Optimal. Leaf size=410 \[ \frac{c^2 d e (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 n (n+1) \left (a e^2+c d^2\right )^2}-\frac{c (1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^2}-\frac{4 c^2 d e^3 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^3}+\frac{c e^2 x \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^3}+\frac{4 c e^4 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{\left (a e^2+c d^2\right )^3}+\frac{e^4 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )^2}+\frac{c x \left (-a e^2+c d^2-2 c d e x^n\right )}{2 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )} \]
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Rubi [A] time = 0.378725, antiderivative size = 410, normalized size of antiderivative = 1., 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.
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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 )^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 = 0.502782, 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.
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Maple [F] time = 0.146, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d+e{x}^{n} \right ) ^{2} \left ( a+c{x}^{2\,n} \right ) ^{2}}}\, 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*}{\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} \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^{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 ) \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 )}^{2}{\left (e x^{n} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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