Integrand size = 19, antiderivative size = 83 \[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)} \]
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Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1432, 251, 371} \[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (n+1)} \]
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Rule 251
Rule 371
Rule 1432
Rubi steps \begin{align*} \text {integral}& = d \int \frac {1}{a+c x^{2 n}} \, dx+e \int \frac {x^n}{a+c x^{2 n}} \, dx \\ & = \frac {d 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}+\frac {e 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 (1+n)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\frac {d x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 n},\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{a (1+n)} \]
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\[\int \frac {d +e \,x^{n}}{a +c \,x^{2 n}}d x\]
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\[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\int { \frac {e x^{n} + d}{c x^{2 \, n} + a} \,d x } \]
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Result contains complex when optimal does not.
Time = 2.16 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.52 \[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\frac {a^{\frac {1}{2 n}} a^{-1 - \frac {1}{2 n}} d x \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} + \frac {a^{- \frac {3}{2} - \frac {1}{2 n}} a^{\frac {1}{2} + \frac {1}{2 n}} e x^{n + 1} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 n^{2} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} \]
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\[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\int { \frac {e x^{n} + d}{c x^{2 \, n} + a} \,d x } \]
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\[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\int { \frac {e x^{n} + d}{c x^{2 \, n} + a} \,d x } \]
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Timed out. \[ \int \frac {d+e x^n}{a+c x^{2 n}} \, dx=\int \frac {d+e\,x^n}{a+c\,x^{2\,n}} \,d x \]
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