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

Optimal. Leaf size=90 \[ 5 a b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )+5 b^2 \sqrt{x} \sqrt{a-b x}-\frac{2 (a-b x)^{5/2}}{3 x^{3/2}}+\frac{10 b (a-b x)^{3/2}}{3 \sqrt{x}} \]

[Out]

5*b^2*Sqrt[x]*Sqrt[a - b*x] + (10*b*(a - b*x)^(3/2))/(3*Sqrt[x]) - (2*(a - b*x)^
(5/2))/(3*x^(3/2)) + 5*a*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]]

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Rubi [A]  time = 0.0669046, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ 5 a b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )+5 b^2 \sqrt{x} \sqrt{a-b x}-\frac{2 (a-b x)^{5/2}}{3 x^{3/2}}+\frac{10 b (a-b x)^{3/2}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x)^(5/2)/x^(5/2),x]

[Out]

5*b^2*Sqrt[x]*Sqrt[a - b*x] + (10*b*(a - b*x)^(3/2))/(3*Sqrt[x]) - (2*(a - b*x)^
(5/2))/(3*x^(3/2)) + 5*a*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]]

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Rubi in Sympy [A]  time = 10.374, size = 82, normalized size = 0.91 \[ 5 a b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )} + 5 b^{2} \sqrt{x} \sqrt{a - b x} + \frac{10 b \left (a - b x\right )^{\frac{3}{2}}}{3 \sqrt{x}} - \frac{2 \left (a - b x\right )^{\frac{5}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a*b**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a - b*x)) + 5*b**2*sqrt(x)*sqrt(a - b*x)
+ 10*b*(a - b*x)**(3/2)/(3*sqrt(x)) - 2*(a - b*x)**(5/2)/(3*x**(3/2))

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Mathematica [A]  time = 0.0763083, size = 69, normalized size = 0.77 \[ \frac{\sqrt{a-b x} \left (-2 a^2+14 a b x+3 b^2 x^2\right )}{3 x^{3/2}}+5 a b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a - b*x]*(-2*a^2 + 14*a*b*x + 3*b^2*x^2))/(3*x^(3/2)) + 5*a*b^(3/2)*ArcTan
[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]]

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Maple [A]  time = 0.032, size = 86, normalized size = 1. \[ -{\frac{-3\,{b}^{2}{x}^{2}-14\,abx+2\,{a}^{2}}{3}\sqrt{-bx+a}{x}^{-{\frac{3}{2}}}}+{\frac{5\,a}{2}{b}^{{\frac{3}{2}}}\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+a)^(5/2)/x^(5/2),x)

[Out]

-1/3*(-b*x+a)^(1/2)*(-3*b^2*x^2-14*a*b*x+2*a^2)/x^(3/2)+5/2*a*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)^(5/2)/x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222688, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a \sqrt{-b} b x^{2} \log \left (-2 \, b x - 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) + 2 \,{\left (3 \, b^{2} x^{2} + 14 \, a b x - 2 \, a^{2}\right )} \sqrt{-b x + a} \sqrt{x}}{6 \, x^{2}}, -\frac{15 \, a b^{\frac{3}{2}} x^{2} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (3 \, b^{2} x^{2} + 14 \, a b x - 2 \, a^{2}\right )} \sqrt{-b x + a} \sqrt{x}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(15*a*sqrt(-b)*b*x^2*log(-2*b*x - 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) +
2*(3*b^2*x^2 + 14*a*b*x - 2*a^2)*sqrt(-b*x + a)*sqrt(x))/x^2, -1/3*(15*a*b^(3/2)
*x^2*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) - (3*b^2*x^2 + 14*a*b*x - 2*a^2)*s
qrt(-b*x + a)*sqrt(x))/x^2]

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Sympy [A]  time = 77.8715, size = 245, normalized size = 2.72 \[ \begin{cases} - \frac{2 a^{2} \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{3 x} + \frac{14 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}}{3} - 5 i a b^{\frac{3}{2}} \log{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + \frac{5 i a b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )}}{2} + 5 a b^{\frac{3}{2}} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} + b^{\frac{5}{2}} x \sqrt{\frac{a}{b x} - 1} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\- \frac{2 i a^{2} \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{3 x} + \frac{14 i a b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{3} + \frac{5 i a b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )}}{2} - 5 i a b^{\frac{3}{2}} \log{\left (\sqrt{- \frac{a}{b x} + 1} + 1 \right )} + i b^{\frac{5}{2}} x \sqrt{- \frac{a}{b x} + 1} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*a**2*sqrt(b)*sqrt(a/(b*x) - 1)/(3*x) + 14*a*b**(3/2)*sqrt(a/(b*x)
- 1)/3 - 5*I*a*b**(3/2)*log(sqrt(a)/(sqrt(b)*sqrt(x))) + 5*I*a*b**(3/2)*log(a/(b
*x))/2 + 5*a*b**(3/2)*asin(sqrt(b)*sqrt(x)/sqrt(a)) + b**(5/2)*x*sqrt(a/(b*x) -
1), Abs(a/(b*x)) > 1), (-2*I*a**2*sqrt(b)*sqrt(-a/(b*x) + 1)/(3*x) + 14*I*a*b**(
3/2)*sqrt(-a/(b*x) + 1)/3 + 5*I*a*b**(3/2)*log(a/(b*x))/2 - 5*I*a*b**(3/2)*log(s
qrt(-a/(b*x) + 1) + 1) + I*b**(5/2)*x*sqrt(-a/(b*x) + 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)^(5/2)/x^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError