Integrand size = 16, antiderivative size = 65 \[ \int x^2 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {e n p x^{3+n} \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{n},2+\frac {3}{n},-\frac {e x^n}{d}\right )}{3 d (3+n)}+\frac {1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2505, 371} \[ \int x^2 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right )-\frac {e n p x^{n+3} \operatorname {Hypergeometric2F1}\left (1,\frac {n+3}{n},2+\frac {3}{n},-\frac {e x^n}{d}\right )}{3 d (n+3)} \]
[In]
[Out]
Rule 371
Rule 2505
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right )-\frac {1}{3} (e n p) \int \frac {x^{2+n}}{d+e x^n} \, dx \\ & = -\frac {e n p x^{3+n} \, _2F_1\left (1,\frac {3+n}{n};2+\frac {3}{n};-\frac {e x^n}{d}\right )}{3 d (3+n)}+\frac {1}{3} x^3 \log \left (c \left (d+e x^n\right )^p\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int x^2 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {1}{3} x^3 \left (-\frac {e n p x^n \operatorname {Hypergeometric2F1}\left (1,\frac {3+n}{n},2+\frac {3}{n},-\frac {e x^n}{d}\right )}{d (3+n)}+\log \left (c \left (d+e x^n\right )^p\right )\right ) \]
[In]
[Out]
\[\int x^{2} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )d x\]
[In]
[Out]
\[ \int x^2 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { x^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 5.88 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.97 \[ \int x^2 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=- \frac {d^{-2 - \frac {3}{n}} d^{1 + \frac {3}{n}} e p x^{n + 3} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {3}{n}\right ) \Gamma \left (1 + \frac {3}{n}\right )}{3 \Gamma \left (2 + \frac {3}{n}\right )} - \frac {d^{-2 - \frac {3}{n}} d^{1 + \frac {3}{n}} e p x^{n + 3} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {3}{n}\right ) \Gamma \left (1 + \frac {3}{n}\right )}{n \Gamma \left (2 + \frac {3}{n}\right )} + \frac {x^{3} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{3} \]
[In]
[Out]
\[ \int x^2 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { x^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
[In]
[Out]
\[ \int x^2 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { x^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int x^2 \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int x^2\,\ln \left (c\,{\left (d+e\,x^n\right )}^p\right ) \,d x \]
[In]
[Out]