Integrand size = 17, antiderivative size = 56 \[ \int (c x)^{2 n} \left (a+b x^n\right )^p \, dx=\frac {(c x)^{1+2 n} \left (a+b x^n\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,3+\frac {1}{n}+p,3+\frac {1}{n},-\frac {b x^n}{a}\right )}{a c (1+2 n)} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {372, 371} \[ \int (c x)^{2 n} \left (a+b x^n\right )^p \, dx=\frac {(c x)^{2 n+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (2+\frac {1}{n},-p,3+\frac {1}{n},-\frac {b x^n}{a}\right )}{c (2 n+1)} \]
[In]
[Out]
Rule 371
Rule 372
Rubi steps \begin{align*} \text {integral}& = \left (\left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int (c x)^{2 n} \left (1+\frac {b x^n}{a}\right )^p \, dx \\ & = \frac {(c x)^{1+2 n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \, _2F_1\left (2+\frac {1}{n},-p;3+\frac {1}{n};-\frac {b x^n}{a}\right )}{c (1+2 n)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.11 \[ \int (c x)^{2 n} \left (a+b x^n\right )^p \, dx=\frac {x (c x)^{2 n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (2+\frac {1}{n},-p,3+\frac {1}{n},-\frac {b x^n}{a}\right )}{1+2 n} \]
[In]
[Out]
\[\int \left (c x \right )^{2 n} \left (a +b \,x^{n}\right )^{p}d x\]
[In]
[Out]
\[ \int (c x)^{2 n} \left (a+b x^n\right )^p \, dx=\int { {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{2 \, n} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 4.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12 \[ \int (c x)^{2 n} \left (a+b x^n\right )^p \, dx=\frac {a^{2 + \frac {1}{n}} a^{p - 2 - \frac {1}{n}} c^{2 n} x^{2 n + 1} \Gamma \left (2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 + \frac {1}{n} \\ 3 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \frac {1}{n}\right )} \]
[In]
[Out]
\[ \int (c x)^{2 n} \left (a+b x^n\right )^p \, dx=\int { {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{2 \, n} \,d x } \]
[In]
[Out]
Exception generated. \[ \int (c x)^{2 n} \left (a+b x^n\right )^p \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int (c x)^{2 n} \left (a+b x^n\right )^p \, dx=\int {\left (c\,x\right )}^{2\,n}\,{\left (a+b\,x^n\right )}^p \,d x \]
[In]
[Out]