3.331 \(\int \frac{(-a+b x)^{5/2}}{x} \, dx\)

Optimal. Leaf size=73 \[ -2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )+2 a^2 \sqrt{b x-a}-\frac{2}{3} a (b x-a)^{3/2}+\frac{2}{5} (b x-a)^{5/2} \]

[Out]

2*a^2*Sqrt[-a + b*x] - (2*a*(-a + b*x)^(3/2))/3 + (2*(-a + b*x)^(5/2))/5 - 2*a^(
5/2)*ArcTan[Sqrt[-a + b*x]/Sqrt[a]]

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Rubi [A]  time = 0.0673964, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )+2 a^2 \sqrt{b x-a}-\frac{2}{3} a (b x-a)^{3/2}+\frac{2}{5} (b x-a)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(-a + b*x)^(5/2)/x,x]

[Out]

2*a^2*Sqrt[-a + b*x] - (2*a*(-a + b*x)^(3/2))/3 + (2*(-a + b*x)^(5/2))/5 - 2*a^(
5/2)*ArcTan[Sqrt[-a + b*x]/Sqrt[a]]

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Rubi in Sympy [A]  time = 9.57803, size = 60, normalized size = 0.82 \[ - 2 a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )} + 2 a^{2} \sqrt{- a + b x} - \frac{2 a \left (- a + b x\right )^{\frac{3}{2}}}{3} + \frac{2 \left (- a + b x\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x-a)**(5/2)/x,x)

[Out]

-2*a**(5/2)*atan(sqrt(-a + b*x)/sqrt(a)) + 2*a**2*sqrt(-a + b*x) - 2*a*(-a + b*x
)**(3/2)/3 + 2*(-a + b*x)**(5/2)/5

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Mathematica [A]  time = 0.0419299, size = 60, normalized size = 0.82 \[ \frac{2}{15} \sqrt{b x-a} \left (23 a^2-11 a b x+3 b^2 x^2\right )-2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-a + b*x)^(5/2)/x,x]

[Out]

(2*Sqrt[-a + b*x]*(23*a^2 - 11*a*b*x + 3*b^2*x^2))/15 - 2*a^(5/2)*ArcTan[Sqrt[-a
+ b*x]/Sqrt[a]]

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Maple [A]  time = 0.009, size = 58, normalized size = 0.8 \[ -{\frac{2\,a}{3} \left ( bx-a \right ) ^{{\frac{3}{2}}}}+{\frac{2}{5} \left ( bx-a \right ) ^{{\frac{5}{2}}}}-2\,{a}^{5/2}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) +2\,{a}^{2}\sqrt{bx-a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x-a)^(5/2)/x,x)

[Out]

-2/3*a*(b*x-a)^(3/2)+2/5*(b*x-a)^(5/2)-2*a^(5/2)*arctan((b*x-a)^(1/2)/a^(1/2))+2
*a^2*(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)^(5/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235498, size = 1, normalized size = 0.01 \[ \left [\sqrt{-a} a^{2} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + \frac{2}{15} \,{\left (3 \, b^{2} x^{2} - 11 \, a b x + 23 \, a^{2}\right )} \sqrt{b x - a}, -2 \, a^{\frac{5}{2}} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + \frac{2}{15} \,{\left (3 \, b^{2} x^{2} - 11 \, a b x + 23 \, a^{2}\right )} \sqrt{b x - a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x - a)^(5/2)/x,x, algorithm="fricas")

[Out]

[sqrt(-a)*a^2*log((b*x - 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) + 2/15*(3*b^2*x^2 -
11*a*b*x + 23*a^2)*sqrt(b*x - a), -2*a^(5/2)*arctan(sqrt(b*x - a)/sqrt(a)) + 2/1
5*(3*b^2*x^2 - 11*a*b*x + 23*a^2)*sqrt(b*x - a)]

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Sympy [A]  time = 6.22501, size = 240, normalized size = 3.29 \[ \begin{cases} \frac{46 a^{\frac{5}{2}} \sqrt{-1 + \frac{b x}{a}}}{15} + i a^{\frac{5}{2}} \log{\left (\frac{b x}{a} \right )} - 2 i a^{\frac{5}{2}} \log{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} + 2 a^{\frac{5}{2}} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} - \frac{22 a^{\frac{3}{2}} b x \sqrt{-1 + \frac{b x}{a}}}{15} + \frac{2 \sqrt{a} b^{2} x^{2} \sqrt{-1 + \frac{b x}{a}}}{5} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\\frac{46 i a^{\frac{5}{2}} \sqrt{1 - \frac{b x}{a}}}{15} + i a^{\frac{5}{2}} \log{\left (\frac{b x}{a} \right )} - 2 i a^{\frac{5}{2}} \log{\left (\sqrt{1 - \frac{b x}{a}} + 1 \right )} - \frac{22 i a^{\frac{3}{2}} b x \sqrt{1 - \frac{b x}{a}}}{15} + \frac{2 i \sqrt{a} b^{2} x^{2} \sqrt{1 - \frac{b x}{a}}}{5} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x-a)**(5/2)/x,x)

[Out]

Piecewise((46*a**(5/2)*sqrt(-1 + b*x/a)/15 + I*a**(5/2)*log(b*x/a) - 2*I*a**(5/2
)*log(sqrt(b)*sqrt(x)/sqrt(a)) + 2*a**(5/2)*asin(sqrt(a)/(sqrt(b)*sqrt(x))) - 22
*a**(3/2)*b*x*sqrt(-1 + b*x/a)/15 + 2*sqrt(a)*b**2*x**2*sqrt(-1 + b*x/a)/5, Abs(
b*x/a) > 1), (46*I*a**(5/2)*sqrt(1 - b*x/a)/15 + I*a**(5/2)*log(b*x/a) - 2*I*a**
(5/2)*log(sqrt(1 - b*x/a) + 1) - 22*I*a**(3/2)*b*x*sqrt(1 - b*x/a)/15 + 2*I*sqrt
(a)*b**2*x**2*sqrt(1 - b*x/a)/5, True))

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GIAC/XCAS [A]  time = 0.207043, size = 77, normalized size = 1.05 \[ -2 \, a^{\frac{5}{2}} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + \frac{2}{5} \,{\left (b x - a\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (b x - a\right )}^{\frac{3}{2}} a + 2 \, \sqrt{b x - a} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x - a)^(5/2)/x,x, algorithm="giac")

[Out]

-2*a^(5/2)*arctan(sqrt(b*x - a)/sqrt(a)) + 2/5*(b*x - a)^(5/2) - 2/3*(b*x - a)^(
3/2)*a + 2*sqrt(b*x - a)*a^2