Integrand size = 18, antiderivative size = 150 \[ \int x^5 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=-\frac {2 d p^2 x^3}{3 e}+\frac {p^2 \left (d+e x^3\right )^2}{12 e^2}+\frac {2 d p \left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e^2}-\frac {p \left (d+e x^3\right )^2 \log \left (c \left (d+e x^3\right )^p\right )}{6 e^2}-\frac {d \left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 e^2}+\frac {\left (d+e x^3\right )^2 \log ^2\left (c \left (d+e x^3\right )^p\right )}{6 e^2} \]
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Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int x^5 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {\left (d+e x^3\right )^2 \log ^2\left (c \left (d+e x^3\right )^p\right )}{6 e^2}-\frac {d \left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 e^2}-\frac {p \left (d+e x^3\right )^2 \log \left (c \left (d+e x^3\right )^p\right )}{6 e^2}+\frac {2 d p \left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e^2}+\frac {p^2 \left (d+e x^3\right )^2}{12 e^2}-\frac {2 d p^2 x^3}{3 e} \]
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Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2436
Rule 2437
Rule 2448
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (-\frac {d \log ^2\left (c (d+e x)^p\right )}{e}+\frac {(d+e x) \log ^2\left (c (d+e x)^p\right )}{e}\right ) \, dx,x,x^3\right ) \\ & = \frac {\text {Subst}\left (\int (d+e x) \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^3\right )}{3 e}-\frac {d \text {Subst}\left (\int \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^3\right )}{3 e} \\ & = \frac {\text {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^3\right )}{3 e^2}-\frac {d \text {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^3\right )}{3 e^2} \\ & = -\frac {d \left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 e^2}+\frac {\left (d+e x^3\right )^2 \log ^2\left (c \left (d+e x^3\right )^p\right )}{6 e^2}-\frac {p \text {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^3\right )}{3 e^2}+\frac {(2 d p) \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^3\right )}{3 e^2} \\ & = -\frac {2 d p^2 x^3}{3 e}+\frac {p^2 \left (d+e x^3\right )^2}{12 e^2}+\frac {2 d p \left (d+e x^3\right ) \log \left (c \left (d+e x^3\right )^p\right )}{3 e^2}-\frac {p \left (d+e x^3\right )^2 \log \left (c \left (d+e x^3\right )^p\right )}{6 e^2}-\frac {d \left (d+e x^3\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{3 e^2}+\frac {\left (d+e x^3\right )^2 \log ^2\left (c \left (d+e x^3\right )^p\right )}{6 e^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.70 \[ \int x^5 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {e p^2 x^3 \left (-6 d+e x^3\right )+2 d^2 p^2 \log \left (d+e x^3\right )+2 p \left (2 d^2+2 d e x^3-e^2 x^6\right ) \log \left (c \left (d+e x^3\right )^p\right )-2 \left (d^2-e^2 x^6\right ) \log ^2\left (c \left (d+e x^3\right )^p\right )}{12 e^2} \]
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Time = 4.79 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.01
method | result | size |
parallelrisch | \(\frac {2 x^{6} {\ln \left (c \left (e \,x^{3}+d \right )^{p}\right )}^{2} e^{2}-2 x^{6} \ln \left (c \left (e \,x^{3}+d \right )^{p}\right ) e^{2} p +e^{2} p^{2} x^{6}+4 x^{3} \ln \left (c \left (e \,x^{3}+d \right )^{p}\right ) d e p -6 d e \,p^{2} x^{3}+10 d^{2} p^{2} \ln \left (e \,x^{3}+d \right )-2 {\ln \left (c \left (e \,x^{3}+d \right )^{p}\right )}^{2} d^{2}-4 \ln \left (c \left (e \,x^{3}+d \right )^{p}\right ) d^{2} p +6 d^{2} p^{2}}{12 e^{2}}\) | \(151\) |
