Optimal. Leaf size=288 \[ -\frac{e (1-n) x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n (n+1)}-\frac{d (1-2 n) x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n}+\frac{x \left (e x^n \left (3 c d^2-a e^2\right )+d \left (c d^2-3 a e^2\right )\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{3 d e^2 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{e^3 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c (n+1)} \]
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Rubi [A] time = 0.252169, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{e (1-n) x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n (n+1)}-\frac{d (1-2 n) x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 c n}+\frac{x \left (e x^n \left (3 c d^2-a e^2\right )+d \left (c d^2-3 a e^2\right )\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{3 d e^2 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c}+\frac{e^3 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a c (n+1)} \]
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 )^3}{\left (a+c x^{2 n}\right )^2} \, dx &=\int \left (\frac{c d^3-3 a d e^2+\left (3 c d^2 e-a e^3\right ) x^n}{c \left (a+c x^{2 n}\right )^2}+\frac{e^2 \left (3 d+e x^n\right )}{c \left (a+c x^{2 n}\right )}\right ) \, dx\\ &=\frac{\int \frac{c d^3-3 a d e^2+\left (3 c d^2 e-a e^3\right ) x^n}{\left (a+c x^{2 n}\right )^2} \, dx}{c}+\frac{e^2 \int \frac{3 d+e x^n}{a+c x^{2 n}} \, dx}{c}\\ &=\frac{x \left (d \left (c d^2-3 a e^2\right )+e \left (3 c d^2-a e^2\right ) x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{\left (3 d e^2\right ) \int \frac{1}{a+c x^{2 n}} \, dx}{c}+\frac{e^3 \int \frac{x^n}{a+c x^{2 n}} \, dx}{c}-\frac{\int \frac{\left (c d^3-3 a d e^2\right ) (1-2 n)+\left (3 c d^2 e-a e^3\right ) (1-n) x^n}{a+c x^{2 n}} \, dx}{2 a c n}\\ &=\frac{x \left (d \left (c d^2-3 a e^2\right )+e \left (3 c d^2-a e^2\right ) x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{3 d 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{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 c (1+n)}-\frac{\left (d \left (c d^2-3 a e^2\right ) (1-2 n)\right ) \int \frac{1}{a+c x^{2 n}} \, dx}{2 a c n}-\frac{\left (e \left (3 c d^2-a e^2\right ) (1-n)\right ) \int \frac{x^n}{a+c x^{2 n}} \, dx}{2 a c n}\\ &=\frac{x \left (d \left (c d^2-3 a e^2\right )+e \left (3 c d^2-a e^2\right ) x^n\right )}{2 a c n \left (a+c x^{2 n}\right )}+\frac{3 d 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{d \left (c d^2-3 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{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 c (1+n)}-\frac{e \left (3 c d^2-a e^2\right ) (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 c n (1+n)}\\ \end{align*}
Mathematica [A] time = 0.282452, size = 188, normalized size = 0.65 \[ \frac{x \left (\frac{e x^n \left (3 c d^2-a e^2\right ) \, _2F_1\left (2,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}+d \left (c d^2-3 a e^2\right ) \, _2F_1\left (2,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+3 a d e^2 \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+\frac{a e^3 x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{n+1}\right )}{a^2 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d+e{x}^{n} \right ) ^{3}}{ \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{{\left (3 \, c d^{2} e - a e^{3}\right )} x x^{n} +{\left (c d^{3} - 3 \, a d e^{2}\right )} x}{2 \,{\left (a c^{2} n x^{2 \, n} + a^{2} c n\right )}} + \int \frac{c d^{3}{\left (2 \, n - 1\right )} + 3 \, a d e^{2} +{\left (a e^{3}{\left (n + 1\right )} + 3 \, c d^{2} e{\left (n - 1\right )}\right )} x^{n}}{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^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}{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 )}^{3}}{{\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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