Optimal. Leaf size=203 \[ -\frac{(1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n}-\frac{d e (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 n (n+1)}+\frac{x \left (-a e^2+c d^2+2 c d e x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{e^2 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c} \]
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Rubi [A] time = 0.167352, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {1437, 1431, 1418, 245, 364} \[ -\frac{(1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n}-\frac{d e (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 n (n+1)}+\frac{x \left (-a e^2+c d^2+2 c d e x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{e^2 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 1437
Rule 1431
Rule 1418
Rule 245
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (d+e x^n\right )^2}{\left (a+c x^{2 n}\right )^2} \, dx &=\int \left (\frac{c d^2-a e^2+2 c d e x^n}{c \left (a+c x^{2 n}\right )^2}+\frac{e^2}{c \left (a+c x^{2 n}\right )}\right ) \, dx\\ &=\frac{\int \frac{c d^2-a e^2+2 c d e x^n}{\left (a+c x^{2 n}\right )^2} \, dx}{c}+\frac{e^2 \int \frac{1}{a+c x^{2 n}} \, dx}{c}\\ &=\frac{x \left (c d^2-a e^2+2 c d e x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{e^2 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 c}-\frac{\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 c n}\\ &=\frac{x \left (c d^2-a e^2+2 c d e x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{e^2 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 c}-\frac{\left (\left (c d^2-a e^2\right ) (1-2 n)\right ) \int \frac{1}{a+c x^{2 n}} \, dx}{2 a c n}-\frac{(d e (1-n)) \int \frac{x^n}{a+c x^{2 n}} \, dx}{a n}\\ &=\frac{x \left (c d^2-a e^2+2 c d e x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{e^2 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 c}-\frac{\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 c n}-\frac{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 n (1+n)}\\ \end{align*}
Mathematica [A] time = 0.160188, size = 136, normalized size = 0.67 \[ \frac{x \left ((n+1) \left (c d^2-a e^2\right ) \, _2F_1\left (2,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+2 c d e x^n \, _2F_1\left (2,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+a e^2 (n+1) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )\right )}{a^2 c (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \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*} \frac{2 \, c d e x x^{n} +{\left (c d^{2} - a e^{2}\right )} x}{2 \,{\left (a c^{2} n x^{2 \, n} + a^{2} c n\right )}} + \int \frac{2 \, c d e{\left (n - 1\right )} x^{n} + c d^{2}{\left (2 \, n - 1\right )} + a e^{2}}{2 \,{\left (a c^{2} n x^{2 \, n} + a^{2} c 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{e^{2} x^{2 \, n} + 2 \, d e x^{n} + d^{2}}{c^{2} x^{4 \, n} + 2 \, a c x^{2 \, n} + a^{2}}, 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{{\left (e x^{n} + d\right )}^{2}}{{\left (c x^{2 \, n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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