Optimal. Leaf size=1708 \[ \frac {x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right ) e^6}{d \left (c d^2-b e d+a e^2\right )^3}-\frac {c \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right ) e^4}{\left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^3}-\frac {c \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right ) e^4}{\left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^3}+\frac {c \left (-e (1-n) b^3+\left (c d-\sqrt {b^2-4 a c} e\right ) (1-n) b^2+c \left (2 a e (2-3 n)+\sqrt {b^2-4 a c} d (1-n)\right ) b-2 a c \left (2 c d (1-2 n)-\sqrt {b^2-4 a c} e (1-n)\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right ) e^2}{a \left (b^2-4 a c\right ) \left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 n}+\frac {c \left (-e (1-n) b^3+\left (c d+\sqrt {b^2-4 a c} e\right ) (1-n) b^2+c \left (2 a e (2-3 n)-\sqrt {b^2-4 a c} d (1-n)\right ) b-2 a c \left (2 c d (1-2 n)+\sqrt {b^2-4 a c} e (1-n)\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right ) e^2}{a \left (b^2-4 a c\right ) \left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 n}+\frac {x \left (c \left (-e b^2+c d b+2 a c e\right ) x^n-2 a c^2 d+b^2 c d-b^3 e+3 a b c e\right ) e^2}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 n \left (b x^n+c x^{2 n}+a\right )}-\frac {c \left (-e \left (2 n^2-3 n+1\right ) b^5+\left (c d-\sqrt {b^2-4 a c} e\right ) \left (2 n^2-3 n+1\right ) b^4+c \left (a e (7-18 n)+\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n) b^3+a c \left (\sqrt {b^2-4 a c} e (5-14 n)-6 c d (1-3 n)\right ) (1-n) b^2-2 a c^2 \left (\sqrt {b^2-4 a c} d \left (7 n^2-9 n+2\right )+2 a e \left (13 n^2-13 n+3\right )\right ) b-4 a^2 c^2 \left (\sqrt {b^2-4 a c} e \left (3 n^2-4 n+1\right )-2 c d \left (8 n^2-6 n+1\right )\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right ) n^2}+\frac {c \left (e \left (2 n^2-3 n+1\right ) b^5-\left (c d+\sqrt {b^2-4 a c} e\right ) \left (2 n^2-3 n+1\right ) b^4-c \left (a e (7-18 n)-\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n) b^3+a c \left (\sqrt {b^2-4 a c} e (5-14 n)+6 c d (1-3 n)\right ) (1-n) b^2-2 a c^2 \left (\sqrt {b^2-4 a c} d \left (7 n^2-9 n+2\right )-2 a e \left (13 n^2-13 n+3\right )\right ) b-4 a^2 c^2 \left (\sqrt {b^2-4 a c} e \left (3 n^2-4 n+1\right )+2 c d \left (8 n^2-6 n+1\right )\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right ) n^2}+\frac {x \left (-c \left (-e (1-2 n) b^4+c d (1-2 n) b^3+a c e (5-14 n) b^2-2 a c^2 d (2-7 n) b-4 a^2 c^2 e (1-3 n)\right ) x^n+2 a^2 b c^2 e (4-11 n)-3 a b^3 c e (2-5 n)-4 a^2 c^3 d (1-4 n)+5 a b^2 c^2 d (1-3 n)-b^4 c d (1-2 n)+b^5 (e-2 e n)\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right ) n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac {x \left (c \left (-e b^2+c d b+2 a c e\right ) x^n-2 a c^2 d+b^2 c d-b^3 e+3 a b c e\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) n \left (b x^n+c x^{2 n}+a\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 5.07, antiderivative size = 1708, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1436, 245, 1430, 1422} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 245
Rule 1422
Rule 1430
Rule 1436
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3} \, dx &=\int \left (\frac {e^6}{\left (c d^2-b d e+a e^2\right )^3 \left (d+e x^n\right )}+\frac {c d-b e-c e x^n}{\left (c d^2-b d e+a e^2\right ) \left (a+b x^n+c x^{2 n}\right )^3}-\frac {e^2 \left (-c d+b e+c e x^n\right )}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x^n+c x^{2 n}\right )^2}-\frac {e^4 \left (-c d+b e+c e x^n\right )}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x^n+c x^{2 n}\right )}\right ) \, dx\\ &=-\frac {e^4 \int \frac {-c d+b e+c e x^n}{a+b x^n+c x^{2 n}} \, dx}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e^6 \int \frac {1}{d+e x^n} \, dx}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^2 \int \frac {-c d+b e+c e x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{\left (c d^2-b d e+a