3.328 \(\int \frac{(-a+b x)^{3/2}}{x} \, dx\)

Optimal. Leaf size=55 \[ 2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )-2 a \sqrt{b x-a}+\frac{2}{3} (b x-a)^{3/2} \]

[Out]

-2*a*Sqrt[-a + b*x] + (2*(-a + b*x)^(3/2))/3 + 2*a^(3/2)*ArcTan[Sqrt[-a + b*x]/S
qrt[a]]

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Rubi [A]  time = 0.0519876, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )-2 a \sqrt{b x-a}+\frac{2}{3} (b x-a)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(-a + b*x)^(3/2)/x,x]

[Out]

-2*a*Sqrt[-a + b*x] + (2*(-a + b*x)^(3/2))/3 + 2*a^(3/2)*ArcTan[Sqrt[-a + b*x]/S
qrt[a]]

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Rubi in Sympy [A]  time = 7.16547, size = 44, normalized size = 0.8 \[ 2 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )} - 2 a \sqrt{- a + b x} + \frac{2 \left (- a + b x\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x-a)**(3/2)/x,x)

[Out]

2*a**(3/2)*atan(sqrt(-a + b*x)/sqrt(a)) - 2*a*sqrt(-a + b*x) + 2*(-a + b*x)**(3/
2)/3

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Mathematica [A]  time = 0.0564319, size = 48, normalized size = 0.87 \[ 2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )+\frac{2}{3} (b x-4 a) \sqrt{b x-a} \]

Antiderivative was successfully verified.

[In]  Integrate[(-a + b*x)^(3/2)/x,x]

[Out]

(2*(-4*a + b*x)*Sqrt[-a + b*x])/3 + 2*a^(3/2)*ArcTan[Sqrt[-a + b*x]/Sqrt[a]]

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Maple [A]  time = 0.01, size = 44, normalized size = 0.8 \[{\frac{2}{3} \left ( bx-a \right ) ^{{\frac{3}{2}}}}+2\,{a}^{3/2}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) -2\,a\sqrt{bx-a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x-a)^(3/2)/x,x)

[Out]

2/3*(b*x-a)^(3/2)+2*a^(3/2)*arctan((b*x-a)^(1/2)/a^(1/2))-2*a*(b*x-a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x - a)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231098, size = 1, normalized size = 0.02 \[ \left [\sqrt{-a} a \log \left (\frac{b x + 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + \frac{2}{3} \, \sqrt{b x - a}{\left (b x - 4 \, a\right )}, 2 \, a^{\frac{3}{2}} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + \frac{2}{3} \, \sqrt{b x - a}{\left (b x - 4 \, a\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x - a)^(3/2)/x,x, algorithm="fricas")

[Out]

[sqrt(-a)*a*log((b*x + 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) + 2/3*sqrt(b*x - a)*(b
*x - 4*a), 2*a^(3/2)*arctan(sqrt(b*x - a)/sqrt(a)) + 2/3*sqrt(b*x - a)*(b*x - 4*
a)]

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Sympy [A]  time = 3.70401, size = 187, normalized size = 3.4 \[ \begin{cases} - \frac{8 a^{\frac{3}{2}} \sqrt{-1 + \frac{b x}{a}}}{3} - i a^{\frac{3}{2}} \log{\left (\frac{b x}{a} \right )} + 2 i a^{\frac{3}{2}} \log{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - 2 a^{\frac{3}{2}} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 \sqrt{a} b x \sqrt{-1 + \frac{b x}{a}}}{3} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{8 i a^{\frac{3}{2}} \sqrt{1 - \frac{b x}{a}}}{3} - i a^{\frac{3}{2}} \log{\left (\frac{b x}{a} \right )} + 2 i a^{\frac{3}{2}} \log{\left (\sqrt{1 - \frac{b x}{a}} + 1 \right )} + \frac{2 i \sqrt{a} b x \sqrt{1 - \frac{b x}{a}}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x-a)**(3/2)/x,x)

[Out]

Piecewise((-8*a**(3/2)*sqrt(-1 + b*x/a)/3 - I*a**(3/2)*log(b*x/a) + 2*I*a**(3/2)
*log(sqrt(b)*sqrt(x)/sqrt(a)) - 2*a**(3/2)*asin(sqrt(a)/(sqrt(b)*sqrt(x))) + 2*s
qrt(a)*b*x*sqrt(-1 + b*x/a)/3, Abs(b*x/a) > 1), (-8*I*a**(3/2)*sqrt(1 - b*x/a)/3
- I*a**(3/2)*log(b*x/a) + 2*I*a**(3/2)*log(sqrt(1 - b*x/a) + 1) + 2*I*sqrt(a)*b
*x*sqrt(1 - b*x/a)/3, True))

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GIAC/XCAS [A]  time = 0.204355, size = 58, normalized size = 1.05 \[ 2 \, a^{\frac{3}{2}} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + \frac{2}{3} \,{\left (b x - a\right )}^{\frac{3}{2}} - 2 \, \sqrt{b x - a} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x - a)^(3/2)/x,x, algorithm="giac")

[Out]

2*a^(3/2)*arctan(sqrt(b*x - a)/sqrt(a)) + 2/3*(b*x - a)^(3/2) - 2*sqrt(b*x - a)*
a