Integrand size = 18, antiderivative size = 145 \[ \int x^3 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {a p^2 x^2}{b}+\frac {p^2 \left (a+b x^2\right )^2}{8 b^2}+\frac {a p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}-\frac {p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac {a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac {\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2} \]
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Time = 0.11 (sec) , antiderivative size = 145, 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} \[ \int x^3 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac {a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}-\frac {p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^2}+\frac {a p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac {p^2 \left (a+b x^2\right )^2}{8 b^2}-\frac {a p^2 x^2}{b} \]
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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}{2} \text {Subst}\left (\int x \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {a \log ^2\left (c (a+b x)^p\right )}{b}+\frac {(a+b x) \log ^2\left (c (a+b x)^p\right )}{b}\right ) \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int (a+b x) \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b}-\frac {a \text {Subst}\left (\int \log ^2\left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 b} \\ & = \frac {\text {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}-\frac {a \text {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2} \\ & = -\frac {a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac {\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac {p \text {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b^2}+\frac {(a p) \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{b^2} \\ & = -\frac {a p^2 x^2}{b}+\frac {p^2 \left (a+b x^2\right )^2}{8 b^2}+\frac {a p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b^2}-\frac {p \left (a+b x^2\right )^2 \log \left (c \left (a+b x^2\right )^p\right )}{4 b^2}-\frac {a \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b^2}+\frac {\left (a+b x^2\right )^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 b^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.72 \[ \int x^3 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {b p^2 x^2 \left (-6 a+b x^2\right )+2 a^2 p^2 \log \left (a+b x^2\right )+2 p \left (2 a^2+2 a b x^2-b^2 x^4\right ) \log \left (c \left (a+b x^2\right )^p\right )-2 \left (a^2-b^2 x^4\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{8 b^2} \]
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Time = 1.33 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {2 x^{4} {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{2} b^{2}-2 x^{4} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) b^{2} p +b^{2} p^{2} x^{4}+4 x^{2} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) a b p -6 a b \,p^{2} x^{2}+10 \ln \left (b \,x^{2}+a \right ) a^{2} p^{2}-2 {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{2} a^{2}-4 \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) a^{2} p +6 a^{2} p^{2}}{8 b^{2}}\) | \(151\) |
risch | \(\text {Expression too large to display}\) | \(24297\) |
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Time = 0.36 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.02 \[ \int x^3 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {b^{2} p^{2} x^{4} + 2 \, b^{2} x^{4} \log \left (c\right )^{2} - 6 \, a b p^{2} x^{2} + 2 \, {\left (b^{2} p^{2} x^{4} - a^{2} p^{2}\right )} \log \left (b x^{2} + a\right )^{2} - 2 \, {\left (b^{2} p^{2} x^{4} - 2 \, a b p^{2} x^{2} - 3 \, a^{2} p^{2} - 2 \, {\left (b^{2} p x^{4} - a^{2} p\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b^{2} p x^{4} - 2 \, a b p x^{2}\right )} \log \left (c\right )}{8 \, b^{2}} \]
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Time = 1.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.96 \[ \int x^3 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx=\begin {cases} \frac {3 a^{2} p \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{4 b^{2}} - \frac {a^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{4 b^{2}} - \frac {3 a p^{2} x^{2}}{4 b} + \frac {a p x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{2 b} + \frac {p^{2} x^{4}}{8} - \frac {p x^{4} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{4} + \frac {x^{4} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{4} & \text {for}\: b \neq 0 \\\frac {x^{4} \log {\left (a^{p} c \right )}^{2}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.83 \[ \int x^3 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {1}{4} \, x^{4} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} - \frac {1}{4} \, b p {\left (\frac {2 \, a^{2} \log \left (b x^{2} + a\right )}{b^{3}} + \frac {b x^{4} - 2 \, a x^{2}}{b^{2}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) + \frac {{\left (b^{2} x^{4} - 6 \, a b x^{2} + 2 \, a^{2} \log \left (b x^{2} + a\right )^{2} + 6 \, a^{2} \log \left (b x^{2} + a\right )\right )} p^{2}}{8 \, b^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.49 \[ \int x^3 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {2 \, {\left (b x^{2} + a\right )}^{2} p^{2} \log \left (b x^{2} + a\right )^{2} - 2 \, {\left (b x^{2} + a\right )}^{2} p^{2} \log \left (b x^{2} + a\right ) + 4 \, {\left (b x^{2} + a\right )}^{2} p \log \left (b x^{2} + a\right ) \log \left (c\right ) + {\left (b x^{2} + a\right )}^{2} p^{2} - 2 \, {\left (b x^{2} + a\right )}^{2} p \log \left (c\right ) + 2 \, {\left (b x^{2} + a\right )}^{2} \log \left (c\right )^{2}}{8 \, b^{2}} - \frac {{\left (2 \, b x^{2} + {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right )^{2} - 2 \, {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + 2 \, a\right )} a p^{2} - 2 \, {\left (b x^{2} - {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + a\right )} a p \log \left (c\right ) + {\left (b x^{2} + a\right )} a \log \left (c\right )^{2}}{2 \, b^{2}} \]
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Time = 1.42 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.69 \[ \int x^3 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {p^2\,x^4}{8}-\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )\,\left (\frac {p\,x^4}{4}-\frac {a\,p\,x^2}{2\,b}\right )+{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2\,\left (\frac {x^4}{4}-\frac {a^2}{4\,b^2}\right )-\frac {3\,a\,p^2\,x^2}{4\,b}+\frac {3\,a^2\,p^2\,\ln \left (b\,x^2+a\right )}{4\,b^2} \]
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