3.2.45 \(\int \frac {x^{-m}}{a^5+x^5} \, dx\) [145]

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

[Out]

x^(1-m)*hypergeom([1, 1/5-1/5*m],[6/5-1/5*m],-x^5/a^5)/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 = {371} \begin {gather*} \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)} \end {gather*}

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], 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 = 0.18, size = 45, normalized size = 0.98 \begin {gather*} -\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 (-1+m)} \end {gather*}

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 = 0.03, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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] Result contains complex when optimal does not.
time = 8.50, size = 92, normalized size = 2.00 \begin {gather*} - \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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