Integrand size = 28, antiderivative size = 376 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^3} \, dx=-\frac {3 b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 (f g-e h)^2 (g+h x)}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}+\frac {3 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h)^2}-\frac {3 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (1+\frac {f g-e h}{h (e+f x)}\right )}{2 h (f g-e h)^2}+\frac {3 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2}+\frac {3 b^3 f^2 p^3 q^3 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}+\frac {3 b^3 f^2 p^3 q^3 \operatorname {PolyLog}\left (3,-\frac {f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2} \]
-3/2*b*f*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/(-e*h+f*g)^2/(h*x+g)-1/ 2*(a+b*ln(c*(d*(f*x+e)^p)^q))^3/h/(h*x+g)^2+3*b^2*f^2*p^2*q^2*(a+b*ln(c*(d *(f*x+e)^p)^q))*ln(f*(h*x+g)/(-e*h+f*g))/h/(-e*h+f*g)^2-3/2*b*f^2*p*q*(a+b *ln(c*(d*(f*x+e)^p)^q))^2*ln(1+(-e*h+f*g)/h/(f*x+e))/h/(-e*h+f*g)^2+3*b^2* f^2*p^2*q^2*(a+b*ln(c*(d*(f*x+e)^p)^q))*polylog(2,(e*h-f*g)/h/(f*x+e))/h/( -e*h+f*g)^2+3*b^3*f^2*p^3*q^3*polylog(2,-h*(f*x+e)/(-e*h+f*g))/h/(-e*h+f*g )^2+3*b^3*f^2*p^3*q^3*polylog(3,(e*h-f*g)/h/(f*x+e))/h/(-e*h+f*g)^2
Time = 0.55 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^3} \, dx=-\frac {-3 b f (f g-e h) p q (g+h x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+3 b (f g-e h)^2 p q \log (e+f x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-3 b f^2 p q (g+h x)^2 \log (e+f x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+(f g-e h)^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3+3 b f^2 p q (g+h x)^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log (g+h x)+3 b^2 p^2 q^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (h (e+f x) (e h-f (2 g+h x)) \log ^2(e+f x)-2 f^2 (g+h x)^2 \log \left (\frac {f (g+h x)}{f g-e h}\right )+2 f (g+h x) \log (e+f x) \left (h (e+f x)+f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )+2 f^2 (g+h x)^2 \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )\right )+b^3 p^3 q^3 \left (h (e+f x) (e h-f (2 g+h x)) \log ^3(e+f x)+3 f (g+h x) \log ^2(e+f x) \left (h (e+f x)+f (g+h x) \log \left (\frac {f (g+h x)}{f g-e h}\right )\right )-6 f^2 (g+h x)^2 \log (e+f x) \left (\log \left (\frac {f (g+h x)}{f g-e h}\right )-\operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )\right )-6 f^2 (g+h x)^2 \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-6 f^2 (g+h x)^2 \operatorname {PolyLog}\left (3,\frac {h (e+f x)}{-f g+e h}\right )\right )}{2 h (f g-e h)^2 (g+h x)^2} \]
-1/2*(-3*b*f*(f*g - e*h)*p*q*(g + h*x)*(a - b*p*q*Log[e + f*x] + b*Log[c*( d*(e + f*x)^p)^q])^2 + 3*b*(f*g - e*h)^2*p*q*Log[e + f*x]*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])^2 - 3*b*f^2*p*q*(g + h*x)^2*Log[e + f*x]*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])^2 + (f*g - e*h) ^2*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])^3 + 3*b*f^2*p*q*( g + h*x)^2*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[g + h*x] + 3*b^2*p^2*q^2*(a - b*p*q*Log[e + f*x] + b*Log[c*(d*(e + f*x)^p)^q] )*(h*(e + f*x)*(e*h - f*(2*g + h*x))*Log[e + f*x]^2 - 2*f^2*(g + h*x)^2*Lo g[(f*(g + h*x))/(f*g - e*h)] + 2*f*(g + h*x)*Log[e + f*x]*(h*(e + f*x) + f *(g + h*x)*Log[(f*(g + h*x))/(f*g - e*h)]) + 2*f^2*(g + h*x)^2*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)]) + b^3*p^3*q^3*(h*(e + f*x)*(e*h - f*(2*g + h*x))*Log[e + f*x]^3 + 3*f*(g + h*x)*Log[e + f*x]^2*(h*(e + f*x) + f*(g + h*x)*Log[(f*(g + h*x))/(f*g - e*h)]) - 6*f^2*(g + h*x)^2*Log[e + f*x]*(Log [(f*(g + h*x))/(f*g - e*h)] - PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)]) - 6*f^2*(g + h*x)^2*PolyLog[2, (h*(e + f*x))/(-(f*g) + e*h)] - 6*f^2*(g + h* x)^2*PolyLog[3, (h*(e + f*x))/(-(f*g) + e*h)]))/(h*(f*g - e*h)^2*(g + h*x) ^2)