e^2\right )^2}+\frac {\int \frac {c d-b e-c e x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx}{c d^2-b d e+a e^2}\\ &=\frac {x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}+\frac {e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (c e^4 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (c e^4 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac {e^2 \int \frac {-a b c e (3-4 n)+2 a c^2 d (1-2 n)-b^2 c d (1-n)+b^3 (e-e n)-c \left (b c d-b^2 e+2 a c e\right ) (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n}-\frac {\int \frac {a b c e-2 a c (c d-b e) (1-4 n)+b^2 (c d-b e) (1-2 n)+c \left (b c d-b^2 e+2 a c e\right ) (1-3 n) x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) n}\\ &=\frac {x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}+\frac {x \left (2 a^2 b c^2 e (4-11 n)-3 a b^3 c e (2-5 n)-4 a^2 c^3 d (1-4 n)+5 a b^2 c^2 d (1-3 n)-b^4 c d (1-2 n)+b^5 (e-2 e n)-c \left (a b^2 c e (5-14 n)-2 a b c^2 d (2-7 n)-4 a^2 c^2 e (1-3 n)+b^3 c d (1-2 n)-b^4 e (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {c e^4 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}-\frac {c e^4 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}+\frac {e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3}+\frac {\int \frac {b^4 c d \left (1-3 n+2 n^2\right )-b^5 e \left (1-3 n+2 n^2\right )+2 a b^3 c e \left (3-11 n+8 n^2\right )+4 a^2 c^3 d \left (1-6 n+8 n^2\right )-a b^2 c^2 d \left (5-21 n+16 n^2\right )-2 a^2 b c^2 e \left (4-17 n+16 n^2\right )+c \left (a b^2 c e (5-14 n)-2 a b c^2 d (2-7 n)-4 a^2 c^2 e (1-3 n)+b^3 c d (1-2 n)-b^4 e (1-2 n)\right ) (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{2 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) n^2}-\frac {\left (c e^2 \left (\left (b c d-b^2 e+2 a c e\right ) (1-n)-\frac {2 a b c e (2-3 n)-4 a c^2 d (1-2 n)+b^2 c d (1-n)-b^3 (e-e n)}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n}+\frac {\left (e^2 \left (-\frac {1}{2} c \left (b c d-b^2 e+2 a c e\right ) (1-n)+\frac {b c \left (b c d-b^2 e+2 a c e\right ) (1-n)+2 c \left (-a b c e (3-4 n)+2 a c^2 d (1-2 n)-b^2 c d (1-n)+b^3 (e-e n)\right )}{2 \sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n}\\ &=\frac {x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}+\frac {x \left (2 a^2 b c^2 e (4-11 n)-3 a b^3 c e (2-5 n)-4 a^2 c^3 d (1-4 n)+5 a b^2 c^2 d (1-3 n)-b^4 c d (1-2 n)+b^5 (e-2 e n)-c \left (a b^2 c e (5-14 n)-2 a b c^2 d (2-7 n)-4 a^2 c^2 e (1-3 n)+b^3 c d (1-2 n)-b^4 e (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {c e^4 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}-\frac {c e^2 \left (b c \left (2 a e (2-3 n)+\sqrt {b^2-4 a c} d (1-n)\right )-2 a c \left (c d (2-4 n)-\sqrt {b^2-4 a c} e (1-n)\right )+b^2 \left (c d-\sqrt {b^2-4 a c} e\right ) (1-n)-b^3 (e-e n)\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}-\frac {c e^4 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}-\frac {c e^2 \left (\left (b c d-b^2 e+2 a c e\right ) (1-n)-\frac {2 a b c e (2-3 n)-4 a c^2 d (1-2 n)+b^2 c d (1-n)-b^3 (e-e n)}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a \left (b^2-4 a c\right ) \left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}+\frac {e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3}+\frac {\left (c \left (a b^2 c \left (\sqrt {b^2-4 a c} e (5-14 n)+6 c d (1-3 n)\right ) (1-n)-b^3 c \left (a e (7-18 n)-\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)+b^5 e \left (1-3 n+2 n^2\right )-b^4 \left (c d+\sqrt {b^2-4 a c} e\right ) \left (1-3 n+2 n^2\right )-4 a^2 c^2 \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )+2 c d \left (1-6 n+8 n^2\right )\right )-2 a b c^2 \left (\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )-2 a e \left (3-13 n+13 n^2\right )\right )\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{4 a^2 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right ) n^2}+\frac {\left (c \left (a b^2 c \left (\sqrt {b^2-4 a c} e (5-14 n)-6 c d (1-3 n)\right ) (1-n)+b^3 