Integrand size = 21, antiderivative size = 250 \[ \int \frac {x^3 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=-\frac {d^2 p x}{e^3}-\frac {a d p x}{2 b e^2}-\frac {a^2 p x}{3 b^2 e}+\frac {d p x^2}{4 e^2}+\frac {a p x^2}{6 b e}-\frac {p x^3}{9 e}+\frac {a^2 d p \log (a+b x)}{2 b^2 e^2}+\frac {a^3 p \log (a+b x)}{3 b^3 e}-\frac {d x^2 \log \left (c (a+b x)^p\right )}{2 e^2}+\frac {x^3 \log \left (c (a+b x)^p\right )}{3 e}+\frac {d^2 (a+b x) \log \left (c (a+b x)^p\right )}{b e^3}-\frac {d^3 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^4}-\frac {d^3 p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^4} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {45, 2463, 2436, 2332, 2442, 2441, 2440, 2438} \[ \int \frac {x^3 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\frac {a^3 p \log (a+b x)}{3 b^3 e}+\frac {a^2 d p \log (a+b x)}{2 b^2 e^2}-\frac {a^2 p x}{3 b^2 e}-\frac {d^3 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^4}+\frac {d^2 (a+b x) \log \left (c (a+b x)^p\right )}{b e^3}-\frac {d x^2 \log \left (c (a+b x)^p\right )}{2 e^2}+\frac {x^3 \log \left (c (a+b x)^p\right )}{3 e}-\frac {d^3 p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^4}-\frac {a d p x}{2 b e^2}+\frac {a p x^2}{6 b e}-\frac {d^2 p x}{e^3}+\frac {d p x^2}{4 e^2}-\frac {p x^3}{9 e} \]
[In]
[Out]
Rule 45
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 \log \left (c (a+b x)^p\right )}{e^3}-\frac {d x \log \left (c (a+b x)^p\right )}{e^2}+\frac {x^2 \log \left (c (a+b x)^p\right )}{e}-\frac {d^3 \log \left (c (a+b x)^p\right )}{e^3 (d+e x)}\right ) \, dx \\ & = \frac {d^2 \int \log \left (c (a+b x)^p\right ) \, dx}{e^3}-\frac {d^3 \int \frac {\log \left (c (a+b x)^p\right )}{d+e x} \, dx}{e^3}-\frac {d \int x \log \left (c (a+b x)^p\right ) \, dx}{e^2}+\frac {\int x^2 \log \left (c (a+b x)^p\right ) \, dx}{e} \\ & = -\frac {d x^2 \log \left (c (a+b x)^p\right )}{2 e^2}+\frac {x^3 \log \left (c (a+b x)^p\right )}{3 e}-\frac {d^3 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^4}+\frac {d^2 \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x\right )}{b e^3}+\frac {\left (b d^3 p\right ) \int \frac {\log \left (\frac {b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{e^4}+\frac {(b d p) \int \frac {x^2}{a+b x} \, dx}{2 e^2}-\frac {(b p) \int \frac {x^3}{a+b x} \, dx}{3 e} \\ & = -\frac {d^2 p x}{e^3}-\frac {d x^2 \log \left (c (a+b x)^p\right )}{2 e^2}+\frac {x^3 \log \left (c (a+b x)^p\right )}{3 e}+\frac {d^2 (a+b x) \log \left (c (a+b x)^p\right )}{b e^3}-\frac {d^3 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^4}+\frac {\left (d^3 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{e^4}+\frac {(b d p) \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{2 e^2}-\frac {(b p) \int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx}{3 e} \\ & = -\frac {d^2 p x}{e^3}-\frac {a d p x}{2 b e^2}-\frac {a^2 p x}{3 b^2 e}+\frac {d p x^2}{4 e^2}+\frac {a p x^2}{6 b e}-\frac {p x^3}{9 e}+\frac {a^2 d p \log (a+b x)}{2 b^2 e^2}+\frac {a^3 p \log (a+b x)}{3 b^3 e}-\frac {d x^2 \log \left (c (a+b x)^p\right )}{2 e^2}+\frac {x^3 \log \left (c (a+b x)^p\right )}{3 e}+\frac {d^2 (a+b x) \log \left (c (a+b x)^p\right )}{b e^3}-\frac {d^3 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^4}-\frac {d^3 p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{e^4} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\frac {6 a^2 e^2 (3 b d+2 a e) p \log (a+b x)+b \left (-e p x \left (12 a^2 e^2-6 a b e (-3 d+e x)+b^2 \left (36 d^2-9 d e x+4 e^2 x^2\right )\right )+6 b \log \left (c (a+b x)^p\right ) \left (6 a d^2 e+b e x \left (6 d^2-3 d e x+2 e^2 x^2\right )-6 b d^3 \log \left (\frac {b (d+e x)}{b d-a e}\right )\right )\right )-36 b^3 d^3 p \operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )}{36 b^3 e^4} \]
