3.607 \(\int \frac{1}{x^{5/2} (a-b x)^{5/2}} \, dx\)

Optimal. Leaf size=88 \[ -\frac{32 b \sqrt{a-b x}}{3 a^4 \sqrt{x}}-\frac{16 \sqrt{a-b x}}{3 a^3 x^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a-b x}}+\frac{2}{3 a x^{3/2} (a-b x)^{3/2}} \]

[Out]

2/(3*a*x^(3/2)*(a - b*x)^(3/2)) + 4/(a^2*x^(3/2)*Sqrt[a - b*x]) - (16*Sqrt[a - b
*x])/(3*a^3*x^(3/2)) - (32*b*Sqrt[a - b*x])/(3*a^4*Sqrt[x])

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Rubi [A]  time = 0.065912, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{32 b \sqrt{a-b x}}{3 a^4 \sqrt{x}}-\frac{16 \sqrt{a-b x}}{3 a^3 x^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a-b x}}+\frac{2}{3 a x^{3/2} (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a - b*x)^(5/2)),x]

[Out]

2/(3*a*x^(3/2)*(a - b*x)^(3/2)) + 4/(a^2*x^(3/2)*Sqrt[a - b*x]) - (16*Sqrt[a - b
*x])/(3*a^3*x^(3/2)) - (32*b*Sqrt[a - b*x])/(3*a^4*Sqrt[x])

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Rubi in Sympy [A]  time = 10.0195, size = 78, normalized size = 0.89 \[ \frac{2}{3 a x^{\frac{3}{2}} \left (a - b x\right )^{\frac{3}{2}}} + \frac{4}{a^{2} x^{\frac{3}{2}} \sqrt{a - b x}} - \frac{16 \sqrt{a - b x}}{3 a^{3} x^{\frac{3}{2}}} - \frac{32 b \sqrt{a - b x}}{3 a^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(-b*x+a)**(5/2),x)

[Out]

2/(3*a*x**(3/2)*(a - b*x)**(3/2)) + 4/(a**2*x**(3/2)*sqrt(a - b*x)) - 16*sqrt(a
- b*x)/(3*a**3*x**(3/2)) - 32*b*sqrt(a - b*x)/(3*a**4*sqrt(x))

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Mathematica [A]  time = 0.0436002, size = 50, normalized size = 0.57 \[ -\frac{2 \left (a^3+6 a^2 b x-24 a b^2 x^2+16 b^3 x^3\right )}{3 a^4 x^{3/2} (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(a - b*x)^(5/2)),x]

[Out]

(-2*(a^3 + 6*a^2*b*x - 24*a*b^2*x^2 + 16*b^3*x^3))/(3*a^4*x^(3/2)*(a - b*x)^(3/2
))

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Maple [A]  time = 0.006, size = 45, normalized size = 0.5 \[ -{\frac{32\,{b}^{3}{x}^{3}-48\,a{b}^{2}{x}^{2}+12\,{a}^{2}bx+2\,{a}^{3}}{3\,{a}^{4}}{x}^{-{\frac{3}{2}}} \left ( -bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(-b*x+a)^(5/2),x)

[Out]

-2/3*(16*b^3*x^3-24*a*b^2*x^2+6*a^2*b*x+a^3)/x^(3/2)/(-b*x+a)^(3/2)/a^4

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Maxima [A]  time = 1.34177, size = 92, normalized size = 1.05 \[ -\frac{2 \,{\left (\frac{9 \, \sqrt{-b x + a} b}{\sqrt{x}} + \frac{{\left (-b x + a\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}}\right )}}{3 \, a^{4}} + \frac{2 \,{\left (b^{3} - \frac{9 \,{\left (b x - a\right )} b^{2}}{x}\right )} x^{\frac{3}{2}}}{3 \,{\left (-b x + a\right )}^{\frac{3}{2}} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + a)^(5/2)*x^(5/2)),x, algorithm="maxima")

[Out]

-2/3*(9*sqrt(-b*x + a)*b/sqrt(x) + (-b*x + a)^(3/2)/x^(3/2))/a^4 + 2/3*(b^3 - 9*
(b*x - a)*b^2/x)*x^(3/2)/((-b*x + a)^(3/2)*a^4)

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Fricas [A]  time = 0.227388, size = 78, normalized size = 0.89 \[ \frac{2 \,{\left (16 \, b^{3} x^{3} - 24 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}\right )}}{3 \,{\left (a^{4} b x^{2} - a^{5} x\right )} \sqrt{-b x + a} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + a)^(5/2)*x^(5/2)),x, algorithm="fricas")

[Out]

2/3*(16*b^3*x^3 - 24*a*b^2*x^2 + 6*a^2*b*x + a^3)/((a^4*b*x^2 - a^5*x)*sqrt(-b*x
+ a)*sqrt(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(-b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.237783, size = 277, normalized size = 3.15 \[ -\frac{\sqrt{-b x + a}{\left (\frac{8 \,{\left (b x - a\right )} a{\left | b \right |}}{b^{2}} + \frac{9 \, a^{2}{\left | b \right |}}{b^{2}}\right )}}{24 \,{\left ({\left (b x - a\right )} b + a b\right )}^{\frac{3}{2}}} - \frac{8 \,{\left (3 \,{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt{-b} b^{3} - 9 \, a{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} \sqrt{-b} b^{4} + 4 \, a^{2} \sqrt{-b} b^{5}\right )}}{3 \,{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} a^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x + a)^(5/2)*x^(5/2)),x, algorithm="giac")

[Out]

-1/24*sqrt(-b*x + a)*(8*(b*x - a)*a*abs(b)/b^2 + 9*a^2*abs(b)/b^2)/((b*x - a)*b
+ a*b)^(3/2) - 8/3*(3*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^4*sqrt
(-b)*b^3 - 9*a*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2*sqrt(-b)*b^
4 + 4*a^2*sqrt(-b)*b^5)/(((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2
- a*b)^3*a^3*abs(b))