Integrand size = 40, antiderivative size = 252 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d i}-\frac {(b c-a d) g^3 (a+b x)^2 \left (3 A+B+3 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^2 i}+\frac {(b c-a d)^2 g^3 (a+b x) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^3 i}+\frac {(b c-a d)^3 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 A+11 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4 i}+\frac {B (b c-a d)^3 g^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i} \]
1/3*g^3*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/d/i-1/6*(-a*d+b*c)*g^3*(b*x+ a)^2*(3*A+B+3*B*ln(e*(b*x+a)/(d*x+c)))/d^2/i+1/6*(-a*d+b*c)^2*g^3*(b*x+a)* (6*A+5*B+6*B*ln(e*(b*x+a)/(d*x+c)))/d^3/i+1/6*(-a*d+b*c)^3*g^3*ln((-a*d+b* c)/b/(d*x+c))*(6*A+11*B+6*B*ln(e*(b*x+a)/(d*x+c)))/d^4/i+B*(-a*d+b*c)^3*g^ 3*polylog(2,d*(b*x+a)/b/(d*x+c))/d^4/i
Time = 0.18 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.40 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g^3 \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)+B (b c-a d) \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 (b d x+(-b c+a d) \log (c+d x))-6 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (i (c+d x))+3 B (b c-a d)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (i (c+d x))\right ) \log (i (c+d x))+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{6 d^4 i} \]
(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)] + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x ))/(c + d*x)]) + 2*d^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*B*(b*c - a*d)^3*Log[c + d*x] + B*(b*c - a*d)*(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*(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)])*Log[i*(c + d*x)] + 3*B*(b*c - a*d)^3*((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 = 0.71 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2962, 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.
| \(\displaystyle \int \frac {(a g+b g x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c i+d i x} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \int \frac {(a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{i}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\int \frac {(a+b x)^2 \left (3 A+B+3 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 d}\right )}{i}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}}{3 d}\right )}{i}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {6 A+11 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{2 d}}{3 d}\right )}{i}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {6 B \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+11 B\right )}{d}}{d}}{2 d}}{3 d}\right )}{i}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {g^3 (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+11 B\right )}{d}-\frac {6 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}}{3 d}\right )}{i}\) |
((b*c - a*d)^3*g^3*(((a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3* d*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^3) - (((a + b*x)^2*(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d*(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^2) - (((a + b*x)*(6*A + 5*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/( d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) - (-(((6*A + 11*B + 6*B*Log[(e* (a + b*x))/(c + d*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d) - (6*B*Pol yLog[2, (d*(a + b*x))/(b*(c + d*x))])/d)/d)/(2*d))/(3*d)))/i
3.1.31.3.1 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_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I ntegersQ[m, q]
Leaf count of result is larger than twice the leaf count of optimal. \(1104\) vs. \(2(244)=488\).
