Optimal. Leaf size=410 \[ \frac {c^2 d e (1-n) x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 n (n+1) \left (a e^2+c d^2\right )^2}-\frac {c (1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^2}-\frac {4 c^2 d e^3 x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^3}+\frac {c e^2 x \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^3}+\frac {c x \left (-a e^2+c d^2-2 c d e x^n\right )}{2 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}+\frac {4 c e^4 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (a e^2+c d^2\right )^3}+\frac {e^4 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.38, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1437, 245, 1431, 1418, 364} \[ \frac {c^2 d e (1-n) x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 n (n+1) \left (a e^2+c d^2\right )^2}-\frac {c (1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^2}-\frac {4 c^2 d e^3 x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^3}+\frac {c e^2 x \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^3}+\frac {4 c e^4 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (a e^2+c d^2\right )^3}+\frac {e^4 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )^2}+\frac {c x \left (-a e^2+c d^2-2 c d e x^n\right )}{2 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 245
Rule 364
Rule 1418
Rule 1431
Rule 1437
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^2} \, dx &=\int \left (\frac {e^4}{\left (c d^2+a e^2\right )^2 \left (d+e x^n\right )^2}+\frac {4 c d e^4}{\left (c d^2+a e^2\right )^3 \left (d+e x^n\right )}-\frac {c \left (-c d^2+a e^2+2 c d e x^n\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^{2 n}\right )^2}-\frac {c e^2 \left (-3 c d^2+a e^2+4 c d e x^n\right )}{\left (c d^2+a e^2\right )^3 \left (a+c x^{2 n}\right )}\right ) \, dx\\ &=-\frac {\left (c e^2\right ) \int \frac {-3 c d^2+a e^2+4 c d e x^n}{a+c x^{2 n}} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {\left (4 c d e^4\right ) \int \frac {1}{d+e x^n} \, dx}{\left (c d^2+a e^2\right )^3}-\frac {c \int \frac {-c d^2+a e^2+2 c d e x^n}{\left (a+c x^{2 n}\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {e^4 \int \frac {1}{\left (d+e x^n\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac {c x \left (c d^2-a e^2-2 c d e x^n\right )}{2 a \left (c d^2+a e^2\right )^2 n \left (a+c x^{2 n}\right )}+\frac {4 c e^4 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (c d^2+a e^2\right )^3}+\frac {e^4 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2+a e^2\right )^2}-\frac {\left (4 c^2 d e^3\right ) \int \frac {x^n}{a+c x^{2 n}} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {\left (c e^2 \left (3 c d^2-a e^2\right )\right ) \int \frac {1}{a+c x^{2 n}} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {c \int \frac {\left (-c d^2+a e^2\right ) (1-2 n)+2 c d e (1-n) x^n}{a+c x^{2 n}} \, dx}{2 a \left (c d^2+a e^2\right )^2 n}\\ &=\frac {c x \left (c d^2-a e^2-2 c d e x^n\right )}{2 a \left (c d^2+a e^2\right )^2 n \left (a+c x^{2 n}\right )}+\frac {c e^2 \left (3 c d^2-a e^2\right ) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^3}+\frac {4 c e^4 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (c d^2+a e^2\right )^3}-\frac {4 c^2 d e^3 x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^3 (1+n)}+\frac {e^4 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2+a e^2\right )^2}-\frac {\left (c \left (c d^2-a e^2\right ) (1-2 n)\right ) \int \frac {1}{a+c x^{2 n}} \, dx}{2 a \left (c d^2+a e^2\right )^2 n}+\frac {\left (c^2 d e (1-n)\right ) \int \frac {x^n}{a+c x^{2 n}} \, dx}{a \left (c d^2+a e^2\right )^2 n}\\ &=\frac {c x \left (c d^2-a e^2-2 c d e x^n\right )}{2 a \left (c d^2+a e^2\right )^2 n \left (a+c x^{2 n}\right )}+\frac {c e^2 \left (3 c d^2-a e^2\right ) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^3}-\frac {c \left (c d^2-a e^2\right ) (1-2 n) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{2 a^2 \left (c d^2+a e^2\right )^2 n}+\frac {4 c e^4 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{\left (c d^2+a e^2\right )^3}-\frac {4 c^2 d e^3 x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a \left (c d^2+a e^2\right )^3 (1+n)}+\frac {c^2 d e (1-n) x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 \left (c d^2+a e^2\right )^2 n (1+n)}+\frac {e^4 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2+a e^2\right )^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.47, size = 298, normalized size = 0.73 \[ \frac {x \left (-\frac {2 c^2 d e x^n \left (a e^2+c d^2\right ) \, _2F_1\left (2,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2 (n+1)}+\frac {c \left (c d^2-a e^2\right ) \left (a e^2+c d^2\right ) \, _2F_1\left (2,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^2}-\frac {4 c^2 d e^3 x^n \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a (n+1)}+\frac {c e^2 \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e^4 \left (a e^2+c d^2\right ) \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2}+4 c e^4 \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )\right )}{\left (a e^2+c d^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.36, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} e^{2} x^{2 \, n} + 2 \, a^{2} d e x^{n} + a^{2} d^{2} + {\left (c^{2} e^{2} x^{2 \, n} + 2 \, c^{2} d e x^{n} + c^{2} d^{2}\right )} x^{4 \, n} + 2 \, {\left (a c e^{2} x^{2 \, n} + 2 \, a c d e x^{n} + a c d^{2}\right )} x^{2 \, n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2 \, n} + a\right )}^{2} {\left (e x^{n} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{n}+d \right )^{2} \left (c \,x^{2 n}+a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (c d^{2} e^{4} {\left (5 \, n - 1\right )} + a e^{6} {\left (n - 1\right )}\right )} \int \frac {1}{c^{3} d^{8} n + 3 \, a c^{2} d^{6} e^{2} n + 3 \, a^{2} c d^{4} e^{4} n + a^{3} d^{2} e^{6} n + {\left (c^{3} d^{7} e n + 3 \, a c^{2} d^{5} e^{3} n + 3 \, a^{2} c d^{3} e^{5} n + a^{3} d e^{7} n\right )} x^{n}}\,{d x} - \frac {2 \, {\left (c^{2} d^{2} e^{2} - a c e^{4}\right )} x x^{2 \, n} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x x^{n} - {\left (c^{2} d^{4} - a c d^{2} e^{2} + 2 \, a^{2} e^{4}\right )} x}{2 \, {\left (a^{2} c^{2} d^{6} n + 2 \, a^{3} c d^{4} e^{2} n + a^{4} d^{2} e^{4} n + {\left (a c^{3} d^{5} e n + 2 \, a^{2} c^{2} d^{3} e^{3} n + a^{3} c d e^{5} n\right )} x^{3 \, n} + {\left (a c^{3} d^{6} n + 2 \, a^{2} c^{2} d^{4} e^{2} n + a^{3} c d^{2} e^{4} n\right )} x^{2 \, n} + {\left (a^{2} c^{2} d^{5} e n + 2 \, a^{3} c d^{3} e^{3} n + a^{4} d e^{5} n\right )} x^{n}\right )}} - \int \frac {a^{2} c e^{4} {\left (4 \, n - 1\right )} - c^{3} d^{4} {\left (2 \, n - 1\right )} - 6 \, a c^{2} d^{2} e^{2} n + 2 \, {\left (a c^{2} d e^{3} {\left (5 \, n - 1\right )} + c^{3} d^{3} e {\left (n - 1\right )}\right )} x^{n}}{2 \, {\left (a^{2} c^{3} d^{6} n + 3 \, a^{3} c^{2} d^{4} e^{2} n + 3 \, a^{4} c d^{2} e^{4} n + a^{5} e^{6} n + {\left (a c^{4} d^{6} n + 3 \, a^{2} c^{3} d^{4} e^{2} n + 3 \, a^{3} c^{2} d^{2} e^{4} n + a^{4} c e^{6} n\right )} x^{2 \, n}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+c\,x^{2\,n}\right )}^2\,{\left (d+e\,x^n\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________