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

Optimal. Leaf size=46 \[ \frac {x^{1-m} \, _2F_1\left (1,\frac {1-m}{5};\frac {6-m}{5};-\frac {x^5}{a^5}\right )}{a^5 (1-m)} \]

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Rubi [A]  time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 45, normalized size = 0.98 \[ -\frac {x^{1-m} \, _2F_1\left (1,\frac {1}{5}-\frac {m}{5};\frac {6}{5}-\frac {m}{5};-\frac {x^5}{a^5}\right )}{a^5 (m-1)} \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{-m}}{a^5+x^5} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^m*(a^5 + x^5)),x]

[Out]

Could not integrate

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fricas [F]  time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{5} + x^{5}\right )} x^{m}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{5} + x^{5}\right )} x^{m}}\,{d x} \]

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)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{-m}}{a^{5}+x^{5}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{5} + x^{5}\right )} x^{m}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^m\,\left (a^5+x^5\right )} \,d x \]

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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sympy [C]  time = 24.44, size = 92, normalized size = 2.00 \[ - \frac {m x x^{- m} \Phi \left (\frac {x^{5} e^{i \pi }}{a^{5}}, 1, \frac {1}{5} - \frac {m}{5}\right ) \Gamma \left (\frac {1}{5} - \frac {m}{5}\right )}{25 a^{5} \Gamma \left (\frac {6}{5} - \frac {m}{5}\right )} + \frac {x x^{- m} \Phi \left (\frac {x^{5} e^{i \pi }}{a^{5}}, 1, \frac {1}{5} - \frac {m}{5}\right ) \Gamma \left (\frac {1}{5} - \frac {m}{5}\right )}{25 a^{5} \Gamma \left (\frac {6}{5} - \frac {m}{5}\right )} \]

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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