Integrand size = 18, antiderivative size = 70 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-b d^2 n x-\frac {1}{2} b d e n x^2-\frac {1}{9} b e^2 n x^3-\frac {b d^3 n \log (x)}{3 e}+\frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {32, 2350, 12, 45} \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {b d^3 n \log (x)}{3 e}-b d^2 n x-\frac {1}{2} b d e n x^2-\frac {1}{9} b e^2 n x^3 \]
[In]
[Out]
Rule 12
Rule 32
Rule 45
Rule 2350
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-(b n) \int \frac {(d+e x)^3}{3 e x} \, dx \\ & = \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {(b n) \int \frac {(d+e x)^3}{x} \, dx}{3 e} \\ & = \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {(b n) \int \left (3 d^2 e+\frac {d^3}{x}+3 d e^2 x+e^3 x^2\right ) \, dx}{3 e} \\ & = -b d^2 n x-\frac {1}{2} b d e n x^2-\frac {1}{9} b e^2 n x^3-\frac {b d^3 n \log (x)}{3 e}+\frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 e} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.10 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{18} x \left (6 a \left (3 d^2+3 d e x+e^2 x^2\right )-b n \left (18 d^2+9 d e x+2 e^2 x^2\right )+6 b \left (3 d^2+3 d e x+e^2 x^2\right ) \log \left (c x^n\right )\right ) \]
[In]
[Out]
Time = 0.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.30
method | result | size |
parallelrisch | \(\frac {b \ln \left (c \,x^{n}\right ) e^{2} x^{3}}{3}-\frac {b \,e^{2} n \,x^{3}}{9}+\frac {a \,e^{2} x^{3}}{3}+b \ln \left (c \,x^{n}\right ) d e \,x^{2}-\frac {b d e n \,x^{2}}{2}+a d e \,x^{2}+x b \ln \left (c \,x^{n}\right ) d^{2}-b \,d^{2} n x +a \,d^{2} x\) | \(91\) |
risch | \(\frac {\left (e x +d \right )^{3} b \ln \left (x^{n}\right )}{3 e}+\frac {i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x}{2}-\frac {i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x}{2}-\frac {i e \pi b d \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}-\frac {i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) x}{2}-\frac {i e^{2} \pi b \,x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{6}+\frac {i e \pi b d \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i e^{2} \pi b \,x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{6}-\frac {i e \pi b d \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i e^{2} \pi b \,x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x}{2}-\frac {i e^{2} \pi b \,x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{6}+\frac {i e \pi b d \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {e^{2} \ln \left (c \right ) b \,x^{3}}{3}-\frac {b \,e^{2} n \,x^{3}}{9}+e \ln \left (c \right ) b d \,x^{2}+\frac {a \,e^{2} x^{3}}{3}-\frac {b d e n \,x^{2}}{2}+\ln \left (c \right ) b \,d^{2} x -\frac {b \,d^{3} n \ln \left (x \right )}{3 e}+a d e \,x^{2}-b \,d^{2} n x +a \,d^{2} x\) | \(414\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.57 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{9} \, {\left (b e^{2} n - 3 \, a e^{2}\right )} x^{3} - \frac {1}{2} \, {\left (b d e n - 2 \, a d e\right )} x^{2} - {\left (b d^{2} n - a d^{2}\right )} x + \frac {1}{3} \, {\left (b e^{2} x^{3} + 3 \, b d e x^{2} + 3 \, b d^{2} x\right )} \log \left (c\right ) + \frac {1}{3} \, {\left (b e^{2} n x^{3} + 3 \, b d e n x^{2} + 3 \, b d^{2} n x\right )} \log \left (x\right ) \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.46 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} - b d^{2} n x + b d^{2} x \log {\left (c x^{n} \right )} - \frac {b d e n x^{2}}{2} + b d e x^{2} \log {\left (c x^{n} \right )} - \frac {b e^{2} n x^{3}}{9} + \frac {b e^{2} x^{3} \log {\left (c x^{n} \right )}}{3} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.29 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{9} \, b e^{2} n x^{3} + \frac {1}{3} \, b e^{2} x^{3} \log \left (c x^{n}\right ) - \frac {1}{2} \, b d e n x^{2} + \frac {1}{3} \, a e^{2} x^{3} + b d e x^{2} \log \left (c x^{n}\right ) - b d^{2} n x + a d e x^{2} + b d^{2} x \log \left (c x^{n}\right ) + a d^{2} x \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.56 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{3} \, b e^{2} n x^{3} \log \left (x\right ) - \frac {1}{9} \, b e^{2} n x^{3} + \frac {1}{3} \, b e^{2} x^{3} \log \left (c\right ) + b d e n x^{2} \log \left (x\right ) - \frac {1}{2} \, b d e n x^{2} + \frac {1}{3} \, a e^{2} x^{3} + b d e x^{2} \log \left (c\right ) + b d^{2} n x \log \left (x\right ) - b d^{2} n x + a d e x^{2} + b d^{2} x \log \left (c\right ) + a d^{2} x \]
[In]
[Out]
Time = 0.36 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.04 \[ \int (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (b\,d^2\,x+b\,d\,e\,x^2+\frac {b\,e^2\,x^3}{3}\right )+\frac {e^2\,x^3\,\left (3\,a-b\,n\right )}{9}+d^2\,x\,\left (a-b\,n\right )+\frac {d\,e\,x^2\,\left (2\,a-b\,n\right )}{2} \]
[In]
[Out]