Integrand size = 31, antiderivative size = 446 \[ \int \frac {x^{-3 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx=\frac {e^2 x^{1-3 n}}{d \left (c d^2-b d e+a e^2\right ) n \left (d+e x^n\right )}+\frac {c \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) x^{1-3 n} \operatorname {Hypergeometric2F1}\left (1,-3+\frac {1}{n},-2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 (1-3 n)}-\frac {c \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) x^{1-3 n} \operatorname {Hypergeometric2F1}\left (1,-3+\frac {1}{n},-2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 (1-3 n)}-\frac {e^2 \left (c d^2 (1-6 n)+e (a e (1-4 n)-b (d-5 d n))\right ) x^{1-3 n} \operatorname {Hypergeometric2F1}\left (1,-3+\frac {1}{n},-2+\frac {1}{n},-\frac {e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^2 (1-3 n) n} \] Output:
e^2*x^(1-3*n)/d/(a*e^2-b*d*e+c*d^2)/n/(d+e*x^n)+c*(2*c^2*d^2+b*(b+(-4*a*c+ b^2)^(1/2))*e^2-2*c*e*(b*d+(-4*a*c+b^2)^(1/2)*d+a*e))*x^(1-3*n)*hypergeom( [1, -3+1/n],[-2+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(1/2)/( b-(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)^2/(1-3*n)-c*(2*c^2*d^2+b*(b-(-4* a*c+b^2)^(1/2))*e^2-2*c*e*(b*d-(-4*a*c+b^2)^(1/2)*d+a*e))*x^(1-3*n)*hyperg eom([1, -3+1/n],[-2+1/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(1/ 2)/(b+(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)^2/(1-3*n)-e^2*(c*d^2*(1-6*n) +e*(a*e*(1-4*n)-b*(-5*d*n+d)))*x^(1-3*n)*hypergeom([1, -3+1/n],[-2+1/n],-e *x^n/d)/d^2/(a*e^2-b*d*e+c*d^2)^2/(1-3*n)/n
Time = 3.31 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.74 \[ \int \frac {x^{-3 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx=\frac {x^{1-3 n} \left (-\frac {c \left (2 c d e-b e^2-\frac {2 c^2 d^2+b^2 e^2-2 c e (b d+a e)}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,-3+\frac {1}{n},-2+\frac {1}{n},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{b-\sqrt {b^2-4 a c}}-\frac {c \left (2 c d e-b e^2+\frac {2 c^2 d^2+b^2 e^2-2 c e (b d+a e)}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,-3+\frac {1}{n},-2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{b+\sqrt {b^2-4 a c}}+\frac {e^2 (2 c d-b e) \operatorname {Hypergeometric2F1}\left (1,-3+\frac {1}{n},-2+\frac {1}{n},-\frac {e x^n}{d}\right )}{d}+\frac {e^2 \left (c d^2+e (-b d+a e)\right ) \operatorname {Hypergeometric2F1}\left (2,-3+\frac {1}{n},-2+\frac {1}{n},-\frac {e x^n}{d}\right )}{d^2}\right )}{\left (c d^2+e (-b d+a e)\right )^2 (1-3 n)} \] Input:
Integrate[1/(x^(3*n)*(d + e*x^n)^2*(a + b*x^n + c*x^(2*n))),x]
Output:
(x^(1 - 3*n)*(-((c*(2*c*d*e - b*e^2 - (2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, -3 + n^(-1), -2 + n^(-1), (2 *c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(b - Sqrt[b^2 - 4*a*c])) - (c*(2*c*d*e - b*e^2 + (2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/Sqrt[b^2 - 4*a*c])*Hyp ergeometric2F1[1, -3 + n^(-1), -2 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a *c])])/(b + Sqrt[b^2 - 4*a*c]) + (e^2*(2*c*d - b*e)*Hypergeometric2F1[1, - 3 + n^(-1), -2 + n^(-1), -((e*x^n)/d)])/d + (e^2*(c*d^2 + e*(-(b*d) + a*e) )*Hypergeometric2F1[2, -3 + n^(-1), -2 + n^(-1), -((e*x^n)/d)])/d^2))/((c* d^2 + e*(-(b*d) + a*e))^2*(1 - 3*n))
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.
