Integrand size = 22, antiderivative size = 90 \[ \int \frac {1}{x^2 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=-\frac {b \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{a (b c-a d) x}+\frac {d \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {d x^n}{c}\right )}{c (b c-a d) x} \] Output:
-b*hypergeom([1, -1/n],[-(1-n)/n],-b*x^n/a)/a/(-a*d+b*c)/x+d*hypergeom([1, -1/n],[-(1-n)/n],-d*x^n/c)/c/(-a*d+b*c)/x
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\frac {b c \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},\frac {-1+n}{n},-\frac {b x^n}{a}\right )-a d \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},\frac {-1+n}{n},-\frac {d x^n}{c}\right )}{a c (-b c+a d) x} \] Input:
Integrate[1/(x^2*(a + b*x^n)*(c + d*x^n)),x]
Output:
(b*c*Hypergeometric2F1[1, -n^(-1), (-1 + n)/n, -((b*x^n)/a)] - a*d*Hyperge ometric2F1[1, -n^(-1), (-1 + n)/n, -((d*x^n)/c)])/(a*c*(-(b*c) + a*d)*x)
Time = 0.37 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1010, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx\) |
\(\Big \downarrow \) 1010 |
\(\displaystyle \frac {b \int \frac {1}{x^2 \left (b x^n+a\right )}dx}{b c-a d}-\frac {d \int \frac {1}{x^2 \left (d x^n+c\right )}dx}{b c-a d}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {d \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {d x^n}{c}\right )}{c x (b c-a d)}-\frac {b \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{a x (b c-a d)}\) |
Input:
Int[1/(x^2*(a + b*x^n)*(c + d*x^n)),x]
Output:
-((b*Hypergeometric2F1[1, -n^(-1), -((1 - n)/n), -((b*x^n)/a)])/(a*(b*c - a*d)*x)) + (d*Hypergeometric2F1[1, -n^(-1), -((1 - n)/n), -((d*x^n)/c)])/( c*(b*c - a*d)*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[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[(e*x)^m/(a + b*x^n), x], x] - Simp[d /(b*c - a*d) Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, n, m}, x] && NeQ[b*c - a*d, 0]
\[\int \frac {1}{x^{2} \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right )}d x\]
Input:
int(1/x^2/(a+b*x^n)/(c+d*x^n),x)
Output:
int(1/x^2/(a+b*x^n)/(c+d*x^n),x)
\[ \int \frac {1}{x^2 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*x^n)/(c+d*x^n),x, algorithm="fricas")
Output:
integral(1/(b*d*x^2*x^(2*n) + (b*c + a*d)*x^2*x^n + a*c*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int \frac {1}{x^{2} \left (a + b x^{n}\right ) \left (c + d x^{n}\right )}\, dx \] Input:
integrate(1/x**2/(a+b*x**n)/(c+d*x**n),x)
Output:
Integral(1/(x**2*(a + b*x**n)*(c + d*x**n)), x)
\[ \int \frac {1}{x^2 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*x^n)/(c+d*x^n),x, algorithm="maxima")
Output:
integrate(1/((b*x^n + a)*(d*x^n + c)*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*x^n)/(c+d*x^n),x, algorithm="giac")
Output:
integrate(1/((b*x^n + a)*(d*x^n + c)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int \frac {1}{x^2\,\left (a+b\,x^n\right )\,\left (c+d\,x^n\right )} \,d x \] Input:
int(1/(x^2*(a + b*x^n)*(c + d*x^n)),x)
Output:
int(1/(x^2*(a + b*x^n)*(c + d*x^n)), x)
\[ \int \frac {1}{x^2 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int \frac {1}{x^{2 n} b d \,x^{2}+x^{n} a d \,x^{2}+x^{n} b c \,x^{2}+a c \,x^{2}}d x \] Input:
int(1/x^2/(a+b*x^n)/(c+d*x^n),x)
Output:
int(1/(x**(2*n)*b*d*x**2 + x**n*a*d*x**2 + x**n*b*c*x**2 + a*c*x**2),x)