3.145 \(\int \frac{x^{-m}}{a^5+x^5} \, dx\)

Optimal. Leaf size=46 \[ \frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{5},\frac{6-m}{5},-\frac{x^5}{a^5}\right )}{a^5 (1-m)} \]

[Out]

(x^(1 - m)*Hypergeometric2F1[1, (1 - m)/5, (6 - m)/5, -(x^5/a^5)])/(a^5*(1 - m))

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Rubi [A]  time = 0.0095126, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {364} \[ \frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{5},\frac{6-m}{5},-\frac{x^5}{a^5}\right )}{a^5 (1-m)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^m*(a^5 + x^5)),x]

[Out]

(x^(1 - m)*Hypergeometric2F1[1, (1 - m)/5, (6 - m)/5, -(x^5/a^5)])/(a^5*(1 - m))

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{x^{-m}}{a^5+x^5} \, dx &=\frac{x^{1-m} \, _2F_1\left (1,\frac{1-m}{5};\frac{6-m}{5};-\frac{x^5}{a^5}\right )}{a^5 (1-m)}\\ \end{align*}

Mathematica [A]  time = 0.0093539, size = 45, normalized size = 0.98 \[ -\frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1}{5}-\frac{m}{5},\frac{6}{5}-\frac{m}{5},-\frac{x^5}{a^5}\right )}{a^5 (m-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^m*(a^5 + x^5)),x]

[Out]

-((x^(1 - m)*Hypergeometric2F1[1, 1/5 - m/5, 6/5 - m/5, -(x^5/a^5)])/(a^5*(-1 + m)))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{m} \left ({a}^{5}+{x}^{5} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^m)/(a^5+x^5),x)

[Out]

int(1/(x^m)/(a^5+x^5),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{5} + x^{5}\right )} x^{m}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^m)/(a^5+x^5),x, algorithm="maxima")

[Out]

integrate(1/((a^5 + x^5)*x^m), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{5} + x^{5}\right )} x^{m}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^m)/(a^5+x^5),x, algorithm="fricas")

[Out]

integral(1/((a^5 + x^5)*x^m), x)

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Sympy [C]  time = 34.013, size = 92, normalized size = 2. \begin{align*} - \frac{m x x^{- m} \Phi \left (\frac{x^{5} e^{i \pi }}{a^{5}}, 1, \frac{1}{5} - \frac{m}{5}\right ) \Gamma \left (\frac{1}{5} - \frac{m}{5}\right )}{25 a^{5} \Gamma \left (\frac{6}{5} - \frac{m}{5}\right )} + \frac{x x^{- m} \Phi \left (\frac{x^{5} e^{i \pi }}{a^{5}}, 1, \frac{1}{5} - \frac{m}{5}\right ) \Gamma \left (\frac{1}{5} - \frac{m}{5}\right )}{25 a^{5} \Gamma \left (\frac{6}{5} - \frac{m}{5}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**m)/(a**5+x**5),x)

[Out]

-m*x*x**(-m)*lerchphi(x**5*exp_polar(I*pi)/a**5, 1, 1/5 - m/5)*gamma(1/5 - m/5)/(25*a**5*gamma(6/5 - m/5)) + x
*x**(-m)*lerchphi(x**5*exp_polar(I*pi)/a**5, 1, 1/5 - m/5)*gamma(1/5 - m/5)/(25*a**5*gamma(6/5 - m/5))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{5} + x^{5}\right )} x^{m}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^m)/(a^5+x^5),x, algorithm="giac")

[Out]

integrate(1/((a^5 + x^5)*x^m), x)