Integrand size = 43, antiderivative size = 269 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c i+d i x} \, dx=\frac {g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d i}-\frac {(b c-a d) g^3 (a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^2 i}+\frac {(b c-a d)^2 g^3 (a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^3 i}+\frac {(b c-a d)^3 g^3 \left (6 A+11 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{6 d^4 i}+\frac {B (b c-a d)^3 g^3 n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i} \]
1/3*g^3*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d/i-1/6*(-a*d+b*c)*g^3*( b*x+a)^2*(3*A+B*n+3*B*ln(e*((b*x+a)/(d*x+c))^n))/d^2/i+1/6*(-a*d+b*c)^2*g^ 3*(b*x+a)*(6*A+5*B*n+6*B*ln(e*((b*x+a)/(d*x+c))^n))/d^3/i+1/6*(-a*d+b*c)^3 *g^3*(6*A+11*B*n+6*B*ln(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b*c)/b/(d*x+c))/d ^4/i+B*(-a*d+b*c)^3*g^3*n*polylog(2,d*(b*x+a)/b/(d*x+c))/d^4/i
Time = 0.18 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.38 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c i+d i x} \, dx=\frac {g^3 \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^3 n \log (c+d x)+B (b c-a d) n \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 n (b d x+(-b c+a d) \log (c+d x))-6 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (i (c+d x))+3 B (b c-a d)^3 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (i (c+d x))\right ) \log (i (c+d x))+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{6 d^4 i} \]
(g^3*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[e*((a + b *x)/(c + d*x))^n]) + 2*d^3*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^ n]) - 6*B*(b*c - a*d)^3*n*Log[c + d*x] + B*(b*c - a*d)*n*(2*b*d*(b*c - a*d )*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + 3*B*(b*c - a*d)^2* n*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) - 6*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[i*(c + d*x)] + 3*B*(b*c - a*d)^3*n*((2*Log[(d*( a + b*x))/(-(b*c) + a*d)] - Log[i*(c + d*x)])*Log[i*(c + d*x)] + 2*PolyLog [2, (b*(c + d*x))/(b*c - a*d)])))/(6*d^4*i)
Time = 0.72 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {2961, 2784, 2784, 2784, 2754, 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.
| \(\displaystyle \int \frac {(a g+b g x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c i+d i x} \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{i}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\int \frac {(a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 d}\right )}{i}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}}{3 d}\right )}{i}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {6 A+11 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{2 d}}{3 d}\right )}{i}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {6 B n \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+11 B n\right )}{d}}{d}}{2 d}}{3 d}\right )}{i}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+11 B n\right )}{d}-\frac {6 B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}}{3 d}\right )}{i}\) |
((b*c - a*d)^3*g^3*(((a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/( 3*d*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^3) - (((a + b*x)^2*(3*A + B* n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d*(c + d*x)^2*(b - (d*(a + b*x ))/(c + d*x))^2) - (((a + b*x)*(6*A + 5*B*n + 6*B*Log[e*((a + b*x)/(c + d* x))^n]))/(d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) - (-(((6*A + 11*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (d*(a + b*x))/(b*(c + d*x))]) /d) - (6*B*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d)/d)/(2*d))/(3*d))) /i
3.2.35.3.1 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_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )/(e*(q + 1))), x] - Simp[f/(e*(q + 1)) Int[(f*x)^(m - 1)*(d + e*x)^(q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]
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[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*L og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
\[\int \frac {\left (b g x +a g \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{d i x +c i}d x\]
\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c i+d i x} \, dx=\int { \frac {{\left (b g x + a g\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{d i x + c i} \,d x } \]
integral((A*b^3*g^3*x^3 + 3*A*a*b^2*g^3*x^2 + 3*A*a^2*b*g^3*x + A*a^3*g^3 + (B*b^3*g^3*x^3 + 3*B*a*b^2*g^3*x^2 + 3*B*a^2*b*g^3*x + B*a^3*g^3)*log(e* ((b*x + a)/(d*x + c))^n))/(d*i*x + c*i), x)
\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c i+d i x} \, dx=\frac {g^{3} \left (\int \frac {A a^{3}}{c + d x}\, dx + \int \frac {A b^{3} x^{3}}{c + d x}\, dx + \int \frac {B a^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx + \int \frac {3 A a b^{2} x^{2}}{c + d x}\, dx + \int \frac {3 A a^{2} b x}{c + d x}\, dx + \int \frac {B b^{3} x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx + \int \frac {3 B a b^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx + \int \frac {3 B a^{2} b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx\right )}{i} \]
g**3*(Integral(A*a**3/(c + d*x), x) + Integral(A*b**3*x**3/(c + d*x), x) + Integral(B*a**3*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c + d*x), x) + I ntegral(3*A*a*b**2*x**2/(c + d*x), x) + Integral(3*A*a**2*b*x/(c + d*x), x ) + Integral(B*b**3*x**3*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c + d*x) , x) + Integral(3*B*a*b**2*x**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c + d*x), x) + Integral(3*B*a**2*b*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n )/(c + d*x), x))/i
Leaf count of result is larger than twice the leaf count of optimal. 1003 vs. \(2 (260) = 520\).
Time = 0.49 (sec) , antiderivative size = 1003, normalized size of antiderivative = 3.73 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c i+d i x} \, dx=3 \, A a^{2} b g^{3} {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} - \frac {1}{6} \, A b^{3} g^{3} {\left (\frac {6 \, c^{3} \log \left (d x + c\right )}{d^{4} i} - \frac {2 \, d^{2} x^{3} - 3 \, c d x^{2} + 6 \, c^{2} x}{d^{3} i}\right )} + \frac {3}{2} \, A a b^{2} g^{3} {\left (\frac {2 \, c^{2} \log \left (d x + c\right )}{d^{3} i} + \frac {d x^{2} - 2 \, c x}{d^{2} i}\right )} + \frac {A a^{3} g^{3} \log \left (d i x + c i\right )}{d i} - \frac {{\left (b^{3} c^{3} g^{3} n - 3 \, a b^{2} c^{2} d g^{3} n + 3 \, a^{2} b c d^{2} g^{3} n - a^{3} d^{3} g^{3} n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{4} i} + \frac {{\left (6 \, a^{3} d^{3} g^{3} \log \left (e\right ) - {\left (11 \, g^{3} n + 6 \, g^{3} \log \left (e\right )\right )} b^{3} c^{3} + 9 \, {\left (3 \, g^{3} n + 2 \, g^{3} \log \left (e\right )\right )} a b^{2} c^{2} d - 18 \, {\left (g^{3} n + g^{3} \log \left (e\right )\right )} a^{2} b c d^{2}\right )} B \log \left (d x + c\right )}{6 \, d^{4} i} + \frac {2 \, B b^{3} d^{3} g^{3} x^{3} \log \left (e\right ) - {\left ({\left (g^{3} n + 3 \, g^{3} \log \left (e\right )\right )} b^{3} c d^{2} - {\left (g^{3} n + 9 \, g^{3} \log \left (e\right )\right )} a b^{2} d^{3}\right )} B x^{2} + 6 \, {\left (b^{3} c^{3} g^{3} n - 3 \, a b^{2} c^{2} d g^{3} n + 3 \, a^{2} b c d^{2} g^{3} n - a^{3} d^{3} g^{3} n\right )} B \log \left (b x + a\right ) \log \left (d x + c\right ) - 3 \, {\left (b^{3} c^{3} g^{3} n - 3 \, a b^{2} c^{2} d g^{3} n + 3 \, a^{2} b c d^{2} g^{3} n - a^{3} d^{3} g^{3} n\right )} B \log \left (d x + c\right )^{2} + {\left ({\left (5 \, g^{3} n + 6 \, g^{3} \log \left (e\right )\right )} b^{3} c^{2} d - 6 \, {\left (2 \, g^{3} n + 3 \, g^{3} \log \left (e\right )\right )} a b^{2} c d^{2} + {\left (7 \, g^{3} n + 18 \, g^{3} \log \left (e\right )\right )} a^{2} b d^{3}\right )} B x + {\left (6 \, a b^{2} c^{2} d g^{3} n - 15 \, a^{2} b c d^{2} g^{3} n + 11 \, a^{3} d^{3} g^{3} n\right )} B \log \left (b x + a\right ) + {\left (2 \, B b^{3} d^{3} g^{3} x^{3} - 3 \, {\left (b^{3} c d^{2} g^{3} - 3 \, a b^{2} d^{3} g^{3}\right )} B x^{2} + 6 \, {\left (b^{3} c^{2} d g^{3} - 3 \, a b^{2} c d^{2} g^{3} + 3 \, a^{2} b d^{3} g^{3}\right )} B x - 6 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (2 \, B b^{3} d^{3} g^{3} x^{3} - 3 \, {\left (b^{3} c d^{2} g^{3} - 3 \, a b^{2} d^{3} g^{3}\right )} B x^{2} + 6 \, {\left (b^{3} c^{2} d g^{3} - 3 \, a b^{2} c d^{2} g^{3} + 3 \, a^{2} b d^{3} g^{3}\right )} B x - 6 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{6 \, d^{4} i} \]
3*A*a^2*b*g^3*(x/(d*i) - c*log(d*x + c)/(d^2*i)) - 1/6*A*b^3*g^3*(6*c^3*lo g(d*x + c)/(d^4*i) - (2*d^2*x^3 - 3*c*d*x^2 + 6*c^2*x)/(d^3*i)) + 3/2*A*a* b^2*g^3*(2*c^2*log(d*x + c)/(d^3*i) + (d*x^2 - 2*c*x)/(d^2*i)) + A*a^3*g^3 *log(d*i*x + c*i)/(d*i) - (b^3*c^3*g^3*n - 3*a*b^2*c^2*d*g^3*n + 3*a^2*b*c *d^2*g^3*n - a^3*d^3*g^3*n)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B/(d^4*i) + 1/6*(6*a^3*d^3*g^3*log (e) - (11*g^3*n + 6*g^3*log(e))*b^3*c^3 + 9*(3*g^3*n + 2*g^3*log(e))*a*b^2 *c^2*d - 18*(g^3*n + g^3*log(e))*a^2*b*c*d^2)*B*log(d*x + c)/(d^4*i) + 1/6 *(2*B*b^3*d^3*g^3*x^3*log(e) - ((g^3*n + 3*g^3*log(e))*b^3*c*d^2 - (g^3*n + 9*g^3*log(e))*a*b^2*d^3)*B*x^2 + 6*(b^3*c^3*g^3*n - 3*a*b^2*c^2*d*g^3*n + 3*a^2*b*c*d^2*g^3*n - a^3*d^3*g^3*n)*B*log(b*x + a)*log(d*x + c) - 3*(b^ 3*c^3*g^3*n - 3*a*b^2*c^2*d*g^3*n + 3*a^2*b*c*d^2*g^3*n - a^3*d^3*g^3*n)*B *log(d*x + c)^2 + ((5*g^3*n + 6*g^3*log(e))*b^3*c^2*d - 6*(2*g^3*n + 3*g^3 *log(e))*a*b^2*c*d^2 + (7*g^3*n + 18*g^3*log(e))*a^2*b*d^3)*B*x + (6*a*b^2 *c^2*d*g^3*n - 15*a^2*b*c*d^2*g^3*n + 11*a^3*d^3*g^3*n)*B*log(b*x + a) + ( 2*B*b^3*d^3*g^3*x^3 - 3*(b^3*c*d^2*g^3 - 3*a*b^2*d^3*g^3)*B*x^2 + 6*(b^3*c ^2*d*g^3 - 3*a*b^2*c*d^2*g^3 + 3*a^2*b*d^3*g^3)*B*x - 6*(b^3*c^3*g^3 - 3*a *b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*B*log(d*x + c))*log((b*x + a)^n) - (2*B*b^3*d^3*g^3*x^3 - 3*(b^3*c*d^2*g^3 - 3*a*b^2*d^3*g^3)*B*x^ 2 + 6*(b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g^3 + 3*a^2*b*d^3*g^3)*B*x - 6*(b^...
Leaf count of result is larger than twice the leaf count of optimal. 3832 vs. \(2 (260) = 520\).
Time = 189.49 (sec) , antiderivative size = 3832, normalized size of antiderivative = 14.25 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c i+d i x} \, dx=\text {Too large to display} \]
-1/120*(6*(B*b^9*c^6*g^3*n - 6*B*a*b^8*c^5*d*g^3*n - 5*(b*x + a)*B*b^8*c^6 *d*g^3*n/(d*x + c) + 15*B*a^2*b^7*c^4*d^2*g^3*n + 30*(b*x + a)*B*a*b^7*c^5 *d^2*g^3*n/(d*x + c) + 10*(b*x + a)^2*B*b^7*c^6*d^2*g^3*n/(d*x + c)^2 - 20 *B*a^3*b^6*c^3*d^3*g^3*n - 75*(b*x + a)*B*a^2*b^6*c^4*d^3*g^3*n/(d*x + c) - 60*(b*x + a)^2*B*a*b^6*c^5*d^3*g^3*n/(d*x + c)^2 - 10*(b*x + a)^3*B*b^6* c^6*d^3*g^3*n/(d*x + c)^3 + 15*B*a^4*b^5*c^2*d^4*g^3*n + 100*(b*x + a)*B*a ^3*b^5*c^3*d^4*g^3*n/(d*x + c) + 150*(b*x + a)^2*B*a^2*b^5*c^4*d^4*g^3*n/( d*x + c)^2 + 60*(b*x + a)^3*B*a*b^5*c^5*d^4*g^3*n/(d*x + c)^3 - 6*B*a^5*b^ 4*c*d^5*g^3*n - 75*(b*x + a)*B*a^4*b^4*c^2*d^5*g^3*n/(d*x + c) - 200*(b*x + a)^2*B*a^3*b^4*c^3*d^5*g^3*n/(d*x + c)^2 - 150*(b*x + a)^3*B*a^2*b^4*c^4 *d^5*g^3*n/(d*x + c)^3 + B*a^6*b^3*d^6*g^3*n + 30*(b*x + a)*B*a^5*b^3*c*d^ 6*g^3*n/(d*x + c) + 150*(b*x + a)^2*B*a^4*b^3*c^2*d^6*g^3*n/(d*x + c)^2 + 200*(b*x + a)^3*B*a^3*b^3*c^3*d^6*g^3*n/(d*x + c)^3 - 5*(b*x + a)*B*a^6*b^ 2*d^7*g^3*n/(d*x + c) - 60*(b*x + a)^2*B*a^5*b^2*c*d^7*g^3*n/(d*x + c)^2 - 150*(b*x + a)^3*B*a^4*b^2*c^2*d^7*g^3*n/(d*x + c)^3 + 10*(b*x + a)^2*B*a^ 6*b*d^8*g^3*n/(d*x + c)^2 + 60*(b*x + a)^3*B*a^5*b*c*d^8*g^3*n/(d*x + c)^3 - 10*(b*x + a)^3*B*a^6*d^9*g^3*n/(d*x + c)^3)*log((b*x + a)/(d*x + c))/(b ^5*d^4*i - 5*(b*x + a)*b^4*d^5*i/(d*x + c) + 10*(b*x + a)^2*b^3*d^6*i/(d*x + c)^2 - 10*(b*x + a)^3*b^2*d^7*i/(d*x + c)^3 + 5*(b*x + a)^4*b*d^8*i/(d* x + c)^4 - (b*x + a)^5*d^9*i/(d*x + c)^5) + (5*B*b^10*c^6*g^3*n - 30*B*...
Timed out. \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c i+d i x} \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^3\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{c\,i+d\,i\,x} \,d x \]