3.2475 \(\int \frac {1}{x^2 (a+b x^n)} \, dx\)

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]

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

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {364} \[ -\frac {\, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a x} \]

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))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a+b x^n\right )} \, dx &=-\frac {\, _2F_1\left (1,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a x}\\ \end {align*}

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Mathematica [A]  time = 0.00, 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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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b x^{2} x^{n} + a x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(b*x^2*x^n + a*x^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^n + a)*x^2), x)

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maple [F]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{n}+a \right ) x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^n + a)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{x^2\,\left (a+b\,x^n\right )} \,d x \]

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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sympy [C]  time = 0.92, size = 39, normalized size = 1.15 \[ - \frac {\Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (- \frac {1}{n}\right )}{a n^{2} x \Gamma \left (1 - \frac {1}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-1/n)/(a*n**2*x*gamma(1 - 1/n))

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