\(\int \frac {1}{x^3 (a+b x^n)} \, dx\) [2476]

Optimal result

Integrand size = 13, antiderivative size = 36 \[ \int \frac {1}{x^3 \left (a+b x^n\right )} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 a x^2} \]

[Out]

-1/2*hypergeom([1, -2/n],[(-2+n)/n],-b*x^n/a)/a/x^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {371} \[ \int \frac {1}{x^3 \left (a+b x^n\right )} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 a x^2} \]

[In]

Int[1/(x^3*(a + b*x^n)),x]

[Out]

-1/2*Hypergeometric2F1[1, -2/n, -((2 - n)/n), -((b*x^n)/a)]/(a*x^2)

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 {\, _2F_1\left (1,-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 a x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^3 \left (a+b x^n\right )} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},1-\frac {2}{n},-\frac {b x^n}{a}\right )}{2 a x^2} \]

[In]

Integrate[1/(x^3*(a + b*x^n)),x]

[Out]

-1/2*Hypergeometric2F1[1, -2/n, 1 - 2/n, -((b*x^n)/a)]/(a*x^2)

Maple [F]

\[\int \frac {1}{x^{3} \left (a +b \,x^{n}\right )}d x\]

[In]

int(1/x^3/(a+b*x^n),x)

[Out]

int(1/x^3/(a+b*x^n),x)

Fricas [F]

\[ \int \frac {1}{x^3 \left (a+b x^n\right )} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(a+b*x^n),x, algorithm="fricas")

[Out]

integral(1/(b*x^3*x^n + a*x^3), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \frac {1}{x^3 \left (a+b x^n\right )} \, dx=- \frac {2 a^{- \frac {2}{n}} a^{-1 + \frac {2}{n}} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {2 e^{i \pi }}{n}\right ) \Gamma \left (- \frac {2}{n}\right )}{n^{2} x^{2} \Gamma \left (1 - \frac {2}{n}\right )} \]

[In]

integrate(1/x**3/(a+b*x**n),x)

[Out]

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

Maxima [F]

\[ \int \frac {1}{x^3 \left (a+b x^n\right )} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(a+b*x^n),x, algorithm="maxima")

[Out]

integrate(1/((b*x^n + a)*x^3), x)

Giac [F]

\[ \int \frac {1}{x^3 \left (a+b x^n\right )} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(a+b*x^n),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + a)*x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (a+b x^n\right )} \, dx=\int \frac {1}{x^3\,\left (a+b\,x^n\right )} \,d x \]

[In]

int(1/(x^3*(a + b*x^n)),x)

[Out]

int(1/(x^3*(a + b*x^n)), x)