Integrand size = 26, antiderivative size = 68 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {b p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h} \]
(a+b*ln(c*(d*(f*x+e)^p)^q))*ln(f*(h*x+g)/(-e*h+f*g))/h+b*p*q*polylog(2,-h* (f*x+e)/(-e*h+f*g))/h
Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.99 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {b p q \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )}{h} \]
((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/h + (b*p *q*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)])/h
Time = 0.40 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2895, 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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x}dx\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f p q \int \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x}dx}{h}\) |
\(\Big \downarrow \) 2840 |
\(\displaystyle \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b p q \int \frac {\log \left (\frac {h (e+f x)}{f g-e h}+1\right )}{e+f x}d(e+f x)}{h}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {b p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h}\) |
((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/h + (b*p *q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h
3.5.24.3.1 Defintions of rubi rules used
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]
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]]
\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{h x +g}d x\]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h x + g} \,d x } \]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{g + h x}\, dx \]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h x + g} \,d x } \]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h x + g} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{g+h\,x} \,d x \]