Integrand size = 33, antiderivative size = 302 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx=\frac {(a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b g-a h) (g+h x)}+\frac {3 B (b c-a d) n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \log \left (1-\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac {6 B^2 (b c-a d) n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}-\frac {6 B^3 (b c-a d) n^3 \operatorname {PolyLog}\left (3,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)} \]
(b*x+a)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(-a*h+b*g)/(h*x+g)+3*B*(-a*d+b *c)*n*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2*ln(1-(-c*h+d*g)*(b*x+a)/(-a*h+b* g)/(d*x+c))/(-a*h+b*g)/(-c*h+d*g)+6*B^2*(-a*d+b*c)*n^2*(A+B*ln(e*(b*x+a)^n /((d*x+c)^n)))*polylog(2,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/(-a*h+b*g) /(-c*h+d*g)-6*B^3*(-a*d+b*c)*n^3*polylog(3,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/( d*x+c))/(-a*h+b*g)/(-c*h+d*g)
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx=\int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx \]
Time = 0.79 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2973, 2953, 2755, 2754, 2821, 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.
| \(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(g+h x)^2} \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(g+h x)^2}dx\) |
| \(\Big \downarrow \) 2953 |
| \(\displaystyle (b c-a d) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{\left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle (b c-a d) \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{(c+d x) (b g-a h) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )}-\frac {3 B n \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{b g-a h}\right )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle (b c-a d) \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{(c+d x) (b g-a h) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )}-\frac {3 B n \left (\frac {2 B n \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d g-c h}-\frac {\log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d g-c h}\right )}{b g-a h}\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle (b c-a d) \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{(c+d x) (b g-a h) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )}-\frac {3 B n \left (\frac {2 B n \left (B n \int \frac {(c+d x) \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}-\operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )\right )}{d g-c h}-\frac {\log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d g-c h}\right )}{b g-a h}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle (b c-a d) \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{(c+d x) (b g-a h) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )}-\frac {3 B n \left (\frac {2 B n \left (B n \operatorname {PolyLog}\left (3,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )-\operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )\right )}{d g-c h}-\frac {\log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d g-c h}\right )}{b g-a h}\right )\) |
(b*c - a*d)*(((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/((b*g - a*h)*(c + d*x)*(b*g - a*h - ((d*g - c*h)*(a + b*x))/(c + d*x))) - (3*B*n*( -(((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - ((d*g - c*h)*(a + b*x) )/((b*g - a*h)*(c + d*x))])/(d*g - c*h)) + (2*B*n*(-((A + B*Log[e*((a + b* x)/(c + d*x))^n])*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x ))]) + B*n*PolyLog[3, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))]))/( d*g - c*h)))/(b*g - a*h))
3.4.13.3.1 Defintions of rubi rules used
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[(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[((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[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]
\[\int \frac {{\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{3}}{\left (h x +g \right )^{2}}d x\]
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (h x + g\right )}^{2}} \,d x } \]
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^2*x^2 + 2*g*h*x + g^2), x)
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (h x + g\right )}^{2}} \,d x } \]
B^3*log((d*x + c)^n)^3/(h^2*x + g*h) + 3*(b*e*n*log(b*x + a)/(b*g*h - a*h^ 2) - d*e*n*log(d*x + c)/(d*g*h - c*h^2) - (b*c*e*n - a*d*e*n)*log(h*x + g) /((d*g*h - c*h^2)*a - (d*g^2 - c*g*h)*b))*A^2*B/e - 3*A^2*B*log((b*x + a)^ n*e/(d*x + c)^n)/(h^2*x + g*h) - A^3/(h^2*x + g*h) + integrate((B^3*c*h*lo g(e)^3 + 3*A*B^2*c*h*log(e)^2 + (B^3*d*h*x + B^3*c*h)*log((b*x + a)^n)^3 + 3*(B^3*c*h*log(e) + A*B^2*c*h + (B^3*d*h*log(e) + A*B^2*d*h)*x)*log((b*x + a)^n)^2 + 3*(A*B^2*c*h - (d*g*n - c*h*log(e))*B^3 - ((h*n - h*log(e))*B^ 3*d - A*B^2*d*h)*x + (B^3*d*h*x + B^3*c*h)*log((b*x + a)^n))*log((d*x + c) ^n)^2 + (B^3*d*h*log(e)^3 + 3*A*B^2*d*h*log(e)^2)*x + 3*(B^3*c*h*log(e)^2 + 2*A*B^2*c*h*log(e) + (B^3*d*h*log(e)^2 + 2*A*B^2*d*h*log(e))*x)*log((b*x + a)^n) - 3*(B^3*c*h*log(e)^2 + 2*A*B^2*c*h*log(e) + (B^3*d*h*x + B^3*c*h )*log((b*x + a)^n)^2 + (B^3*d*h*log(e)^2 + 2*A*B^2*d*h*log(e))*x + 2*(B^3* c*h*log(e) + A*B^2*c*h + (B^3*d*h*log(e) + A*B^2*d*h)*x)*log((b*x + a)^n)) *log((d*x + c)^n))/(d*h^3*x^3 + c*g^2*h + (2*d*g*h^2 + c*h^3)*x^2 + (d*g^2 *h + 2*c*g*h^2)*x), x)
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (h x + g\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx=\int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3}{{\left (g+h\,x\right )}^2} \,d x \]