Integrand size = 21, antiderivative size = 141 \[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\frac {3 d e^2 x}{c}+\frac {e^3 x^{1+n}}{c (1+n)}+\frac {d \left (c d^2-3 a e^2\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {e \left (3 c d^2-a e^2\right ) x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c (1+n)} \]
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Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1439, 1432, 251, 371} \[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\frac {e x^{n+1} \left (3 c d^2-a e^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c (n+1)}+\frac {d x \left (c d^2-3 a e^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {3 d e^2 x}{c}+\frac {e^3 x^{n+1}}{c (n+1)} \]
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Rule 251
Rule 371
Rule 1432
Rule 1439
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 d e^2}{c}+\frac {e^3 x^n}{c}+\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 )}\right ) \, dx \\ & = \frac {3 d e^2 x}{c}+\frac {e^3 x^{1+n}}{c (1+n)}+\frac {\int \frac {c d^3-3 a d e^2+\left (3 c d^2 e-a e^3\right ) x^n}{a+c x^{2 n}} \, dx}{c} \\ & = \frac {3 d e^2 x}{c}+\frac {e^3 x^{1+n}}{c (1+n)}+\frac {\left (d \left (c d^2-3 a e^2\right )\right ) \int \frac {1}{a+c x^{2 n}} \, dx}{c}+\frac {\left (e \left (3 c d^2-a e^2\right )\right ) \int \frac {x^n}{a+c x^{2 n}} \, dx}{c} \\ & = \frac {3 d e^2 x}{c}+\frac {e^3 x^{1+n}}{c (1+n)}+\frac {d \left (c d^2-3 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 c}+\frac {e \left (3 c d^2-a e^2\right ) 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)} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\frac {x \left (d \left (c d^2-3 a e^2\right ) (1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+e \left (a e \left (3 d (1+n)+e x^n\right )+\left (3 c d^2-a e^2\right ) x^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )\right )\right )}{a c (1+n)} \]
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\[\int \frac {\left (d +e \,x^{n}\right )^{3}}{a +c \,x^{2 n}}d x\]
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\[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{c x^{2 \, n} + a} \,d x } \]
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Result contains complex when optimal does not.
Time = 4.01 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.30 \[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\frac {a^{\frac {1}{2 n}} a^{-1 - \frac {1}{2 n}} d^{3} x \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {3 a^{- \frac {5}{2} - \frac {1}{2 n}} a^{\frac {3}{2} + \frac {1}{2 n}} e^{3} x^{3 n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {3}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )}{4 n \Gamma \left (\frac {5}{2} + \frac {1}{2 n}\right )} + \frac {a^{- \frac {5}{2} - \frac {1}{2 n}} a^{\frac {3}{2} + \frac {1}{2 n}} e^{3} x^{3 n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {3}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (\frac {5}{2} + \frac {1}{2 n}\right )} + \frac {3 a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} d^{2} e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} + \frac {3 a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} d^{2} e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} - \frac {3 a^{- \frac {1}{2 n}} a^{1 + \frac {1}{2 n}} c^{\frac {1}{2 n}} c^{-1 - \frac {1}{2 n}} d e^{2} x \Phi \left (\frac {a x^{- 2 n} e^{i \pi }}{c}, 1, \frac {e^{i \pi }}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 a n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} \]
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\[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{c x^{2 \, n} + a} \,d x } \]
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\[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{c x^{2 \, n} + a} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx=\int \frac {{\left (d+e\,x^n\right )}^3}{a+c\,x^{2\,n}} \,d x \]
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