Optimal. Leaf size=97 \[ -\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d}-\frac {p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {p \text {Li}_2\left (\frac {b x}{a}+1\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {36, 29, 31, 2416, 2394, 2315, 2393, 2391} \[ -\frac {p \text {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {p \text {PolyLog}\left (2,\frac {b x}{a}+1\right )}{d}-\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rubi steps
\begin {align*} \int \frac {\log \left (c (a+b x)^p\right )}{x (d+e x)} \, dx &=\int \left (\frac {\log \left (c (a+b x)^p\right )}{d x}-\frac {e \log \left (c (a+b x)^p\right )}{d (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx}{d}-\frac {e \int \frac {\log \left (c (a+b x)^p\right )}{d+e x} \, dx}{d}\\ &=\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d}-\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {(b p) \int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx}{d}+\frac {(b p) \int \frac {\log \left (\frac {b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{d}\\ &=\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d}-\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {p \text {Li}_2\left (1+\frac {b x}{a}\right )}{d}+\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d}-\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {p \text {Li}_2\left (1+\frac {b x}{a}\right )}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 98, normalized size = 1.01 \[ -\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d}-\frac {p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {p \text {Li}_2\left (\frac {a+b x}{a}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{e x^{2} + d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{{\left (e x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.27, size = 420, normalized size = 4.33 \[ -\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \ln \relax (x )}{2 d}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{2 d}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \relax (x )}{2 d}-\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 d}+\frac {i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \relax (x )}{2 d}-\frac {i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 d}-\frac {i \pi \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3} \ln \relax (x )}{2 d}+\frac {i \pi \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3} \ln \left (e x +d \right )}{2 d}-\frac {p \ln \relax (x ) \ln \left (\frac {b x +a}{a}\right )}{d}+\frac {p \ln \left (\frac {a e -b d +\left (e x +d \right ) b}{a e -b d}\right ) \ln \left (e x +d \right )}{d}-\frac {p \dilog \left (\frac {b x +a}{a}\right )}{d}+\frac {p \dilog \left (\frac {a e -b d +\left (e x +d \right ) b}{a e -b d}\right )}{d}+\frac {\ln \relax (c ) \ln \relax (x )}{d}-\frac {\ln \relax (c ) \ln \left (e x +d \right )}{d}+\frac {\ln \relax (x ) \ln \left (\left (b x +a \right )^{p}\right )}{d}-\frac {\ln \left (\left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.65, size = 123, normalized size = 1.27 \[ -b p {\left (\frac {\log \left (\frac {b x}{a} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {b x}{a}\right )}{b d} - \frac {\log \left (e x + d\right ) \log \left (-\frac {b e x + b d}{b d - a e} + 1\right ) + {\rm Li}_2\left (\frac {b e x + b d}{b d - a e}\right )}{b d}\right )} - {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \relax (x)}{d}\right )} \log \left ({\left (b x + a\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{x\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c \left (a + b x\right )^{p} \right )}}{x \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________