Time = 2.20 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2895, 2845, 2858, 27, 2789, 2755, 2754, 2779, 2821, 2838, 7143}
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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^3} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^3}dx\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {3 b f p q \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(e+f x) (g+h x)^2}dx}{2 h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {3 b p q \int \frac {f^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(e+f x) \left (f \left (g-\frac {e h}{f}\right )+h (e+f x)\right )^2}d(e+f x)}{2 h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 b f^2 p q \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(e+f x) (f g-e h+h (e+f x))^2}d(e+f x)}{2 h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {3 b f^2 p q \left (\frac {\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(e+f x) (f g-e h+h (e+f x))}d(e+f x)}{f g-e h}-\frac {h \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(f g-e h+h (e+f x))^2}d(e+f x)}{f g-e h}\right )}{2 h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle \frac {3 b f^2 p q \left (\frac {\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(e+f x) (f g-e h+h (e+f x))}d(e+f x)}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {2 b p q \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{f g-e h+h (e+f x)}d(e+f x)}{f g-e h}\right )}{f g-e h}\right )}{2 h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {3 b f^2 p q \left (\frac {\int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(e+f x) (f g-e h+h (e+f x))}d(e+f x)}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {2 b p q \left (\frac {\log \left (\frac {h (e+f x)}{f g-e h}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {b p q \int \frac {\log \left (\frac {h (e+f x)}{f g-e h}+1\right )}{e+f x}d(e+f x)}{h}\right )}{f g-e h}\right )}{f g-e h}\right )}{2 h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {3 b f^2 p q \left (\frac {\frac {2 b p q \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac {f g-e h}{h (e+f x)}+1\right )}{e+f x}d(e+f x)}{f g-e h}-\frac {\log \left (\frac {f g-e h}{h (e+f x)}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{f g-e h}}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {2 b p q \left (\frac {\log \left (\frac {h (e+f x)}{f g-e h}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {b p q \int \frac {\log \left (\frac {h (e+f x)}{f g-e h}+1\right )}{e+f x}d(e+f x)}{h}\right )}{f g-e h}\right )}{f g-e h}\right )}{2 h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {3 b f^2 p q \left (\frac {\frac {2 b p q \left (\operatorname {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )-b p q \int \frac {\operatorname {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right )}{e+f x}d(e+f x)\right )}{f g-e h}-\frac {\log \left (\frac {f g-e h}{h (e+f x)}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{f g-e h}}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {2 b p q \left (\frac {\log \left (\frac {h (e+f x)}{f g-e h}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}-\frac {b p q \int \frac {\log \left (\frac {h (e+f x)}{f g-e h}+1\right )}{e+f x}d(e+f x)}{h}\right )}{f g-e h}\right )}{f g-e h}\right )}{2 h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3 b f^2 p q \left (\frac {\frac {2 b p q \left (\operatorname {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )-b p q \int \frac {\operatorname {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right )}{e+f x}d(e+f x)\right )}{f g-e h}-\frac {\log \left (\frac {f g-e h}{h (e+f x)}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{f g-e h}}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {2 b p q \left (\frac {\log \left (\frac {h (e+f x)}{f g-e h}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}+\frac {b