Integrand size = 19, antiderivative size = 91 \[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=-\frac {p x}{e}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}-\frac {d p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^2} \]
-p*x/e+(b*x+a)*ln(c*(b*x+a)^p)/b/e-d*ln(c*(b*x+a)^p)*ln(b*(e*x+d)/(-a*e+b* d))/e^2-d*p*polylog(2,-e*(b*x+a)/(-a*e+b*d))/e^2
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.87 \[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\frac {-b e p x+\log \left (c (a+b x)^p\right ) \left (a e+b e x-b d \log \left (\frac {b (d+e x)}{b d-a e}\right )\right )-b d p \operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )}{b e^2} \]
(-(b*e*p*x) + Log[c*(a + b*x)^p]*(a*e + b*e*x - b*d*Log[(b*(d + e*x))/(b*d - a*e)]) - b*d*p*PolyLog[2, (e*(a + b*x))/(-(b*d) + a*e)])/(b*e^2)
Time = 0.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2863, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx\) |
\(\Big \downarrow \) 2863 |
\(\displaystyle \int \left (\frac {\log \left (c (a+b x)^p\right )}{e}-\frac {d \log \left (c (a+b x)^p\right )}{e (d+e x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {d p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^2}-\frac {p x}{e}\) |
-((p*x)/e) + ((a + b*x)*Log[c*(a + b*x)^p])/(b*e) - (d*Log[c*(a + b*x)^p]* Log[(b*(d + e*x))/(b*d - a*e)])/e^2 - (d*p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/e^2
3.3.21.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Time = 1.71 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.71
method | result | size |
parts | \(\frac {x \ln \left (c \left (b x +a \right )^{p}\right )}{e}-\frac {\ln \left (c \left (b x +a \right )^{p}\right ) d \ln \left (e x +d \right )}{e^{2}}-\frac {p b \left (\frac {e x +d}{e b}-\frac {a \ln \left (\left (e x +d \right ) b +a e -b d \right )}{b^{2}}-\frac {d \left (\frac {\operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{b}+\frac {\ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{b}\right )}{e}\right )}{e}\) | \(156\) |
risch | \(\frac {\ln \left (\left (b x +a \right )^{p}\right ) x}{e}-\frac {\ln \left (\left (b x +a \right )^{p}\right ) d \ln \left (e x +d \right )}{e^{2}}-\frac {p x}{e}-\frac {p d}{e^{2}}+\frac {p a \ln \left (\left (e x +d \right ) b +a e -b d \right )}{b e}+\frac {p d \operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e^{2}}+\frac {p d \ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e^{2}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {x}{e}-\frac {d \ln \left (e x +d \right )}{e^{2}}\right )\) | \(270\) |
x*ln(c*(b*x+a)^p)/e-ln(c*(b*x+a)^p)*d/e^2*ln(e*x+d)-p*b/e*(1/e*(e*x+d)/b-a /b^2*ln((e*x+d)*b+a*e-b*d)-1/e*d*(dilog(((e*x+d)*b+a*e-b*d)/(a*e-b*d))/b+l n(e*x+d)*ln(((e*x+d)*b+a*e-b*d)/(a*e-b*d))/b))
\[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int { \frac {x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d} \,d x } \]
\[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int \frac {x \log {\left (c \left (a + b x\right )^{p} \right )}}{d + e x}\, dx \]
\[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int { \frac {x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d} \,d x } \]
\[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int { \frac {x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d} \,d x } \]
Timed out. \[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int \frac {x\,\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{d+e\,x} \,d x \]