3.2468 \(\int \frac{1}{x^2 \left (a+b x^n\right )} \, dx\)

Optimal. Leaf size=34 \[ -\frac{\, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0355479, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^n)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.02507, size = 20, normalized size = 0.59 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-hyper((1, -1/n), ((n - 1)/n,), -b*x**n/a)/(a*x)

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Mathematica [A]  time = 0.0123853, size = 31, normalized size = 0.91 \[ -\frac{\, _2F_1\left (1,-\frac{1}{n};1-\frac{1}{n};-\frac{b x^n}{a}\right )}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*x^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b x^{2} x^{n} + a x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(b*x^2*x^n + a*x^2), x)

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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),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)*x^2), x)