Integrand size = 22, antiderivative size = 95 \[ \int \frac {1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=-\frac {b \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 a (b c-a d) x^2}+\frac {d \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},-\frac {2-n}{n},-\frac {d x^n}{c}\right )}{2 c (b c-a d) x^2} \] Output:
-1/2*b*hypergeom([1, -2/n],[-(2-n)/n],-b*x^n/a)/a/(-a*d+b*c)/x^2+1/2*d*hyp ergeom([1, -2/n],[-(2-n)/n],-d*x^n/c)/c/(-a*d+b*c)/x^2
Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\frac {b c \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},\frac {-2+n}{n},-\frac {b x^n}{a}\right )-a d \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},\frac {-2+n}{n},-\frac {d x^n}{c}\right )}{2 a c (-b c+a d) x^2} \] Input:
Integrate[1/(x^3*(a + b*x^n)*(c + d*x^n)),x]
Output:
(b*c*Hypergeometric2F1[1, -2/n, (-2 + n)/n, -((b*x^n)/a)] - a*d*Hypergeome tric2F1[1, -2/n, (-2 + n)/n, -((d*x^n)/c)])/(2*a*c*(-(b*c) + a*d)*x^2)
Time = 0.38 (sec) , antiderivative size = 95, 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.
\(\displaystyle \int \frac {1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx\) |
\(\Big \downarrow \) 1010 |
\(\displaystyle \frac {b \int \frac {1}{x^3 \left (b x^n+a\right )}dx}{b c-a d}-\frac {d \int \frac {1}{x^3 \left (d x^n+c\right )}dx}{b c-a d}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {d \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},-\frac {2-n}{n},-\frac {d x^n}{c}\right )}{2 c x^2 (b c-a d)}-\frac {b \operatorname {Hypergeometric2F1}\left (1,-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 a x^2 (b c-a d)}\) |
Input:
Int[1/(x^3*(a + b*x^n)*(c + d*x^n)),x]
Output:
-1/2*(b*Hypergeometric2F1[1, -2/n, -((2 - n)/n), -((b*x^n)/a)])/(a*(b*c - a*d)*x^2) + (d*Hypergeometric2F1[1, -2/n, -((2 - n)/n), -((d*x^n)/c)])/(2* c*(b*c - a*d)*x^2)
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]
\[\int \frac {1}{x^{3} \left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right )}d x\]
Input:
int(1/x^3/(a+b*x^n)/(c+d*x^n),x)
Output:
int(1/x^3/(a+b*x^n)/(c+d*x^n),x)
\[ \int \frac {1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )} x^{3}} \,d x } \] Input:
integrate(1/x^3/(a+b*x^n)/(c+d*x^n),x, algorithm="fricas")
Output:
integral(1/(b*d*x^3*x^(2*n) + (b*c + a*d)*x^3*x^n + a*c*x^3), x)
Exception generated. \[ \int \frac {1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate(1/x**3/(a+b*x**n)/(c+d*x**n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )} x^{3}} \,d x } \] Input:
integrate(1/x^3/(a+b*x^n)/(c+d*x^n),x, algorithm="maxima")
Output:
integrate(1/((b*x^n + a)*(d*x^n + c)*x^3), x)
\[ \int \frac {1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )} x^{3}} \,d x } \] Input:
integrate(1/x^3/(a+b*x^n)/(c+d*x^n),x, algorithm="giac")
Output:
integrate(1/((b*x^n + a)*(d*x^n + c)*x^3), x)
Timed out. \[ \int \frac {1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int \frac {1}{x^3\,\left (a+b\,x^n\right )\,\left (c+d\,x^n\right )} \,d x \] Input:
int(1/(x^3*(a + b*x^n)*(c + d*x^n)),x)
Output:
int(1/(x^3*(a + b*x^n)*(c + d*x^n)), x)
\[ \int \frac {1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int \frac {1}{x^{2 n} b d \,x^{3}+x^{n} a d \,x^{3}+x^{n} b c \,x^{3}+a c \,x^{3}}d x \] Input:
int(1/x^3/(a+b*x^n)/(c+d*x^n),x)
Output:
int(1/(x**(2*n)*b*d*x**3 + x**n*a*d*x**3 + x**n*b*c*x**3 + a*c*x**3),x)