Integrand size = 27, antiderivative size = 63 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (-\frac {e (g+f x)}{d f-e g}\right )}{f}+\frac {b n \operatorname {PolyLog}\left (2,\frac {f (d+e x)}{d f-e g}\right )}{f} \]
Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (g+f x)}{-d f+e g}\right )}{f}+\frac {b n \operatorname {PolyLog}\left (2,\frac {f (d+e x)}{d f-e g}\right )}{f} \]
((a + b*Log[c*(d + e*x)^n])*Log[(e*(g + f*x))/(-(d*f) + e*g)])/f + (b*n*Po lyLog[2, (f*(d + e*x))/(d*f - e*g)])/f
Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2005, 2841, 2840, 2838}
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.
\(\displaystyle \int \frac {a+b \log \left (c (d+e x)^n\right )}{x \left (f+\frac {g}{x}\right )} \, dx\) |
\(\Big \downarrow \) 2005 |
\(\displaystyle \int \frac {a+b \log \left (c (d+e x)^n\right )}{f x+g}dx\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle \frac {\log \left (-\frac {e (f x+g)}{d f-e g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}-\frac {b e n \int \frac {\log \left (-\frac {e (g+f x)}{d f-e g}\right )}{d+e x}dx}{f}\) |
\(\Big \downarrow \) 2840 |
\(\displaystyle \frac {\log \left (-\frac {e (f x+g)}{d f-e g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}-\frac {b n \int \frac {\log \left (1-\frac {f (d+e x)}{d f-e g}\right )}{d+e x}d(d+e x)}{f}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\log \left (-\frac {e (f x+g)}{d f-e g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}+\frac {b n \operatorname {PolyLog}\left (2,\frac {f (d+e x)}{d f-e g}\right )}{f}\) |
((a + b*Log[c*(d + e*x)^n])*Log[-((e*(g + f*x))/(d*f - e*g))])/f + (b*n*Po lyLog[2, (f*(d + e*x))/(d*f - e*g)])/f
3.4.6.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ Symbol] :> Simp[1/g Subst[Int[(a + b*Log[1 + c*e*(x/g)])/x, x], x, f + g* x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c *(e*f - d*g), 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_ )), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x )^n])/g), x] - Simp[b*e*(n/g) Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.24 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.44
method | result | size |
risch | \(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (f x +g \right )}{f}-\frac {b n \operatorname {dilog}\left (\frac {\left (f x +g \right ) e +d f -e g}{d f -e g}\right )}{f}-\frac {b n \ln \left (f x +g \right ) \ln \left (\frac {\left (f x +g \right ) e +d f -e g}{d f -e g}\right )}{f}+\frac {\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \ln \left (f x +g \right )}{f}\) | \(217\) |
b*ln((e*x+d)^n)*ln(f*x+g)/f-b/f*n*dilog(((f*x+g)*e+d*f-e*g)/(d*f-e*g))-b/f *n*ln(f*x+g)*ln(((f*x+g)*e+d*f-e*g)/(d*f-e*g))+(-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)*ln(f*x+g)/f
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (f + \frac {g}{x}\right )} x} \,d x } \]
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{f x + g}\, dx \]
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (f + \frac {g}{x}\right )} x} \,d x } \]
\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (f + \frac {g}{x}\right )} x} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{x\,\left (f+\frac {g}{x}\right )} \,d x \]