\(\int \frac {1}{x^3 (a+b x^n)^3} \, dx\) [2486]

Optimal result

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

[Out]

-1/2*hypergeom([3, -2/n],[(-2+n)/n],-b*x^n/a)/a^3/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} \[ \int \frac {1}{x^3 \left (a+b x^n\right )^3} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (3,-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 a^3 x^2} \]

[In]

Int[1/(x^3*(a + b*x^n)^3),x]

[Out]

-1/2*Hypergeometric2F1[3, -2/n, -((2 - n)/n), -((b*x^n)/a)]/(a^3*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 (3,-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 a^3 x^2} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[1/(x^3*(a + b*x^n)^3),x]

[Out]

-1/2*Hypergeometric2F1[3, -2/n, 1 - 2/n, -((b*x^n)/a)]/(a^3*x^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(1/(b^3*x^3*x^(3*n) + 3*a*b^2*x^3*x^(2*n) + 3*a^2*b*x^3*x^n + a^3*x^3), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.36 (sec) , antiderivative size = 1479, normalized size of antiderivative = 41.08 \[ \int \frac {1}{x^3 \left (a+b x^n\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-2*a**2*a**(-3 + 2/n)*n**2*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**2*a**(2/
n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a**(2/n)*b*n**4*x**2*x**n*gamma(1 - 2/n) + a**(2/n)*b**2*n**4*x**2*x**(2*n)*
gamma(1 - 2/n)) - 3*a**2*a**(-3 + 2/n)*n**2*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a**(2/n)
*b*n**4*x**2*x**n*gamma(1 - 2/n) + a**(2/n)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)) - 6*a**2*a**(-3 + 2/n)*n*l
erchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n)
+ 2*a*a**(2/n)*b*n**4*x**2*x**n*gamma(1 - 2/n) + a**(2/n)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)) - 2*a**2*a**
(-3 + 2/n)*n*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a**(2/n)*b*n**4*x**2*x**n*gamma(1 - 2/n
) + a**(2/n)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)) - 4*a**2*a**(-3 + 2/n)*lerchphi(b*x**n*exp_polar(I*pi)/a,
1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a**(2/n)*b*n**4*x**2*x**n*g
amma(1 - 2/n) + a**(2/n)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)) - 4*a*a**(-3 + 2/n)*b*n**2*x**n*lerchphi(b*x*
*n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a**(2/
n)*b*n**4*x**2*x**n*gamma(1 - 2/n) + a**(2/n)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)) - 2*a*a**(-3 + 2/n)*b*n*
*2*x**n*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a**(2/n)*b*n**4*x**2*x**n*gamma(1 - 2/n) + a
**(2/n)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)) - 12*a*a**(-3 + 2/n)*b*n*x**n*lerchphi(b*x**n*exp_polar(I*pi)/
a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a**(2/n)*b*n**4*x**2*x**n
*gamma(1 - 2/n) + a**(2/n)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)) - 2*a*a**(-3 + 2/n)*b*n*x**n*gamma(-2/n)/(a
**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a**(2/n)*b*n**4*x**2*x**n*gamma(1 - 2/n) + a**(2/n)*b**2*n**4*x**2
*x**(2*n)*gamma(1 - 2/n)) - 8*a*a**(-3 + 2/n)*b*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n
)*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a**(2/n)*b*n**4*x**2*x**n*gamma(1 - 2/n) + a**(2/n
)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)) - 2*a**(-3 + 2/n)*b**2*n**2*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)
/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a**(2/n)*b*n**4*x**2*x**
n*gamma(1 - 2/n) + a**(2/n)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)) - 6*a**(-3 + 2/n)*b**2*n*x**(2*n)*lerchphi
(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*gamma(1 - 2/n) + 2*a*a
**(2/n)*b*n**4*x**2*x**n*gamma(1 - 2/n) + a**(2/n)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)) - 4*a**(-3 + 2/n)*b
**2*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2*exp_polar(I*pi)/n)*gamma(-2/n)/(a**2*a**(2/n)*n**4*x**2*g
amma(1 - 2/n) + 2*a*a**(2/n)*b*n**4*x**2*x**n*gamma(1 - 2/n) + a**(2/n)*b**2*n**4*x**2*x**(2*n)*gamma(1 - 2/n)
)

Maxima [F]

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

[In]

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

[Out]

(n^2 + 3*n + 2)*integrate(1/(a^2*b*n^2*x^3*x^n + a^3*n^2*x^3), x) + 1/2*(2*b*(n + 1)*x^n + a*(3*n + 2))/(a^2*b
^2*n^2*x^2*x^(2*n) + 2*a^3*b*n^2*x^2*x^n + a^4*n^2*x^2)

Giac [F]

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

[In]

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

[Out]

integrate(1/((b*x^n + a)^3*x^3), x)

Mupad [F(-1)]

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

[In]

int(1/(x^3*(a + b*x^n)^3),x)

[Out]

int(1/(x^3*(a + b*x^n)^3), x)