Integrand size = 16, antiderivative size = 57 \[ \int x \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {b n x}{2 c}-\frac {n x^2}{2}-\frac {b^2 n \log (b+c x)}{2 c^2}+\frac {1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2605, 78} \[ \int x \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=-\frac {b^2 n \log (b+c x)}{2 c^2}+\frac {1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )+\frac {b n x}{2 c}-\frac {n x^2}{2} \]
[In]
[Out]
Rule 78
Rule 2605
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )-\frac {1}{2} n \int \frac {x (b+2 c x)}{b+c x} \, dx \\ & = \frac {1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )-\frac {1}{2} n \int \left (-\frac {b}{c}+2 x+\frac {b^2}{c (b+c x)}\right ) \, dx \\ & = \frac {b n x}{2 c}-\frac {n x^2}{2}-\frac {b^2 n \log (b+c x)}{2 c^2}+\frac {1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int x \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=-\frac {1}{2} n \left (-\frac {b x}{c}+x^2+\frac {b^2 \log (b+c x)}{c^2}\right )+\frac {1}{2} x^2 \log \left (d (x (b+c x))^n\right ) \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93
method | result | size |
parts | \(\frac {x^{2} \ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )}{2}-\frac {n \left (-\frac {-c \,x^{2}+b x}{c}+\frac {b^{2} \ln \left (x c +b \right )}{c^{2}}\right )}{2}\) | \(53\) |
parallelrisch | \(\frac {x^{2} \ln \left (d \left (x \left (x c +b \right )\right )^{n}\right ) c^{2} n -x^{2} c^{2} n^{2}+\ln \left (x \right ) b^{2} n^{2}+x b c \,n^{2}-\ln \left (d \left (x \left (x c +b \right )\right )^{n}\right ) b^{2} n -b^{2} n^{2}}{2 c^{2} n}\) | \(83\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.04 \[ \int x \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {c^{2} n x^{2} \log \left (c x^{2} + b x\right ) - c^{2} n x^{2} + c^{2} x^{2} \log \left (d\right ) + b c n x - b^{2} n \log \left (c x + b\right )}{2 \, c^{2}} \]
[In]
[Out]
Time = 0.72 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.23 \[ \int x \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\begin {cases} - \frac {b^{2} n \log {\left (b + c x \right )}}{2 c^{2}} + \frac {b n x}{2 c} - \frac {n x^{2}}{2} + \frac {x^{2} \log {\left (d \left (b x + c x^{2}\right )^{n} \right )}}{2} & \text {for}\: c \neq 0 \\- \frac {n x^{2}}{4} + \frac {x^{2} \log {\left (d \left (b x\right )^{n} \right )}}{2} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int x \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {1}{2} \, x^{2} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) - \frac {1}{2} \, n {\left (\frac {b^{2} \log \left (c x + b\right )}{c^{2}} + \frac {c x^{2} - b x}{c}\right )} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int x \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {1}{2} \, n x^{2} \log \left (c x^{2} + b x\right ) - \frac {1}{2} \, {\left (n - \log \left (d\right )\right )} x^{2} + \frac {b n x}{2 \, c} - \frac {b^{2} n \log \left (c x + b\right )}{2 \, c^{2}} \]
[In]
[Out]
Time = 1.52 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int x \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {x^2\,\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )}{2}-\frac {n\,x^2}{2}+\frac {b\,n\,x}{2\,c}-\frac {b^2\,n\,\ln \left (b+c\,x\right )}{2\,c^2} \]
[In]
[Out]