[In]
[Out]
Time = 1.84 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.19
method | result | size |
parts | \(\frac {x^{3} \ln \left (c \left (b x +a \right )^{p}\right )}{3 e}-\frac {d \,x^{2} \ln \left (c \left (b x +a \right )^{p}\right )}{2 e^{2}}+\frac {\ln \left (c \left (b x +a \right )^{p}\right ) d^{2} x}{e^{3}}-\frac {\ln \left (c \left (b x +a \right )^{p}\right ) d^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {p b \left (-\frac {-\frac {\frac {2 \left (e x +d \right )^{3} b^{2}}{3}-\left (e x +d \right )^{2} a b e -\frac {7 \left (e x +d \right )^{2} b^{2} d}{2}+2 \left (e x +d \right ) a^{2} e^{2}+5 \left (e x +d \right ) a b d e +11 \left (e x +d \right ) b^{2} d^{2}}{b^{3}}+\frac {a e \left (2 a^{2} e^{2}+3 a d e b +6 b^{2} d^{2}\right ) \ln \left (\left (e x +d \right ) b +a e -b d \right )}{b^{4}}}{6 e^{3}}-\frac {d^{3} \left (\frac {\operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{b}+\frac {\ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{b}\right )}{e^{3}}\right )}{e}\) | \(297\) |
risch | \(\frac {\ln \left (\left (b x +a \right )^{p}\right ) x^{3}}{3 e}-\frac {\ln \left (\left (b x +a \right )^{p}\right ) d \,x^{2}}{2 e^{2}}+\frac {\ln \left (\left (b x +a \right )^{p}\right ) x \,d^{2}}{e^{3}}-\frac {\ln \left (\left (b x +a \right )^{p}\right ) d^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {p \,x^{3}}{9 e}+\frac {d p \,x^{2}}{4 e^{2}}-\frac {d^{2} p x}{e^{3}}-\frac {49 p \,d^{3}}{36 e^{4}}+\frac {a p \,x^{2}}{6 b e}-\frac {a d p x}{2 b \,e^{2}}-\frac {2 p a \,d^{2}}{3 b \,e^{3}}-\frac {a^{2} p x}{3 b^{2} e}-\frac {p \,a^{2} d}{3 b^{2} e^{2}}+\frac {p \,a^{3} \ln \left (\left (e x +d \right ) b +a e -b d \right )}{3 b^{3} e}+\frac {p \,a^{2} \ln \left (\left (e x +d \right ) b +a e -b d \right ) d}{2 b^{2} e^{2}}+\frac {p a \ln \left (\left (e x +d \right ) b +a e -b d \right ) d^{2}}{b \,e^{3}}+\frac {p \,d^{3} \operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e^{4}}+\frac {p \,d^{3} \ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e^{4}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\frac {1}{3} e^{2} x^{3}-\frac {1}{2} d e \,x^{2}+d^{2} x}{e^{3}}-\frac {d^{3} \ln \left (e x +d \right )}{e^{4}}\right )\) | \(482\) |
[In]
[Out]
\[ \int \frac {x^3 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d} \,d x } \]
[In]
[Out]
\[ \int \frac {x^3 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int \frac {x^{3} \log {\left (c \left (a + b x\right )^{p} \right )}}{d + e x}\, dx \]
[In]
[Out]
\[ \int \frac {x^3 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d} \,d x } \]
[In]
[Out]
\[ \int \frac {x^3 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^3 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int \frac {x^3\,\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{d+e\,x} \,d x \]
[In]
[Out]