risch | \(\text {Expression too large to display}\) | \(24313\) |
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Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.99 \[ \int x^5 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {e^{2} p^{2} x^{6} + 2 \, e^{2} x^{6} \log \left (c\right )^{2} - 6 \, d e p^{2} x^{3} + 2 \, {\left (e^{2} p^{2} x^{6} - d^{2} p^{2}\right )} \log \left (e x^{3} + d\right )^{2} - 2 \, {\left (e^{2} p^{2} x^{6} - 2 \, d e p^{2} x^{3} - 3 \, d^{2} p^{2} - 2 \, {\left (e^{2} p x^{6} - d^{2} p\right )} \log \left (c\right )\right )} \log \left (e x^{3} + d\right ) - 2 \, {\left (e^{2} p x^{6} - 2 \, d e p x^{3}\right )} \log \left (c\right )}{12 \, e^{2}} \]
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Time = 3.67 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.91 \[ \int x^5 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\begin {cases} \frac {d^{2} p \log {\left (c \left (d + e x^{3}\right )^{p} \right )}}{2 e^{2}} - \frac {d^{2} \log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{6 e^{2}} - \frac {d p^{2} x^{3}}{2 e} + \frac {d p x^{3} \log {\left (c \left (d + e x^{3}\right )^{p} \right )}}{3 e} + \frac {p^{2} x^{6}}{12} - \frac {p x^{6} \log {\left (c \left (d + e x^{3}\right )^{p} \right )}}{6} + \frac {x^{6} \log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}}{6} & \text {for}\: e \neq 0 \\\frac {x^{6} \log {\left (c d^{p} \right )}^{2}}{6} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.80 \[ \int x^5 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {1}{6} \, x^{6} \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2} - \frac {1}{6} \, e p {\left (\frac {2 \, d^{2} \log \left (e x^{3} + d\right )}{e^{3}} + \frac {e x^{6} - 2 \, d x^{3}}{e^{2}}\right )} \log \left ({\left (e x^{3} + d\right )}^{p} c\right ) + \frac {{\left (e^{2} x^{6} - 6 \, d e x^{3} + 2 \, d^{2} \log \left (e x^{3} + d\right )^{2} + 6 \, d^{2} \log \left (e x^{3} + d\right )\right )} p^{2}}{12 \, e^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.44 \[ \int x^5 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {2 \, {\left (e x^{3} + d\right )}^{2} p^{2} \log \left (e x^{3} + d\right )^{2} - 2 \, {\left (e x^{3} + d\right )}^{2} p^{2} \log \left (e x^{3} + d\right ) + 4 \, {\left (e x^{3} + d\right )}^{2} p \log \left (e x^{3} + d\right ) \log \left (c\right ) + {\left (e x^{3} + d\right )}^{2} p^{2} - 2 \, {\left (e x^{3} + d\right )}^{2} p \log \left (c\right ) + 2 \, {\left (e x^{3} + d\right )}^{2} \log \left (c\right )^{2}}{12 \, e^{2}} - \frac {{\left (2 \, e x^{3} + {\left (e x^{3} + d\right )} \log \left (e x^{3} + d\right )^{2} - 2 \, {\left (e x^{3} + d\right )} \log \left (e x^{3} + d\right ) + 2 \, d\right )} d p^{2} - 2 \, {\left (e x^{3} - {\left (e x^{3} + d\right )} \log \left (e x^{3} + d\right ) + d\right )} d p \log \left (c\right ) + {\left (e x^{3} + d\right )} d \log \left (c\right )^{2}}{3 \, e^{2}} \]
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Time = 1.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.67 \[ \int x^5 \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx=\frac {p^2\,x^6}{12}-\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )\,\left (\frac {p\,x^6}{6}-\frac {d\,p\,x^3}{3\,e}\right )+{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2\,\left (\frac {x^6}{6}-\frac {d^2}{6\,e^2}\right )-\frac {d\,p^2\,x^3}{2\,e}+\frac {d^2\,p^2\,\ln \left (e\,x^3+d\right )}{2\,e^2} \]
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