Integrand size = 33, antiderivative size = 425 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx=-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{h}+\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \log \left (1-\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {3 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{h}+\frac {3 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {6 B^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{h}-\frac {6 B^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (3,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{h}-\frac {6 B^3 n^3 \operatorname {PolyLog}\left (4,\frac {d (a+b x)}{b (c+d x)}\right )}{h}+\frac {6 B^3 n^3 \operatorname {PolyLog}\left (4,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{h} \]
-ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/h+(A+B*ln(e* (b*x+a)^n/((d*x+c)^n)))^3*ln(1-(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/h-3* B*n*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2*polylog(2,d*(b*x+a)/b/(d*x+c))/h+3 *B*n*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2*polylog(2,(-c*h+d*g)*(b*x+a)/(-a* h+b*g)/(d*x+c))/h+6*B^2*n^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))*polylog(3,d* (b*x+a)/b/(d*x+c))/h-6*B^2*n^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))*polylog(3 ,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/h-6*B^3*n^3*polylog(4,d*(b*x+a)/b/ (d*x+c))/h+6*B^3*n^3*polylog(4,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/h
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx=\int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx \]
Time = 0.94 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.20, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2973, 2953, 2804, 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.
\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{g+h x} \, dx\) |
\(\Big \downarrow \) 2973 |
\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{g+h x}dx\) |
\(\Big \downarrow \) 2953 |
\(\displaystyle (b c-a d) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{\left (b-\frac {d (a+b x)}{c+d x}\right ) \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}\) |
\(\Big \downarrow \) 2804 |
\(\displaystyle (b c-a d) \int \left (\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(b c-a d) h \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(c h-d g) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(b c-a d) h \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle (b c-a d) \left (-\frac {6 B^2 n^2 \operatorname {PolyLog}\left (3,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{h (b c-a d)}+\frac {6 B^2 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{h (b c-a d)}+\frac {3 B n \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{h (b c-a d)}+\frac {\log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{h (b c-a d)}-\frac {3 B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{h (b c-a d)}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{h (b c-a d)}+\frac {6 B^3 n^3 \operatorname {PolyLog}\left (4,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{h (b c-a d)}-\frac {6 B^3 n^3 \operatorname {PolyLog}\left (4,\frac {d (a+b x)}{b (c+d x)}\right )}{h (b c-a d)}\right )\) |
(b*c - a*d)*(-(((A + B*Log[e*((a + b*x)/(c + d*x))^n])^3*Log[1 - (d*(a + b *x))/(b*(c + d*x))])/((b*c - a*d)*h)) + ((A + B*Log[e*((a + b*x)/(c + d*x) )^n])^3*Log[1 - ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*c - a*d)*h) - (3*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/((b*c - a*d)*h) + (3*B*n*(A + B*Log[e*((a + b*x)/ (c + d*x))^n])^2*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x) )])/((b*c - a*d)*h) + (6*B^2*n^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Po lyLog[3, (d*(a + b*x))/(b*(c + d*x))])/((b*c - a*d)*h) - (6*B^2*n^2*(A + B *Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[3, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*c - a*d)*h) - (6*B^3*n^3*PolyLog[4, (d*(a + b*x))/ (b*(c + d*x))])/((b*c - a*d)*h) + (6*B^3*n^3*PolyLog[4, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*c - a*d)*h))
3.4.12.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / ; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d) Sub st[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2 )), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n} , x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] && !Intege rQ[n]
\[\int \frac {{\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{3}}{h x +g}d x\]
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{h x + g} \,d x } \]
integral((B^3*log((b*x + a)^n*e/(d*x + c)^n)^3 + 3*A*B^2*log((b*x + a)^n*e /(d*x + c)^n)^2 + 3*A^2*B*log((b*x + a)^n*e/(d*x + c)^n) + A^3)/(h*x + g), x)
Exception generated. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{h x + g} \,d x } \]
A^3*log(h*x + g)/h - integrate(-(B^3*log((b*x + a)^n)^3 - B^3*log((d*x + c )^n)^3 + B^3*log(e)^3 + 3*A*B^2*log(e)^2 + 3*A^2*B*log(e) + 3*(B^3*log(e) + A*B^2)*log((b*x + a)^n)^2 + 3*(B^3*log((b*x + a)^n) + B^3*log(e) + A*B^2 )*log((d*x + c)^n)^2 + 3*(B^3*log(e)^2 + 2*A*B^2*log(e) + A^2*B)*log((b*x + a)^n) - 3*(B^3*log((b*x + a)^n)^2 + B^3*log(e)^2 + 2*A*B^2*log(e) + A^2* B + 2*(B^3*log(e) + A*B^2)*log((b*x + a)^n))*log((d*x + c)^n))/(h*x + g), x)
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{h x + g} \,d x } \]
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx=\int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3}{g+h\,x} \,d x \]