p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h}\right )}{f g-e h}\right )}{f g-e h}\right )}{2 h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3 b f^2 p q \left (\frac {\frac {2 b p q \left (\operatorname {PolyLog}\left (2,-\frac {f g-e h}{h (e+f x)}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )+b p q \operatorname {PolyLog}\left (3,-\frac {f g-e h}{h (e+f x)}\right )\right )}{f g-e h}-\frac {\log \left (\frac {f g-e h}{h (e+f x)}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{f g-e h}}{f g-e h}-\frac {h \left (\frac {(e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(f g-e h) (h (e+f x)-e h+f g)}-\frac {2 b p q \left (\frac {\log \left (\frac {h (e+f x)}{f g-e h}+1\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}+\frac {b p q \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h}\right )}{f g-e h}\right )}{f g-e h}\right )}{2 h}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2}\) |
-1/2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3/(h*(g + h*x)^2) + (3*b*f^2*p*q*(-( (h*(((e + f*x)*(a + b*Log[c*d^q*(e + f*x)^(p*q)])^2)/((f*g - e*h)*(f*g - e *h + h*(e + f*x))) - (2*b*p*q*(((a + b*Log[c*d^q*(e + f*x)^(p*q)])*Log[1 + (h*(e + f*x))/(f*g - e*h)])/h + (b*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h))/(f*g - e*h)))/(f*g - e*h)) + (-(((a + b*Log[c*d^q*(e + f*x)^(p *q)])^2*Log[1 + (f*g - e*h)/(h*(e + f*x))])/(f*g - e*h)) + (2*b*p*q*((a + b*Log[c*d^q*(e + f*x)^(p*q)])*PolyLog[2, -((f*g - e*h)/(h*(e + f*x)))] + b *p*q*PolyLog[3, -((f*g - e*h)/(h*(e + f*x)))]))/(f*g - e*h))/(f*g - e*h))) /(2*h)
3.5.40.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
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_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
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[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{3}}{\left (h x +g \right )^{3}}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^3} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{{\left (h x + g\right )}^{3}} \,d x } \]
integral((b^3*log(((f*x + e)^p*d)^q*c)^3 + 3*a*b^2*log(((f*x + e)^p*d)^q*c )^2 + 3*a^2*b*log(((f*x + e)^p*d)^q*c) + a^3)/(h^3*x^3 + 3*g*h^2*x^2 + 3*g ^2*h*x + g^3), x)
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3}}{\left (g + h x\right )^{3}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^3} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{{\left (h x + g\right )}^{3}} \,d x } \]
3/2*a^2*b*f*p*q*(f*log(f*x + e)/(f^2*g^2*h - 2*e*f*g*h^2 + e^2*h^3) - f*lo g(h*x + g)/(f^2*g^2*h - 2*e*f*g*h^2 + e^2*h^3) + 1/(f*g^2*h - e*g*h^2 + (f *g*h^2 - e*h^3)*x)) - 1/2*b^3*log(((f*x + e)^p)^q)^3/(h^3*x^2 + 2*g*h^2*x + g^2*h) - 3/2*a^2*b*log(((f*x + e)^p*d)^q*c)/(h^3*x^2 + 2*g*h^2*x + g^2*h ) - 1/2*a^3/(h^3*x^2 + 2*g*h^2*x + g^2*h) + integrate(1/2*(6*(e*h*q^2*log( d)^2 + 2*e*h*q*log(c)*log(d) + e*h*log(c)^2)*a*b^2 + 2*(e*h*q^3*log(d)^3 + 3*e*h*q^2*log(c)*log(d)^2 + 3*e*h*q*log(c)^2*log(d) + e*h*log(c)^3)*b^3 + 3*(2*a*b^2*e*h + (f*g*p*q + 2*e*h*q*log(d) + 2*e*h*log(c))*b^3 + (2*a*b^2 *f*h + (f*h*p*q + 2*f*h*q*log(d) + 2*f*h*log(c))*b^3)*x)*log(((f*x + e)^p) ^q)^2 + 2*(3*(f*h*q^2*log(d)^2 + 2*f*h*q*log(c)*log(d) + f*h*log(c)^2)*a*b ^2 + (f*h*q^3*log(d)^3 + 3*f*h*q^2*log(c)*log(d)^2 + 3*f*h*q*log(c)^2*log( d) + f*h*log(c)^3)*b^3)*x + 6*(2*(e*h*q*log(d) + e*h*log(c))*a*b^2 + (e*h* q^2*log(d)^2 + 2*e*h*q*log(c)*log(d) + e*h*log(c)^2)*b^3 + (2*(f*h*q*log(d ) + f*h*log(c))*a*b^2 + (f*h*q^2*log(d)^2 + 2*f*h*q*log(c)*log(d) + f*h*lo g(c)^2)*b^3)*x)*log(((f*x + e)^p)^q))/(f*h^4*x^4 + e*g^3*h + (3*f*g*h^3 + e*h^4)*x^3 + 3*(f*g^2*h^2 + e*g*h^3)*x^2 + (f*g^3*h + 3*e*g^2*h^2)*x), x)
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^3} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{{\left (h x + g\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3}{{\left (g+h\,x\right )}^3} \,d x \]