| \(\displaystyle \int \frac {x^{-3 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx\) |
| \(\Big \downarrow \) 1880 |
| \(\displaystyle \frac {2 c \int \frac {x^{-3 n}}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )^2}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {x^{-3 n}}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )^2}dx}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 1006 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (-\frac {\int -\frac {x^{-3 n} \left (2 c e (1-4 n) x^n+b e (1-4 n)-\sqrt {b^2-4 a c} e (1-4 n)+2 c d n\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {\int -\frac {x^{-3 n} \left (-2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)-2 c d n-b (e-4 e n)\right )}{\left (2 c x^n+b-\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (e \sqrt {b^2-4 a c}-b e+2 c d\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {2 c \left (\frac {\int \frac {x^{-3 n} \left (2 c e (1-4 n) x^n+\sqrt {b^2-4 a c} e (1-4 n)+2 c d n+b (e-4 e n)\right )}{\left (2 c x^n+b+\sqrt {b^2-4 a c}\right ) \left (e x^n+d\right )}dx}{d n \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {e x^{1-3 n}}{d n \left (d+e x^n\right ) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}\right )}{\sqrt {b^2-4 a c}}\) |
Input:
Int[1/(x^(3*n)*(d + e*x^n)^2*(a + b*x^n + c*x^(2*n))),x]
Output:
$Aborted
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomi alQ[a, b, c, d, e, m, n, p, q, x]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^( n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[ 2*(c/r) Int[(f*x)^m*((d + e*x^n)^q/(b - r + 2*c*x^n)), x], x] - Simp[2*(c /r) Int[(f*x)^m*((d + e*x^n)^q/(b + r + 2*c*x^n)), x], x]] /; FreeQ[{a, b , c, d, e, f, m, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && !Rati onalQ[n]
\[\int \frac {x^{-3 n}}{\left (d +e \,x^{n}\right )^{2} \left (a +b \,x^{n}+c \,x^{2 n}\right )}d x\]
Input:
int(1/(x^(3*n))/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n)),x)
Output:
int(1/(x^(3*n))/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n)),x)
\[ \int \frac {x^{-3 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (e x^{n} + d\right )}^{2} x^{3 \, n}} \,d x } \] Input:
integrate(1/(x^(3*n))/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas ")
Output:
integral(1/((b*e^2*x^(3*n) + a*d^2 + (c*e^2*x^(2*n) + 2*c*d*e*x^n + c*d^2) *x^(2*n) + (2*b*d*e + a*e^2)*x^(2*n) + (b*d^2 + 2*a*d*e)*x^n)*x^(3*n)), x)
Exception generated. \[ \int \frac {x^{-3 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate(1/(x**(3*n))/(d+e*x**n)**2/(a+b*x**n+c*x**(2*n)),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {x^{-3 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (e x^{n} + d\right )}^{2} x^{3 \, n}} \,d x } \] Input:
integrate(1/(x^(3*n))/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima ")
Output:
