Iran.
The nuclear-armed neighbours have a long history of diplomatic spats, and Delhi and Islamabad often accuse each other of sending spies into their territories.

Inflation dropped to 2.6% in June from 2.9% the month before, whereas economists had expected no change.
Sterling had been trading above $1.31 at one point before the inflation figures were released.
However, the inflation data sent the pound down to $1.3038 at the end of London business.
The pound also fell 1% against the euro to 1.1251 euros.
"Today's slowdown in price growth should squash any speculation of a rate hike for the time being," said Neil Wilson, senior market analyst at ETX Capital.
"An August rate hike now looks highly unlikely, but we should remember that the Bank has only limited tolerance for continued above-target inflation and may yet seek to push rates back up to 0.5% this year, if conditions in the wider economy improve whilst inflation remains above 2%."
The fall in the pound initially lifted the FTSE 100 share index. The FTSE 100 often moves inversely to sterling, as a weaker pound boosts the value of overseas earnings of the multinational companies in the index.
Having been more than 40 points lower at one point in the morning, the FTSE 100 closed down 13.91 at 7,390.22.
Royal Mail shares rose 3% after the company reported a 1% rise in revenues in the three months to 25 June, helped by its Europe-wide parcels business GLS.
In the UK, the volume of letter deliveries fell again, although general election political mailings helped to slow the decline.
Five people have been injured following a crash between a bin lorry, a bus and two cars.
[NEXT_CONCEPT]
A former Indian navy officer who has been sentenced to death on charges of spying in Pakistan has filed a mercy petition, the Pakistan military says.
[NEXT_CONCEPT]
The pound fell against key currencies after an unexpected slip in inflation - seen as easing pressure on the Bank of England to lift interest rates.