Integrand size = 19, antiderivative size = 84 \[ \int \frac {\left (c+d x^n\right )^2}{a+b x^n} \, dx=-\frac {d (a d (1+n)-b (c+2 c n)) x}{b^2 (1+n)}+\frac {d x \left (c+d x^n\right )}{b (1+n)}+\frac {(b c-a d)^2 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a b^2} \]
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Time = 0.07 (sec) , antiderivative size = 84, 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 = {427, 396, 251} \[ \int \frac {\left (c+d x^n\right )^2}{a+b x^n} \, dx=\frac {x (b c-a d)^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a b^2}-\frac {d x (a d (n+1)-b (2 c n+c))}{b^2 (n+1)}+\frac {d x \left (c+d x^n\right )}{b (n+1)} \]
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Rule 251
Rule 396
Rule 427
Rubi steps \begin{align*} \text {integral}& = \frac {d x \left (c+d x^n\right )}{b (1+n)}+\frac {\int \frac {-c (a d-b c (1+n))-d (a d (1+n)-b (c+2 c n)) x^n}{a+b x^n} \, dx}{b (1+n)} \\ & = -\frac {d (a d (1+n)-b (c+2 c n)) x}{b^2 (1+n)}+\frac {d x \left (c+d x^n\right )}{b (1+n)}+\frac {(b c-a d)^2 \int \frac {1}{a+b x^n} \, dx}{b^2} \\ & = -\frac {d (a d (1+n)-b (c+2 c n)) x}{b^2 (1+n)}+\frac {d x \left (c+d x^n\right )}{b (1+n)}+\frac {(b c-a d)^2 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a b^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.49 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+d x^n\right )^2}{a+b x^n} \, dx=\frac {x \left (2 c d x^n \Phi \left (-\frac {b x^n}{a},1,1+\frac {1}{n}\right )+d^2 x^{2 n} \Phi \left (-\frac {b x^n}{a},1,2+\frac {1}{n}\right )+c^2 \Phi \left (-\frac {b x^n}{a},1,\frac {1}{n}\right )\right )}{a n} \]
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\[\int \frac {\left (c +d \,x^{n}\right )^{2}}{a +b \,x^{n}}d x\]
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\[ \int \frac {\left (c+d x^n\right )^2}{a+b x^n} \, dx=\int { \frac {{\left (d x^{n} + c\right )}^{2}}{b x^{n} + a} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.81 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.80 \[ \int \frac {\left (c+d x^n\right )^2}{a+b x^n} \, dx=\frac {a^{\frac {1}{n}} a^{-1 - \frac {1}{n}} c^{2} x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{n^{2} \Gamma \left (1 + \frac {1}{n}\right )} + \frac {2 a^{-3 - \frac {1}{n}} a^{2 + \frac {1}{n}} d^{2} x^{2 n + 1} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 2 + \frac {1}{n}\right ) \Gamma \left (2 + \frac {1}{n}\right )}{n \Gamma \left (3 + \frac {1}{n}\right )} + \frac {a^{-3 - \frac {1}{n}} a^{2 + \frac {1}{n}} d^{2} x^{2 n + 1} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 2 + \frac {1}{n}\right ) \Gamma \left (2 + \frac {1}{n}\right )}{n^{2} \Gamma \left (3 + \frac {1}{n}\right )} - \frac {2 a^{- \frac {1}{n}} a^{1 + \frac {1}{n}} b^{\frac {1}{n}} b^{-1 - \frac {1}{n}} c d x \Phi \left (\frac {a x^{- n} e^{i \pi }}{b}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a n^{2} \Gamma \left (1 + \frac {1}{n}\right )} \]
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\[ \int \frac {\left (c+d x^n\right )^2}{a+b x^n} \, dx=\int { \frac {{\left (d x^{n} + c\right )}^{2}}{b x^{n} + a} \,d x } \]
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\[ \int \frac {\left (c+d x^n\right )^2}{a+b x^n} \, dx=\int { \frac {{\left (d x^{n} + c\right )}^{2}}{b x^{n} + a} \,d x } \]
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Timed out. \[ \int \frac {\left (c+d x^n\right )^2}{a+b x^n} \, dx=\int \frac {{\left (c+d\,x^n\right )}^2}{a+b\,x^n} \,d x \]
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