3.2508 \(\int \frac{1}{x^2 \left (a+b x^n\right )^{5/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\, _2F_1\left (1,-\frac{3}{2}-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x \left (a+b x^n\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0664841, antiderivative size = 61, 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 \[ -\frac{\sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{5}{2},-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^2 x \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.14974, size = 46, normalized size = 0.94 \[ - \frac{\sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{3} x \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.152693, size = 101, normalized size = 2.06 \[ \frac{-\left (3 n^2+8 n+4\right ) \left (a+b x^n\right ) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )+2 (3 n+2) \left (a+b x^n\right )+2 a n}{3 a^2 n^2 x \left (a+b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*n + 2*(2 + 3*n)*(a + b*x^n) - (4 + 8*n + 3*n^2)*(a + b*x^n)*Sqrt[1 + (b*x^n
)/a]*Hypergeometric2F1[1/2, -n^(-1), (-1 + n)/n, -((b*x^n)/a)])/(3*a^2*n^2*x*(a
+ b*x^n)^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*x^n)^(5/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)^(5/2)*x^2), 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^2),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**2/(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. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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