Integrand size = 29, antiderivative size = 155 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g h-f i}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (h+i x)}{e h-d i}\right )}{g h-f i}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i} \]
(a+b*ln(c*(e*x+d)^n))*ln(e*(g*x+f)/(-d*g+e*f))/(-f*i+g*h)-(a+b*ln(c*(e*x+d )^n))*ln(e*(i*x+h)/(-d*i+e*h))/(-f*i+g*h)+b*n*polylog(2,-g*(e*x+d)/(-d*g+e *f))/(-f*i+g*h)-b*n*polylog(2,-i*(e*x+d)/(-d*i+e*h))/(-f*i+g*h)
Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right )-\log \left (\frac {e (h+i x)}{e h-d i}\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-b n \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{-e h+d i}\right )}{g h-f i} \]
((a + b*Log[c*(d + e*x)^n])*(Log[(e*(f + g*x))/(e*f - d*g)] - Log[(e*(h + i*x))/(e*h - d*i)]) + b*n*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - b*n*P olyLog[2, (i*(d + e*x))/(-(e*h) + d*i)])/(g*h - f*i)
Time = 0.40 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2865, 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.
\(\displaystyle \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx\) |
\(\Big \downarrow \) 2865 |
\(\displaystyle \int \left (\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x) (g h-f i)}-\frac {i \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x) (g h-f i)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}-\frac {\log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}\) |
((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i) - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i) + (b*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i) - (b*n*PolyLog[ 2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)
3.3.21.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ RFx, x] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.12 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.44
method | result | size |
risch | \(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (i x +h \right )}{f i -g h}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{f i -g h}-\frac {b n \operatorname {dilog}\left (\frac {\left (i x +h \right ) e +d i -e h}{d i -e h}\right )}{f i -g h}-\frac {b n \ln \left (i x +h \right ) \ln \left (\frac {\left (i x +h \right ) e +d i -e h}{d i -e h}\right )}{f i -g h}+\frac {b n \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f i -g h}+\frac {b n \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f i -g h}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\ln \left (i x +h \right )}{f i -g h}-\frac {\ln \left (g x +f \right )}{f i -g h}\right )\) | \(378\) |
b*ln((e*x+d)^n)/(f*i-g*h)*ln(i*x+h)-b*ln((e*x+d)^n)/(f*i-g*h)*ln(g*x+f)-b* n/(f*i-g*h)*dilog(((i*x+h)*e+d*i-e*h)/(d*i-e*h))-b*n/(f*i-g*h)*ln(i*x+h)*l n(((i*x+h)*e+d*i-e*h)/(d*i-e*h))+b*n/(f*i-g*h)*dilog(((g*x+f)*e+d*g-e*f)/( d*g-e*f))+b*n/(f*i-g*h)*ln(g*x+f)*ln(((g*x+f)*e+d*g-e*f)/(d*g-e*f))+(-1/2* I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/2*I*b*Pi*csgn(I*c )*csgn(I*c*(e*x+d)^n)^2+1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2 -1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3+b*ln(c)+a)*(1/(f*i-g*h)*ln(i*x+h)-1/(f*i -g*h)*ln(g*x+f))
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} {\left (i x + h\right )}} \,d x } \]
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx=\int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{\left (f + g x\right ) \left (h + i x\right )}\, dx \]
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} {\left (i x + h\right )}} \,d x } \]
a*(log(g*x + f)/(g*h - f*i) - log(i*x + h)/(g*h - f*i)) + b*integrate((log ((e*x + d)^n) + log(c))/(g*i*x^2 + f*h + (g*h + f*i)*x), x)
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} {\left (i x + h\right )}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x) (h+i x)} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\left (f+g\,x\right )\,\left (h+i\,x\right )} \,d x \]