Optimal. Leaf size=582 \[ -\frac{c e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 n^2 (n+1) \left (a e^2+c d^2\right )}+\frac{c d (1-4 n) (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 n^2 \left (a e^2+c d^2\right )}+\frac{c e^3 (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n (n+1) \left (a e^2+c d^2\right )^2}-\frac{c d e^2 (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^2}-\frac{c x \left (d (1-4 n)-e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a e^2+c d^2\right ) \left (a+c x^{2 n}\right )}-\frac{c e^5 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^3}+\frac{c d e^4 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^3}+\frac{e^6 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2+c d^2\right )^3}+\frac{c e^2 x \left (d-e x^n\right )}{2 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}+\frac{c x \left (d-e x^n\right )}{4 a n \left (a e^2+c d^2\right ) \left (a+c x^{2 n}\right )^2} \]
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Rubi [A] time = 0.416065, antiderivative size = 582, normalized size of antiderivative = 1., number of steps used = 15, 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} \[ -\frac{c e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 n^2 (n+1) \left (a e^2+c d^2\right )}+\frac{c d (1-4 n) (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 n^2 \left (a e^2+c d^2\right )}+\frac{c e^3 (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n (n+1) \left (a e^2+c d^2\right )^2}-\frac{c d e^2 (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^2}-\frac{c x \left (d (1-4 n)-e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a e^2+c d^2\right ) \left (a+c x^{2 n}\right )}-\frac{c e^5 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^3}+\frac{c d e^4 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^3}+\frac{e^6 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2+c d^2\right )^3}+\frac{c e^2 x \left (d-e x^n\right )}{2 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}+\frac{c x \left (d-e x^n\right )}{4 a n \left (a e^2+c d^2\right ) \left (a+c x^{2 n}\right )^2} \]
Antiderivative was successfully verified.
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Rule 1437
Rule 245
Rule 1431
Rule 1418
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^n\right ) \left (a+c x^{2 n}\right )^3} \, dx &=\int \left (\frac{e^6}{\left (c d^2+a e^2\right )^3 \left (d+e x^n\right )}-\frac{c \left (-d+e x^n\right )}{\left (c d^2+a e^2\right ) \left (a+c x^{2 n}\right )^3}-\frac{c e^2 \left (-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^4 \left (-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^4\right ) \int \frac{-d+e x^n}{a+c x^{2 n}} \, dx}{\left (c d^2+a e^2\right )^3}+\frac{e^6 \int \frac{1}{d+e x^n} \, dx}{\left (c d^2+a e^2\right )^3}-\frac{\left (c e^2\right ) \int \frac{-d+e x^n}{\left (a+c x^{2 n}\right )^2} \, dx}{\left (c d^2+a e^2\right )^2}-\frac{c \int \frac{-d+e x^n}{\left (a+c x^{2 n}\right )^3} \, dx}{c d^2+a e^2}\\ &=\frac{c x \left (d-e x^n\right )}{4 a \left (c d^2+a e^2\right ) n \left (a+c x^{2 n}\right )^2}+\frac{c e^2 x \left (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{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+a e^2\right )^3}+\frac{\left (c d e^4\right ) \int \frac{1}{a+c x^{2 n}} \, dx}{\left (c d^2+a e^2\right )^3}-\frac{\left (c e^5\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\right ) \int \frac{-d (1-2 n)+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 \int \frac{-d (1-4 n)+e (1-3 n) x^n}{\left (a+c x^{2 n}\right )^2} \, dx}{4 a \left (c d^2+a e^2\right ) n}\\ &=\frac{c x \left (d-e x^n\right )}{4 a \left (c d^2+a e^2\right ) n \left (a+c x^{2 n}\right )^2}+\frac{c e^2 x \left (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 x \left (d (1-4 n)-e (1-3 n) x^n\right )}{8 a^2 \left (c d^2+a e^2\right ) n^2 \left (a+c x^{2 n}\right )}+\frac{c d e^4 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{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+a e^2\right )^3}-\frac{c e^5 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 \int \frac{-d (1-4 n) (1-2 n)+e (1-3 n) (1-n) x^n}{a+c x^{2 n}} \, dx}{8 a^2 \left (c d^2+a e^2\right ) n^2}-\frac{\left (c d e^2 (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 e^3 (1-n)\right ) \int \frac{x^n}{a+c x^{2 n}} \, dx}{2 a \left (c d^2+a e^2\right )^2 n}\\ &=\frac{c x \left (d-e x^n\right )}{4 a \left (c d^2+a e^2\right ) n \left (a+c x^{2 n}\right )^2}+\frac{c e^2 x \left (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 x \left (d (1-4 n)-e (1-3 n) x^n\right )}{8 a^2 \left (c d^2+a