3.2503 \(\int \frac{1}{x^2 \left (a+b x^n\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0671232, 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 \[ -\frac{\sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{3}{2},-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.11889, size = 46, normalized size = 0.94 \[ - \frac{\sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} x \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0935102, size = 67, normalized size = 1.37 \[ \frac{2-(n+2) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )}{a n x \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*x^n)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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