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

Optimal. Leaf size=84 \[ \frac{32 b \sqrt{a+b x}}{3 a^4 \sqrt{x}}-\frac{16 \sqrt{a+b x}}{3 a^3 x^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a+b x}}+\frac{2}{3 a x^{3/2} (a+b x)^{3/2}} \]

[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 = 0.0629755, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{32 b \sqrt{a+b x}}{3 a^4 \sqrt{x}}-\frac{16 \sqrt{a+b x}}{3 a^3 x^{3/2}}+\frac{4}{a^2 x^{3/2} \sqrt{a+b x}}+\frac{2}{3 a x^{3/2} (a+b x)^{3/2}} \]

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 = 9.22177, size = 78, normalized size = 0.93 \[ \frac{2}{3 a x^{\frac{3}{2}} \left (a + b x\right )^{\frac{3}{2}}} + \frac{4}{a^{2} x^{\frac{3}{2}} \sqrt{a + b x}} - \frac{16 \sqrt{a + b x}}{3 a^{3} x^{\frac{3}{2}}} + \frac{32 b \sqrt{a + b x}}{3 a^{4} \sqrt{x}} \]

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 = 0.0366153, size = 49, normalized size = 0.58 \[ -\frac{2 \left (a^3-6 a^2 b x-24 a b^2 x^2-16 b^3 x^3\right )}{3 a^4 x^{3/2} (a+b x)^{3/2}} \]

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 = 0.007, size = 44, 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 = 1.34954, size = 86, normalized size = 1.02 \[ \frac{2 \,{\left (\frac{9 \, \sqrt{b x + a} b}{\sqrt{x}} - \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}}\right )}}{3 \, a^{4}} - \frac{2 \,{\left (b^{3} - \frac{9 \,{\left (b x + a\right )} b^{2}}{x}\right )} x^{\frac{3}{2}}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}} \]

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 = 0.230459, size = 78, normalized size = 0.93 \[ \frac{2 \,{\left (16 \, b^{3} x^{3} + 24 \, a b^{2} x^{2} + 6 \, a^{2} b x - a^{3}\right )}}{3 \,{\left (a^{4} b x^{2} + a^{5} x\right )} \sqrt{b x + a} \sqrt{x}} \]

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 = 0.229639, size = 234, normalized size = 2.79 \[ -\frac{\sqrt{b x + a}{\left (\frac{8 \,{\left (b x + a\right )} a{\left | b \right |}}{b^{2}} - \frac{9 \, a^{2}{\left | b \right |}}{b^{2}}\right )}}{24 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{3}{2}}} + \frac{8 \,{\left (3 \,{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{7}{2}} + 9 \, a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{9}{2}} + 4 \, a^{2} b^{\frac{11}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{3}{\left | b \right |}} \]

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*b^(7/2)
+ 9*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(9/2) + 4*a^2*b^(11
/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*a^3*abs(b))