3.532 \(\int \frac{(a-b x)^{3/2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ 2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )-\frac{2 (a-b x)^{3/2}}{3 x^{3/2}}+\frac{2 b \sqrt{a-b x}}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0508722, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ 2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )-\frac{2 (a-b x)^{3/2}}{3 x^{3/2}}+\frac{2 b \sqrt{a-b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x)^(3/2)/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.50989, size = 60, normalized size = 0.9 \[ 2 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )} + \frac{2 b \sqrt{a - b x}}{\sqrt{x}} - \frac{2 \left (a - b x\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x+a)**(3/2)/x**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0538307, size = 55, normalized size = 0.82 \[ 2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )-\frac{2 (a-4 b x) \sqrt{a-b x}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x)^(3/2)/x^(5/2),x]

[Out]

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

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Maple [A]  time = 0.029, size = 71, normalized size = 1.1 \[ -{\frac{-8\,bx+2\,a}{3}\sqrt{-bx+a}{x}^{-{\frac{3}{2}}}}+{1{b}^{{\frac{3}{2}}}\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+a)^(3/2)/x^(5/2),x)

[Out]

-2/3*(-b*x+a)^(1/2)*(-4*b*x+a)/x^(3/2)+b^(3/2)*arctan(b^(1/2)*(x-1/2*a/b)/(-b*x^
2+a*x)^(1/2))*(x*(-b*x+a))^(1/2)/x^(1/2)/(-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)^(3/2)/x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220202, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{-b} b x^{2} \log \left (-2 \, b x - 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) + 2 \,{\left (4 \, b x - a\right )} \sqrt{-b x + a} \sqrt{x}}{3 \, x^{2}}, -\frac{2 \,{\left (3 \, b^{\frac{3}{2}} x^{2} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (4 \, b x - a\right )} \sqrt{-b x + a} \sqrt{x}\right )}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x + a)^(3/2)/x^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*sqrt(-b)*b*x^2*log(-2*b*x - 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) + 2*(
4*b*x - a)*sqrt(-b*x + a)*sqrt(x))/x^2, -2/3*(3*b^(3/2)*x^2*arctan(sqrt(-b*x + a
)/(sqrt(b)*sqrt(x))) - (4*b*x - a)*sqrt(-b*x + a)*sqrt(x))/x^2]

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Sympy [A]  time = 28.028, size = 187, normalized size = 2.79 \[ \begin{cases} - \frac{2 a \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{3 x} + \frac{8 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}}{3} - 2 i b^{\frac{3}{2}} \log{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + i b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )} + 2 b^{\frac{3}{2}} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\- \frac{2 i a \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{3 x} + \frac{8 i b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{3} + i b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )} - 2 i b^{\frac{3}{2}} \log{\left (\sqrt{- \frac{a}{b x} + 1} + 1 \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x+a)**(3/2)/x**(5/2),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x + a)^(3/2)/x^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError