Integrand size = 33, antiderivative size = 393 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx=\frac {B (b c-a d) h n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b g-a h)^2 (d g-c h) (g+h x)}+\frac {b^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 h (b g-a h)^2}-\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 h (g+h x)^2}+\frac {B^2 (b c-a d)^2 h n^2 \log \left (\frac {g+h x}{c+d x}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {B (b c-a d) (2 b d g-b c h-a d h) n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2} \]
B*(-a*d+b*c)*h*n*(b*x+a)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*h+b*g)^2/(- c*h+d*g)/(h*x+g)+1/2*b^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/h/(-a*h+b*g)^ 2-1/2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/h/(h*x+g)^2+B^2*(-a*d+b*c)^2*h*n ^2*ln((h*x+g)/(d*x+c))/(-a*h+b*g)^2/(-c*h+d*g)^2+B*(-a*d+b*c)*(-a*d*h-b*c* h+2*b*d*g)*n*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))*ln(1-(-c*h+d*g)*(b*x+a)/(-a *h+b*g)/(d*x+c))/(-a*h+b*g)^2/(-c*h+d*g)^2+B^2*(-a*d+b*c)*(-a*d*h-b*c*h+2* b*d*g)*n^2*polylog(2,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/(-a*h+b*g)^2/( -c*h+d*g)^2
Leaf count is larger than twice the leaf count of optimal. \(13182\) vs. \(2(393)=786\).
Time = 5.23 (sec) , antiderivative size = 13182, normalized size of antiderivative = 33.54 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx=\text {Result too large to show} \]
Time = 1.04 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.32, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2973, 2953, 2798, 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.
| \(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(g+h x)^3} \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(g+h x)^3}dx\) |
| \(\Big \downarrow \) 2953 |
| \(\displaystyle (b c-a d) \int \frac {\left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{\left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2798 |
| \(\displaystyle (b c-a d) \left (\frac {B n \int \frac {(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{h (b c-a d)}-\frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 h (b c-a d) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^2}\right )\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle (b c-a d) \left (\frac {B n \int \left (\frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{(b g-a h)^2 (a+b x)}+\frac {(b c-a d) h (-2 b d g+b c h+a d h) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b g-a h)^2 (d g-c h) \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )}+\frac {(b c-a d)^2 h^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b g-a h) (d g-c h) \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^2}\right )d\frac {a+b x}{c+d x}}{h (b c-a d)}-\frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 h (b c-a d) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (b c-a d) \left (\frac {B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 B n (b g-a h)^2}+\frac {h^2 (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(c+d x) (b g-a h)^2 (d g-c h) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )}+\frac {h (b c-a d) (-a d h-b c h+2 b d g) \log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {B h^2 n (b c-a d)^2 \log \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {B h n (b c-a d) (-a d h-b c h+2 b d g) \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}\right )}{h (b c-a d)}-\frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 h (b c-a d) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^2}\right )\) |
(b*c - a*d)*(-1/2*((b - (d*(a + b*x))/(c + d*x))^2*(A + B*Log[e*((a + b*x) /(c + d*x))^n])^2)/((b*c - a*d)*h*(b*g - a*h - ((d*g - c*h)*(a + b*x))/(c + d*x))^2) + (B*n*(((b*c - a*d)^2*h^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*g - a*h)^2*(d*g - c*h)*(c + d*x)*(b*g - a*h - ((d*g - c* h)*(a + b*x))/(c + d*x))) + (b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2) /(2*B*(b*g - a*h)^2*n) + (B*(b*c - a*d)^2*h^2*n*Log[b*g - a*h - ((d*g - c* h)*(a + b*x))/(c + d*x)])/((b*g - a*h)^2*(d*g - c*h)^2) + ((b*c - a*d)*h*( 2*b*d*g - b*c*h - a*d*h)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (( d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h) ^2) + (B*(b*c - a*d)*h*(2*b*d*g - b*c*h - a*d*h)*n*PolyLog[2, ((d*g - c*h) *(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2)))/((b* c - a*d)*h))
3.4.8.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_)*(( f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/((q + 1)*(e*f - d*g))), x] - Simp[b*n*(p/((q + 1) *(e*f - d*g))) Int[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[e*f - d*g, 0] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
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]
\[\int \frac {{\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{2}}{\left (h x +g \right )^{3}}d x\]
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (h x + g\right )}^{3}} \,d x } \]
integral((B^2*log((b*x + a)^n*e/(d*x + c)^n)^2 + 2*A*B*log((b*x + a)^n*e/( d*x + c)^n) + A^2)/(h^3*x^3 + 3*g*h^2*x^2 + 3*g^2*h*x + g^3), x)
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (h x + g\right )}^{3}} \,d x } \]
-1/2*B^2*(log((d*x + c)^n)^2/(h^3*x^2 + 2*g*h^2*x + g^2*h) + 2*integrate(- (d*h*x*log(e)^2 + c*h*log(e)^2 + (d*h*x + c*h)*log((b*x + a)^n)^2 + 2*(d*h *x*log(e) + c*h*log(e))*log((b*x + a)^n) + (d*g*n + (h*n - 2*h*log(e))*d*x - 2*c*h*log(e) - 2*(d*h*x + c*h)*log((b*x + a)^n))*log((d*x + c)^n))/(d*h ^4*x^4 + c*g^3*h + (3*d*g*h^3 + c*h^4)*x^3 + 3*(d*g^2*h^2 + c*g*h^3)*x^2 + (d*g^3*h + 3*c*g^2*h^2)*x), x)) + (b^2*e*n*log(b*x + a)/(b^2*g^2*h - 2*a* b*g*h^2 + a^2*h^3) - d^2*e*n*log(d*x + c)/(d^2*g^2*h - 2*c*d*g*h^2 + c^2*h ^3) - (2*a*b*d^2*e*g*n - a^2*d^2*e*h*n - (2*c*d*e*g*n - c^2*e*h*n)*b^2)*lo g(h*x + g)/((d^2*g^2*h^2 - 2*c*d*g*h^3 + c^2*h^4)*a^2 - 2*(d^2*g^3*h - 2*c *d*g^2*h^2 + c^2*g*h^3)*a*b + (d^2*g^4 - 2*c*d*g^3*h + c^2*g^2*h^2)*b^2) + (b*c*e*n - a*d*e*n)/((d*g^2*h - c*g*h^2)*a - (d*g^3 - c*g^2*h)*b + ((d*g* h^2 - c*h^3)*a - (d*g^2*h - c*g*h^2)*b)*x))*A*B/e - A*B*log((b*x + a)^n*e/ (d*x + c)^n)/(h^3*x^2 + 2*g*h^2*x + g^2*h) - 1/2*A^2/(h^3*x^2 + 2*g*h^2*x + g^2*h)
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (h x + g\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx=\int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^2}{{\left (g+h\,x\right )}^3} \,d x \]