Integrand size = 16, antiderivative size = 93 \[ \int x \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=-3 p^3 x^2+\frac {3 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac {3 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}+\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b} \]
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Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2504, 2436, 2333, 2332} \[ \int x \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {3 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}+\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b}-\frac {3 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}-3 p^3 x^2 \]
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Rule 2332
Rule 2333
Rule 2436
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \log ^3\left (c (a+b x)^p\right ) \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int \log ^3\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b} \\ & = \frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b}-\frac {(3 p) \text {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b} \\ & = -\frac {3 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}+\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b}+\frac {\left (3 p^2\right ) \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{b} \\ & = -3 p^3 x^2+\frac {3 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac {3 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 b}+\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94 \[ \int x \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {-6 b p^3 x^2+6 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-3 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 b} \]
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Time = 1.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.62
method | result | size |
parallelrisch | \(\frac {x^{2} {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3} a b p -3 x^{2} {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{2} a b \,p^{2}+6 x^{2} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) a b \,p^{3}-6 x^{2} a b \,p^{4}+{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3} a^{2} p -3 {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{2} a^{2} p^{2}+6 \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) a^{2} p^{3}}{2 a b p}\) | \(151\) |
risch | \(\text {Expression too large to display}\) | \(3925\) |
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Time = 0.35 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.89 \[ \int x \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {6 \, b p^{3} x^{2} - 6 \, b p^{2} x^{2} \log \left (c\right ) + 3 \, b p x^{2} \log \left (c\right )^{2} - b x^{2} \log \left (c\right )^{3} - {\left (b p^{3} x^{2} + a p^{3}\right )} \log \left (b x^{2} + a\right )^{3} + 3 \, {\left (b p^{3} x^{2} + a p^{3} - {\left (b p^{2} x^{2} + a p^{2}\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right )^{2} - 3 \, {\left (2 \, b p^{3} x^{2} + 2 \, a p^{3} + {\left (b p x^{2} + a p\right )} \log \left (c\right )^{2} - 2 \, {\left (b p^{2} x^{2} + a p^{2}\right )} \log \left (c\right )\right )} \log \left (b x^{2} + a\right )}{2 \, b} \]
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Time = 0.79 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.54 \[ \int x \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\begin {cases} \frac {3 a p^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{b} - \frac {3 a p \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{2 b} + \frac {a \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{2 b} - 3 p^{3} x^{2} + 3 p^{2} x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )} - \frac {3 p x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{2} + \frac {x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{2} & \text {for}\: b \neq 0 \\\frac {x^{2} \log {\left (a^{p} c \right )}^{3}}{2} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.76 \[ \int x \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {3}{2} \, b p {\left (\frac {x^{2}}{b} - \frac {a \log \left (b x^{2} + a\right )}{b^{2}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2} + \frac {1}{2} \, x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} + \frac {1}{2} \, b p {\left (\frac {{\left (a \log \left (b x^{2} + a\right )^{3} - 6 \, b x^{2} + 3 \, a \log \left (b x^{2} + a\right )^{2} + 6 \, a \log \left (b x^{2} + a\right )\right )} p^{2}}{b^{2}} + \frac {3 \, {\left (2 \, b x^{2} - a \log \left (b x^{2} + a\right )^{2} - 2 \, a \log \left (b x^{2} + a\right )\right )} p \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{b^{2}}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.82 \[ \int x \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {{\left ({\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right )^{3} - 6 \, b x^{2} - 3 \, {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right )^{2} + 6 \, {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) - 6 \, a\right )} p^{3} + 3 \, {\left (2 \, b x^{2} + {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right )^{2} - 2 \, {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + 2 \, a\right )} p^{2} \log \left (c\right ) - 3 \, {\left (b x^{2} - {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + a\right )} p \log \left (c\right )^{2} + {\left (b x^{2} + a\right )} \log \left (c\right )^{3}}{2 \, b} \]
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Time = 1.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.11 \[ \int x \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx={\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3\,\left (\frac {a}{2\,b}+\frac {x^2}{2}\right )-{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2\,\left (\frac {3\,p\,x^2}{2}+\frac {3\,a\,p}{2\,b}\right )-3\,p^3\,x^2+3\,p^2\,x^2\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )+\frac {3\,a\,p^3\,\ln \left (b\,x^2+a\right )}{b} \]
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