-(c*d^2*e^5*(6*n - 1) - b*d*e^6*(5*n - 1) + a*e^7*(4*n - 1))*integrate(1/( c^2*d^9*n - 2*b*c*d^8*e*n + b^2*d^7*e^2*n + a^2*d^5*e^4*n + 2*(c*d^7*e^2*n - b*d^6*e^3*n)*a + (c^2*d^8*e*n - 2*b*c*d^7*e^2*n + b^2*d^6*e^3*n + a^2*d ^4*e^5*n + 2*(c*d^6*e^3*n - b*d^5*e^4*n)*a)*x^n), x) - (a^3*e^5*(4*n - 1) + b^2*c*d^4*e*n - b^3*d^3*e^2*n + (2*c*d^2*e^3*n - b*d*e^4*n)*a^2 - (c^2*d ^4*e*n - 3*b*c*d^3*e^2*n + b^2*d^2*e^3*n)*a)*x/((n^2 - n)*a^4*d^5*e^2 + (( n^2 - n)*c*d^7 - (n^2 - n)*b*d^6*e)*a^3 + ((n^2 - n)*a^4*d^4*e^3 + ((n^2 - n)*c*d^6*e - (n^2 - n)*b*d^5*e^2)*a^3)*x^n) - integrate((b^3*c^2*d^2 - 2* b^4*c*d*e + b^5*e^2 - (2*c^3*d*e - 3*b*c^2*e^2)*a^2 - 2*(b*c^3*d^2 - 3*b^2 *c^2*d*e + 2*b^3*c*e^2)*a + (b^2*c^3*d^2 - 2*b^3*c^2*d*e + b^4*c*e^2 + a^2 *c^3*e^2 - (c^4*d^2 - 4*b*c^3*d*e + 3*b^2*c^2*e^2)*a)*x^n)/(a^6*e^4 + 2*(c *d^2*e^2 - b*d*e^3)*a^5 + (c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*a^4 + (a^5 *c*e^4 + 2*(c^2*d^2*e^2 - b*c*d*e^3)*a^4 + (c^3*d^4 - 2*b*c^2*d^3*e + b^2* c*d^2*e^2)*a^3)*x^(2*n) + (a^5*b*e^4 + 2*(b*c*d^2*e^2 - b^2*d*e^3)*a^4 + ( b*c^2*d^4 - 2*b^2*c*d^3*e + b^3*d^2*e^2)*a^3)*x^n), x)
\[ \int \frac {x^{-3 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (e x^{n} + d\right )}^{2} x^{3 \, n}} \,d x } \] Input:
integrate(1/(x^(3*n))/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")
Output:
integrate(1/((c*x^(2*n) + b*x^n + a)*(e*x^n + d)^2*x^(3*n)), x)
Timed out. \[ \int \frac {x^{-3 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx=\int \frac {1}{x^{3\,n}\,{\left (d+e\,x^n\right )}^2\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \] Input:
int(1/(x^(3*n)*(d + e*x^n)^2*(a + b*x^n + c*x^(2*n))),x)
Output:
int(1/(x^(3*n)*(d + e*x^n)^2*(a + b*x^n + c*x^(2*n))), x)
\[ \int \frac {x^{-3 n}}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx=\text {too large to display} \] Input:
int(1/(x^(3*n))/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n)),x)
Output:
(48*x**(4*n)*int(x**(2*n)/(3*x**(4*n)*c*e**2*n**2 - 4*x**(4*n)*c*e**2*n + x**(4*n)*c*e**2 + 3*x**(3*n)*b*e**2*n**2 - 4*x**(3*n)*b*e**2*n + x**(3*n)* b*e**2 + 6*x**(3*n)*c*d*e*n**2 - 8*x**(3*n)*c*d*e*n + 2*x**(3*n)*c*d*e + 3 *x**(2*n)*a*e**2*n**2 - 4*x**(2*n)*a*e**2*n + x**(2*n)*a*e**2 + 6*x**(2*n) *b*d*e*n**2 - 8*x**(2*n)*b*d*e*n + 2*x**(2*n)*b*d*e + 3*x**(2*n)*c*d**2*n* *2 - 4*x**(2*n)*c*d**2*n + x**(2*n)*c*d**2 + 6*x**n*a*d*e*n**2 - 8*x**n*a* d*e*n + 2*x**n*a*d*e + 3*x**n*b*d**2*n**2 - 4*x**n*b*d**2*n + x**n*b*d**2 + 3*a*d**2*n**2 - 4*a*d**2*n + a*d**2),x)*b*c*e**3*n**5 - 124*x**(4*n)*int (x**(2*n)/(3*x**(4*n)*c*e**2*n**2 - 4*x**(4*n)*c*e**2*n + x**(4*n)*c*e**2 + 3*x**(3*n)*b*e**2*n**2 - 4*x**(3*n)*b*e**2*n + x**(3*n)*b*e**2 + 6*x**(3 *n)*c*d*e*n**2 - 8*x**(3*n)*c*d*e*n + 2*x**(3*n)*c*d*e + 3*x**(2*n)*a*e**2 *n**2 - 4*x**(2*n)*a*e**2*n + x**(2*n)*a*e**2 + 6*x**(2*n)*b*d*e*n**2 - 8* x**(2*n)*b*d*e*n + 2*x**(2*n)*b*d*e + 3*x**(2*n)*c*d**2*n**2 - 4*x**(2*n)* c*d**2*n + x**(2*n)*c*d**2 + 6*x**n*a*d*e*n**2 - 8*x**n*a*d*e*n + 2*x**n*a *d*e + 3*x**n*b*d**2*n**2 - 4*x**n*b*d**2*n + x**n*b*d**2 + 3*a*d**2*n**2 - 4*a*d**2*n + a*d**2),x)*b*c*e**3*n**4 + 120*x**(4*n)*int(x**(2*n)/(3*x** (4*n)*c*e**2*n**2 - 4*x**(4*n)*c*e**2*n + x**(4*n)*c*e**2 + 3*x**(3*n)*b*e **2*n**2 - 4*x**(3*n)*b*e**2*n + x**(3*n)*b*e**2 + 6*x**(3*n)*c*d*e*n**2 - 8*x**(3*n)*c*d*e*n + 2*x**(3*n)*c*d*e + 3*x**(2*n)*a*e**2*n**2 - 4*x**(2* n)*a*e**2*n + x**(2*n)*a*e**2 + 6*x**(2*n)*b*d*e*n**2 - 8*x**(2*n)*b*d*...