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

Optimal result

Integrand size = 15, antiderivative size = 46 \[ \int \frac {x^{-m}}{a^5+x^5} \, dx=\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{5},\frac {6-m}{5},-\frac {x^5}{a^5}\right )}{a^5 (1-m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {371} \[ \int \frac {x^{-m}}{a^5+x^5} \, dx=\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{5},\frac {6-m}{5},-\frac {x^5}{a^5}\right )}{a^5 (1-m)} \]

[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*} \text {integral}& = \frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{5},\frac {6-m}{5},-\frac {x^5}{a^5}\right )}{a^5 (1-m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {x^{-m}}{a^5+x^5} \, dx=-\frac {x^{1-m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{5}-\frac {m}{5},\frac {6}{5}-\frac {m}{5},-\frac {x^5}{a^5}\right )}{a^5 (-1+m)} \]

[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)))

Maple [F]

\[\int \frac {x^{-m}}{a^{5}+x^{5}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^{-m}}{a^5+x^5} \, dx=\int { \frac {1}{{\left (a^{5} + x^{5}\right )} x^{m}} \,d x } \]

[In]

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

[Out]

integral(1/((a^5 + x^5)*x^m), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.41 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.00 \[ \int \frac {x^{-m}}{a^5+x^5} \, dx=- \frac {m x^{1 - m} \Phi \left (\frac {x^{5} e^{i \pi }}{a^{5}}, 1, \frac {1}{5} - \frac {m}{5}\right ) \Gamma \left (\frac {1}{5} - \frac {m}{5}\right )}{25 a^{5} \Gamma \left (\frac {6}{5} - \frac {m}{5}\right )} + \frac {x^{1 - m} \Phi \left (\frac {x^{5} e^{i \pi }}{a^{5}}, 1, \frac {1}{5} - \frac {m}{5}\right ) \Gamma \left (\frac {1}{5} - \frac {m}{5}\right )}{25 a^{5} \Gamma \left (\frac {6}{5} - \frac {m}{5}\right )} \]

[In]

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

[Out]

-m*x**(1 - 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**(1 - 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))

Maxima [F]

\[ \int \frac {x^{-m}}{a^5+x^5} \, dx=\int { \frac {1}{{\left (a^{5} + x^{5}\right )} x^{m}} \,d x } \]

[In]

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

[Out]

integrate(1/((a^5 + x^5)*x^m), x)

Giac [F]

\[ \int \frac {x^{-m}}{a^5+x^5} \, dx=\int { \frac {1}{{\left (a^{5} + x^{5}\right )} x^{m}} \,d x } \]

[In]

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

[Out]

integrate(1/((a^5 + x^5)*x^m), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-m}}{a^5+x^5} \, dx=\int \frac {1}{x^m\,\left (a^5+x^5\right )} \,d x \]

[In]

int(1/(x^m*(a^5 + x^5)),x)

[Out]

int(1/(x^m*(a^5 + x^5)), x)