3.2509 \(\int \frac{1}{x^3 \left (a+b x^n\right )^{5/2}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{\, _2F_1\left (1,-\frac{3}{2}-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2 \left (a+b x^n\right )^{3/2}} \]

[Out]

-Hypergeometric2F1[1, -3/2 - 2/n, -((2 - n)/n), -((b*x^n)/a)]/(2*a*x^2*(a + b*x^
n)^(3/2))

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Rubi [A]  time = 0.0685791, antiderivative size = 63, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{5}{2},-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a^2 x^2 \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^n)^(5/2)),x]

[Out]

-(Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[5/2, -2/n, -((2 - n)/n), -((b*x^n)/a)])/
(2*a^2*x^2*Sqrt[a + b*x^n])

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Rubi in Sympy [A]  time = 7.08746, size = 49, normalized size = 0.96 \[ - \frac{\sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{3} x^{2} \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a+b*x**n)**(5/2),x)

[Out]

-sqrt(a + b*x**n)*hyper((5/2, -2/n), ((n - 2)/n,), -b*x**n/a)/(2*a**3*x**2*sqrt(
1 + b*x**n/a))

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Mathematica [A]  time = 0.152639, size = 101, normalized size = 1.98 \[ \frac{-\left (3 n^2+16 n+16\right ) \left (a+b x^n\right ) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )+4 (3 n+4) \left (a+b x^n\right )+4 a n}{6 a^2 n^2 x^2 \left (a+b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^n)^(5/2)),x]

[Out]

(4*a*n + 4*(4 + 3*n)*(a + b*x^n) - (16 + 16*n + 3*n^2)*(a + b*x^n)*Sqrt[1 + (b*x
^n)/a]*Hypergeometric2F1[1/2, -2/n, (-2 + n)/n, -((b*x^n)/a)])/(6*a^2*n^2*x^2*(a
+ b*x^n)^(3/2))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( a+b{x}^{n} \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b*x^n)^(5/2),x)

[Out]

int(1/x^3/(a+b*x^n)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^(5/2)*x^3),x, algorithm="maxima")

[Out]

integrate(1/((b*x^n + a)^(5/2)*x^3), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^(5/2)*x^3),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a+b*x**n)**(5/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^(5/2)*x^3),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + a)^(5/2)*x^3), x)