Integrand size = 33, antiderivative size = 165 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h i-g j}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (i+j x)}{f i-e j}\right )}{h i-g j}+\frac {b p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {b p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j} \]
(a+b*ln(c*(d*(f*x+e)^p)^q))*ln(f*(h*x+g)/(-e*h+f*g))/(-g*j+h*i)-(a+b*ln(c* (d*(f*x+e)^p)^q))*ln(f*(j*x+i)/(-e*j+f*i))/(-g*j+h*i)+b*p*q*polylog(2,-h*( f*x+e)/(-e*h+f*g))/(-g*j+h*i)-b*p*q*polylog(2,-j*(f*x+e)/(-e*j+f*i))/(-g*j +h*i)
Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.71 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (\log \left (\frac {f (g+h x)}{f g-e h}\right )-\log \left (\frac {f (i+j x)}{f i-e j}\right )\right )+b p q \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-b p q \operatorname {PolyLog}\left (2,\frac {j (e+f x)}{-f i+e j}\right )}{h i-g j} \]
((a + b*Log[c*(d*(e + f*x)^p)^q])*(Log[(f*(g + h*x))/(f*g - e*h)] - Log[(f *(i + j*x))/(f*i - e*j)]) + b*p*q*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)] - b*p*q*PolyLog[2, (j*(e + f*x))/(-(f*i) + e*j)])/(h*i - g*j)
Time = 0.67 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2895, 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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)}dx\) |
\(\Big \downarrow \) 2865 |
\(\displaystyle \int \left (\frac {h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(g+h x) (h i-g j)}-\frac {j \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(i+j x) (h i-g j)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac {\log \left (\frac {f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac {b p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac {b p q \operatorname {PolyLog}\left (2,-\frac {j (e+f x)}{f i-e j}\right )}{h i-g j}\) |
((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/(h*i - g *j) - ((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(i + j*x))/(f*i - e*j)])/(h *i - g*j) + (b*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j) - (b*p*q*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)
3.6.27.3.1 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]
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right ) \left (j x +i \right )}d x\]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}} \,d x } \]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\left (g + h x\right ) \left (i + j x\right )}\, dx \]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}} \,d x } \]
a*(log(h*x + g)/(h*i - g*j) - log(j*x + i)/(h*i - g*j)) + b*integrate((q*l og(d) + log(((f*x + e)^p)^q) + log(c))/(h*j*x^2 + g*i + (h*i + g*j)*x), x)
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{{\left (h x + g\right )} {\left (j x + i\right )}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x) (i+j x)} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\left (g+h\,x\right )\,\left (i+j\,x\right )} \,d x \]