e^2\right ) n^2 \left (a+c x^{2 n}\right )}+\frac{c d e^4 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 d e^2 (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{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+a e^2\right )^3}-\frac{c e^5 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 e^3 (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 )}{2 a^2 \left (c d^2+a e^2\right )^2 n (1+n)}+\frac{(c d (1-4 n) (1-2 n)) \int \frac{1}{a+c x^{2 n}} \, dx}{8 a^2 \left (c d^2+a e^2\right ) n^2}-\frac{(c e (1-3 n) (1-n)) \int \frac{x^n}{a+c x^{2 n}} \, dx}{8 a^2 \left (c d^2+a e^2\right ) n^2}\\ &=\frac{c x \left (d-e x^n\right )}{4 a \left (c d^2+a e^2\right ) n \left (a+c x^{2 n}\right )^2}+\frac{c e^2 x \left (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 x \left (d (1-4 n)-e (1-3 n) x^n\right )}{8 a^2 \left (c d^2+a e^2\right ) n^2 \left (a+c x^{2 n}\right )}+\frac{c d e^4 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 d (1-4 n) (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 )}{8 a^3 \left (c d^2+a e^2\right ) n^2}-\frac{c d e^2 (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{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+a e^2\right )^3}-\frac{c e^5 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 e (1-3 n) (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 )}{8 a^3 \left (c d^2+a e^2\right ) n^2 (1+n)}+\frac{c e^3 (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 )}{2 a^2 \left (c d^2+a e^2\right )^2 n (1+n)}\\ \end{align*}
Mathematica [A] time = 0.464, size = 346, normalized size = 0.59 \[ \frac{x \left (-\frac{c e^3 x^n \left (a e^2+c d^2\right ) \, _2F_1\left (2,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 (n+1)}-\frac{c e x^n \left (a e^2+c d^2\right )^2 \, _2F_1\left (3,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^3 (n+1)}+\frac{c d e^2 \left (a e^2+c d^2\right ) \, _2F_1\left (2,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2}+\frac{c d \left (a e^2+c d^2\right )^2 \, _2F_1\left (3,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^3}+\frac{c d e^4 \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a}-\frac{c e^5 x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1)}+\frac{e^6 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d}\right )}{\left (a e^2+c d^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.17, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d+e{x}^{n} \right ) \left ( a+c{x}^{2\,n} \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{6} \int \frac{1}{c^{3} d^{7} + 3 \, a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} + a^{3} d e^{6} +{\left (c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x^{n}}\,{d x} - \frac{{\left (a c^{2} e^{3}{\left (7 \, n - 1\right )} + c^{3} d^{2} e{\left (3 \, n - 1\right )}\right )} x x^{3 \, n} -{\left (a c^{2} d e^{2}{\left (8 \, n - 1\right )} + c^{3} d^{3}{\left (4 \, n - 1\right )}\right )} x x^{2 \, n} +{\left (a^{2} c e^{3}{\left (9 \, n - 1\right )} + a c^{2} d^{2} e{\left (5 \, n - 1\right )}\right )} x x^{n} -{\left (a^{2} c d e^{2}{\left (10 \, n - 1\right )} + a c^{2} d^{3}{\left (6 \, n - 1\right )}\right )} x}{8 \,{\left (a^{4} c^{2} d^{4} n^{2} + 2 \, a^{5} c d^{2} e^{2} n^{2} + a^{6} e^{4} n^{2} +{\left (a^{2} c^{4} d^{4} n^{2} + 2 \, a^{3} c^{3} d^{2} e^{2} n^{2} + a^{4} c^{2} e^{4} n^{2}\right )} x^{4 \, n} + 2 \,{\left (a^{3} c^{3} d^{4} n^{2} + 2 \, a^{4} c^{2} d^{2} e^{2} n^{2} + a^{5} c e^{4} n^{2}\right )} x^{2 \, n}\right )}} - \int -\frac{{\left (8 \, n^{2} - 6 \, n + 1\right )} c^{3} d^{5} + 2 \,{\left (12 \, n^{2} - 8 \, n + 1\right )} a c^{2} d^{3} e^{2} +{\left (24 \, n^{2} - 10 \, n + 1\right )} a^{2} c d e^{4} -{\left ({\left (3 \, n^{2} - 4 \, n + 1\right )} c^{3} d^{4} e + 2 \,{\left (5 \, n^{2} - 6 \, n + 1\right )} a c^{2} d^{2} e^{3} +{\left (15 \, n^{2} - 8 \, n + 1\right )} a^{2} c e^{5}\right )} x^{n}}{8 \,{\left (a^{3} c^{3} d^{6} n^{2} + 3 \, a^{4} c^{2} d^{4} e^{2} n^{2} + 3 \, a^{5} c d^{2} e^{4} n^{2} + a^{6} e^{6} n^{2} +{\left (a^{2} c^{4} d^{6} n^{2} + 3 \, a^{3} c^{3} d^{4} e^{2} n^{2} + 3 \, a^{4} c^{2} d^{2} e^{4} n^{2} + a^{5} c e^{6} n^{2}\right )} x^{2 \, n}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{3} e x^{n} + a^{3} d +{\left (c^{3} e x^{n} + c^{3} d\right )} x^{6 \, n} + 3 \,{\left (a c^{2} e x^{n} + a c^{2} d\right )} x^{4 \, n} + 3 \,{\left (a^{2} c e x^{n} + a^{2} c d\right )} x^{2 \, n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2 \, n} + a\right )}^{3}{\left (e x^{n} + d\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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