Integrand size = 13, antiderivative size = 34 \[ \int \frac {1}{x^2 \left (a+b x^n\right )^2} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (2,-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{a^2 x} \] Output:
-hypergeom([2, -1/n],[-(1-n)/n],-b*x^n/a)/a^2/x
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^2 \left (a+b x^n\right )^2} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (2,-\frac {1}{n},1-\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2 x} \] Input:
Integrate[1/(x^2*(a + b*x^n)^2),x]
Output:
-(Hypergeometric2F1[2, -n^(-1), 1 - n^(-1), -((b*x^n)/a)]/(a^2*x))
Time = 0.27 (sec) , antiderivative size = 34, 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 = {888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^2 \left (a+b x^n\right )^2} \, dx\) |
\(\Big \downarrow \) 888 |
\(\displaystyle -\frac {\operatorname {Hypergeometric2F1}\left (2,-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{a^2 x}\) |
Input:
Int[1/(x^2*(a + b*x^n)^2),x]
Output:
-(Hypergeometric2F1[2, -n^(-1), -((1 - n)/n), -((b*x^n)/a)]/(a^2*x))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
\[\int \frac {1}{x^{2} \left (a +b \,x^{n}\right )^{2}}d x\]
Input:
int(1/x^2/(a+b*x^n)^2,x)
Output:
int(1/x^2/(a+b*x^n)^2,x)
\[ \int \frac {1}{x^2 \left (a+b x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*x^n)^2,x, algorithm="fricas")
Output:
integral(1/(b^2*x^2*x^(2*n) + 2*a*b*x^2*x^n + a^2*x^2), x)
Result contains complex when optimal does not.
Time = 0.94 (sec) , antiderivative size = 367, normalized size of antiderivative = 10.79 \[ \int \frac {1}{x^2 \left (a+b x^n\right )^2} \, dx=- \frac {a a^{-2 + \frac {1}{n}} n \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (- \frac {1}{n}\right )}{a a^{\frac {1}{n}} n^{3} x \Gamma \left (1 - \frac {1}{n}\right ) + a^{\frac {1}{n}} b n^{3} x x^{n} \Gamma \left (1 - \frac {1}{n}\right )} - \frac {a a^{-2 + \frac {1}{n}} n \Gamma \left (- \frac {1}{n}\right )}{a a^{\frac {1}{n}} n^{3} x \Gamma \left (1 - \frac {1}{n}\right ) + a^{\frac {1}{n}} b n^{3} x x^{n} \Gamma \left (1 - \frac {1}{n}\right )} - \frac {a a^{-2 + \frac {1}{n}} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (- \frac {1}{n}\right )}{a a^{\frac {1}{n}} n^{3} x \Gamma \left (1 - \frac {1}{n}\right ) + a^{\frac {1}{n}} b n^{3} x x^{n} \Gamma \left (1 - \frac {1}{n}\right )} - \frac {a^{-2 + \frac {1}{n}} b n x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (- \frac {1}{n}\right )}{a a^{\frac {1}{n}} n^{3} x \Gamma \left (1 - \frac {1}{n}\right ) + a^{\frac {1}{n}} b n^{3} x x^{n} \Gamma \left (1 - \frac {1}{n}\right )} - \frac {a^{-2 + \frac {1}{n}} b x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (- \frac {1}{n}\right )}{a a^{\frac {1}{n}} n^{3} x \Gamma \left (1 - \frac {1}{n}\right ) + a^{\frac {1}{n}} b n^{3} x x^{n} \Gamma \left (1 - \frac {1}{n}\right )} \] Input:
integrate(1/x**2/(a+b*x**n)**2,x)
Output:
-a*a**(-2 + 1/n)*n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n )*gamma(-1/n)/(a*a**(1/n)*n**3*x*gamma(1 - 1/n) + a**(1/n)*b*n**3*x*x**n*g amma(1 - 1/n)) - a*a**(-2 + 1/n)*n*gamma(-1/n)/(a*a**(1/n)*n**3*x*gamma(1 - 1/n) + a**(1/n)*b*n**3*x*x**n*gamma(1 - 1/n)) - a*a**(-2 + 1/n)*lerchphi (b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-1/n)/(a*a**(1/n)*n **3*x*gamma(1 - 1/n) + a**(1/n)*b*n**3*x*x**n*gamma(1 - 1/n)) - a**(-2 + 1 /n)*b*n*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamm a(-1/n)/(a*a**(1/n)*n**3*x*gamma(1 - 1/n) + a**(1/n)*b*n**3*x*x**n*gamma(1 - 1/n)) - a**(-2 + 1/n)*b*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_ polar(I*pi)/n)*gamma(-1/n)/(a*a**(1/n)*n**3*x*gamma(1 - 1/n) + a**(1/n)*b* n**3*x*x**n*gamma(1 - 1/n))
\[ \int \frac {1}{x^2 \left (a+b x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*x^n)^2,x, algorithm="maxima")
Output:
(n + 1)*integrate(1/(a*b*n*x^2*x^n + a^2*n*x^2), x) + 1/(a*b*n*x*x^n + a^2 *n*x)
\[ \int \frac {1}{x^2 \left (a+b x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*x^n)^2,x, algorithm="giac")
Output:
integrate(1/((b*x^n + a)^2*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^n\right )^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,x^n\right )}^2} \,d x \] Input:
int(1/(x^2*(a + b*x^n)^2),x)
Output:
int(1/(x^2*(a + b*x^n)^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^n\right )^2} \, dx=\int \frac {1}{x^{2 n} b^{2} x^{2}+2 x^{n} a b \,x^{2}+a^{2} x^{2}}d x \] Input:
int(1/x^2/(a+b*x^n)^2,x)
Output:
int(1/(x**(2*n)*b**2*x**2 + 2*x**n*a*b*x**2 + a**2*x**2),x)