Integrand size = 16, antiderivative size = 59 \[ \int x^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {a p x^2}{4 b}-\frac {p x^4}{8}-\frac {a^2 p \log \left (a+b x^2\right )}{4 b^2}+\frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2442, 45} \[ \int x^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {a^2 p \log \left (a+b x^2\right )}{4 b^2}+\frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right )+\frac {a p x^2}{4 b}-\frac {p x^4}{8} \]
[In]
[Out]
Rule 45
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x \log \left (c (a+b x)^p\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right )-\frac {1}{4} (b p) \text {Subst}\left (\int \frac {x^2}{a+b x} \, dx,x,x^2\right ) \\ & = \frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right )-\frac {1}{4} (b p) \text {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {a p x^2}{4 b}-\frac {p x^4}{8}-\frac {a^2 p \log \left (a+b x^2\right )}{4 b^2}+\frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int x^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {a p x^2}{4 b}-\frac {p x^4}{8}-\frac {a^2 p \log \left (a+b x^2\right )}{4 b^2}+\frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right ) \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97
method | result | size |
parts | \(\frac {x^{4} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{4}-\frac {p b \left (-\frac {-\frac {1}{2} b \,x^{4}+x^{2} a}{2 b^{2}}+\frac {a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}}\right )}{2}\) | \(57\) |
parallelrisch | \(-\frac {-2 x^{4} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) b^{2}+b^{2} p \,x^{4}-2 a b p \,x^{2}+2 \ln \left (b \,x^{2}+a \right ) a^{2} p +2 a^{2} p}{8 b^{2}}\) | \(63\) |
risch | \(\text {Expression too large to display}\) | \(1190\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int x^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=-\frac {b^{2} p x^{4} - 2 \, b^{2} x^{4} \log \left (c\right ) - 2 \, a b p x^{2} - 2 \, {\left (b^{2} p x^{4} - a^{2} p\right )} \log \left (b x^{2} + a\right )}{8 \, b^{2}} \]
[In]
[Out]
Time = 0.81 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.10 \[ \int x^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\begin {cases} - \frac {a^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{4 b^{2}} + \frac {a p x^{2}}{4 b} - \frac {p x^{4}}{8} + \frac {x^{4} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{4} & \text {for}\: b \neq 0 \\\frac {x^{4} \log {\left (a^{p} c \right )}}{4} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.93 \[ \int x^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {1}{4} \, x^{4} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) - \frac {1}{8} \, b p {\left (\frac {2 \, a^{2} \log \left (b x^{2} + a\right )}{b^{3}} + \frac {b x^{4} - 2 \, a x^{2}}{b^{2}}\right )} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.64 \[ \int x^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {2 \, {\left (b x^{2} + a\right )}^{2} p \log \left (b x^{2} + a\right ) - {\left (b x^{2} + a\right )}^{2} p + 2 \, {\left (b x^{2} + a\right )}^{2} \log \left (c\right )}{8 \, b^{2}} + \frac {{\left (b x^{2} - {\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + a\right )} a p - {\left (b x^{2} + a\right )} a \log \left (c\right )}{2 \, b^{2}} \]
[In]
[Out]
Time = 1.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int x^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {x^4\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{4}-\frac {p\,x^4}{8}-\frac {a^2\,p\,\ln \left (b\,x^2+a\right )}{4\,b^2}+\frac {a\,p\,x^2}{4\,b} \]
[In]
[Out]