Integrand size = 12, antiderivative size = 27 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} b n x^2+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2341} \[ \int x \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n x^2 \]
[In]
[Out]
Rule 2341
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4} b n x^2+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a x^2}{2}-\frac {1}{4} b n x^2+\frac {1}{2} b x^2 \log \left (c x^n\right ) \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {x^{2} b \ln \left (c \,x^{n}\right )}{2}-\frac {b n \,x^{2}}{4}+\frac {x^{2} a}{2}\) | \(27\) |
norman | \(\left (-\frac {b n}{4}+\frac {a}{2}\right ) x^{2}+\frac {b \,x^{2} \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )}{2}\) | \(28\) |
default | \(\frac {x^{2} a}{2}+\frac {b \,x^{2} \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )}{2}-\frac {b n \,x^{2}}{4}\) | \(29\) |
parts | \(\frac {x^{2} a}{2}+\frac {b \,x^{2} \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )}{2}-\frac {b n \,x^{2}}{4}\) | \(29\) |
risch | \(\frac {b \,x^{2} \ln \left (x^{n}\right )}{2}+\frac {x^{2} \left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )-b n +2 a \right )}{4}\) | \(112\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{2} \, b n x^{2} \log \left (x\right ) + \frac {1}{2} \, b x^{2} \log \left (c\right ) - \frac {1}{4} \, {\left (b n - 2 \, a\right )} x^{2} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a x^{2}}{2} - \frac {b n x^{2}}{4} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} \, b n x^{2} + \frac {1}{2} \, b x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a x^{2} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{2} \, b n x^{2} \log \left (x\right ) - \frac {1}{4} \, b n x^{2} + \frac {1}{2} \, b x^{2} \log \left (c\right ) + \frac {1}{2} \, a x^{2} \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \, dx=x^2\,\left (\frac {a}{2}-\frac {b\,n}{4}\right )+\frac {b\,x^2\,\ln \left (c\,x^n\right )}{2} \]
[In]
[Out]