c \left (a e (7-18 n)+\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)-b^5 e \left (1-3 n+2 n^2\right )+b^4 \left (c d-\sqrt {b^2-4 a c} e\right ) \left (1-3 n+2 n^2\right )-4 a^2 c^2 \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )-2 c d \left (1-6 n+8 n^2\right )\right )-2 a b c^2 \left (\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )+2 a e \left (3-13 n+13 n^2\right )\right )\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{4 a^2 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right ) n^2}\\ &=\frac {x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}+\frac {x \left (2 a^2 b c^2 e (4-11 n)-3 a b^3 c e (2-5 n)-4 a^2 c^3 d (1-4 n)+5 a b^2 c^2 d (1-3 n)-b^4 c d (1-2 n)+b^5 (e-2 e n)-c \left (a b^2 c e (5-14 n)-2 a b c^2 d (2-7 n)-4 a^2 c^2 e (1-3 n)+b^3 c d (1-2 n)-b^4 e (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {c e^4 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}-\frac {c e^2 \left (b c \left (2 a e (2-3 n)+\sqrt {b^2-4 a c} d (1-n)\right )-2 a c \left (c d (2-4 n)-\sqrt {b^2-4 a c} e (1-n)\right )+b^2 \left (c d-\sqrt {b^2-4 a c} e\right ) (1-n)-b^3 (e-e n)\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}+\frac {c \left (a b^2 c \left (\sqrt {b^2-4 a c} e (5-14 n)-6 c d (1-3 n)\right ) (1-n)+b^3 c \left (a e (7-18 n)+\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)-b^5 e \left (1-3 n+2 n^2\right )+b^4 \left (c d-\sqrt {b^2-4 a c} e\right ) \left (1-3 n+2 n^2\right )-4 a^2 c^2 \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )-2 c d \left (1-6 n+8 n^2\right )\right )-2 a b c^2 \left (\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )+2 a e \left (3-13 n+13 n^2\right )\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{5/2} \left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-e (b d-a e)\right ) n^2}-\frac {c e^4 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}-\frac {c e^2 \left (\left (b c d-b^2 e+2 a c e\right ) (1-n)-\frac {2 a b c e (2-3 n)-4 a c^2 d (1-2 n)+b^2 c d (1-n)-b^3 (e-e n)}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a \left (b^2-4 a c\right ) \left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}+\frac {c \left (a b^2 c \left (\sqrt {b^2-4 a c} e (5-14 n)+6 c d (1-3 n)\right ) (1-n)-b^3 c \left (a e (7-18 n)-\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)+b^5 e \left (1-3 n+2 n^2\right )-b^4 \left (c d+\sqrt {b^2-4 a c} e\right ) \left (1-3 n+2 n^2\right )-4 a^2 c^2 \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )+2 c d \left (1-6 n+8 n^2\right )\right )-2 a b c^2 \left (\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )-2 a e \left (3-13 n+13 n^2\right )\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{5/2} \left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-e (b d-a e)\right ) n^2}+\frac {e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3}\\ \end {align*}
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Mathematica [B] time = 8.53, size = 43535, normalized size = 25.49 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 20.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{3} e x^{4 \, n} + a^{3} d + {\left (c^{3} e x^{n} + c^{3} d\right )} x^{6 \, n} + 3 \, {\left (b c^{2} e x^{2 \, n} + a c^{2} d + {\left (b c^{2} d + a c^{2} e\right )} x^{n}\right )} x^{4 \, n} + {\left (b^{3} d + 3 \, a b^{2} e\right )} x^{3 \, n} + 3 \, {\left (b^{2} c e x^{3 \, n} + a^{2} c d + {\left (b^{2} c d + 2 \, a b c e\right )} x^{2 \, n} + {\left (2 \, a b c d + a^{2} c e\right )} x^{n}\right )} x^{2 \, n} + 3 \, {\left (a b^{2} d + a^{2} b e\right )} x^{2 \, n} + {\left (3 \, a^{2} b d + a^{3} e\right )} x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3} {\left (e x^{n} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{n}+d \right ) \left (b \,x^{n}+c \,x^{2 n}+a \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\left (d+e\,x^n\right )\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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