Time = 1.65 (sec) , antiderivative size = 1105, normalized size of antiderivative = 4.38
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1105\) |
| default | \(\text {Expression too large to display}\) | \(1105\) |
| parts | \(\text {Expression too large to display}\) | \(1116\) |
| risch | \(\text {Expression too large to display}\) | \(3731\) |
-1/d^2*e*(a*d-b*c)*(A*d^2*g^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)/e/i*(-3/2*b^2/d^ 4*e^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2+1/d^4*ln(b*e-(b*e/d+(a*d-b*c )*e/d/(d*x+c))*d)+1/3*b^3*e^3/d^4/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3+ 3*b*e/d^4/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d))+B*d^2*g^3*(a^2*d^2-2*a*b* c*d+b^2*c^2)/e/i*(3*b^2/d^3*e^2*(-1/2/b^2/e^2/d*ln(b*e-(b*e/d+(a*d-b*c)*e/ d/(d*x+c))*d)+1/2/b/e/d/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)-1/2*ln(b*e/d +(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(2*b*e-(b*e/d+(a*d-b *c)*e/d/(d*x+c))*d)/b^2/e^2/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^2)+1/d^3 *(dilog(-((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/b/e)/d+ln(b*e/d+(a*d-b*c)*e /d/(d*x+c))*ln(-((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/b/e)/d)+b^3*e^3/d^3* (1/3/b^3/e^3/d*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)-1/3/b^2/e^2/d/(b*e- (b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)-1/6/b/e/d/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+ c))*d)^2+1/3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c)) *(3*e^2*b^2-3*(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d*b*e+d^2*(b*e/d+(a*d-b*c)*e/d /(d*x+c))^2)/b^3/e^3/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)^3)+3*b*e/d^3*(1 /b/e/d*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)+ln(b*e/d+(a*d-b*c)*e/d/(d*x +c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b/e/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))* d))))
\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\int { \frac {{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{d i x + c i} \,d x } \]
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((b *e*x + a*e)/(d*x + c)))/(d*i*x + c*i), x)
\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g^{3} \left (\int \frac {A a^{3}}{c + d x}\, dx + \int \frac {A b^{3} x^{3}}{c + d x}\, dx + \int \frac {B a^{3} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {3 A a b^{2} x^{2}}{c + d x}\, dx + \int \frac {3 A a^{2} b x}{c + d x}\, dx + \int \frac {B b^{3} x^{3} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {3 B a b^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {3 B a^{2} b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx\right )}{i} \]
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(a*e/(c + d*x) + b*e*x/(c + d*x))/(c + d*x), x) + Inte gral(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(a*e/(c + d*x) + b*e*x/(c + d*x))/(c + d*x), x) + Integral(3*B*a*b**2*x**2*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(c + d*x), x) + Integral(3*B*a**2*b*x*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(c + d*x), x))/i
Leaf count of result is larger than twice the leaf count of optimal. 790 vs. \(2 (243) = 486\).
Time = 0.26 (sec) , antiderivative size = 790, normalized size of antiderivative = 3.13 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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} - 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 )} {\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 (6 \, g^{3} \log \left (e\right ) + 11 \, g^{3}\right )} b^{3} c^{3} + 9 \, {\left (2 \, g^{3} \log \left (e\right ) + 3 \, g^{3}\right )} a b^{2} c^{2} d - 18 \, {\left (g^{3} \log \left (e\right ) + g^{3}\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 (3 \, g^{3} \log \left (e\right ) + g^{3}\right )} b^{3} c d^{2} - {\left (9 \, g^{3} \log \left (e\right ) + g^{3}\right )} a b^{2} d^{3}\right )} B x^{2} + 3 \, {\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 )^{2} + {\left ({\left (6 \, g^{3} \log \left (e\right ) + 5 \, g^{3}\right )} b^{3} c^{2} d - 6 \, {\left (3 \, g^{3} \log \left (e\right ) + 2 \, g^{3}\right )} a b^{2} c d^{2} + {\left (18 \, g^{3} \log \left (e\right ) + 7 \, g^{3}\right )} a^{2} b d^{3}\right )} B x + {\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 + {\left (6 \, a b^{2} c^{2} d g^{3} - 15 \, a^{2} b c d^{2} g^{3} + 11 \, a^{3} d^{3} g^{3}\right )} B\right )} \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\right )} \log \left (d x + c\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 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2 *g^3 - a^3*d^3*g^3)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dil og(-(b*d*x + a*d)/(b*c - a*d)))*B/(d^4*i) + 1/6*(6*a^3*d^3*g^3*log(e) - (6 *g^3*log(e) + 11*g^3)*b^3*c^3 + 9*(2*g^3*log(e) + 3*g^3)*a*b^2*c^2*d - 18* (g^3*log(e) + g^3)*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) - ((3*g^3*log(e) + g^3)*b^3*c*d^2 - (9*g^3*log(e) + g^3)*a* b^2*d^3)*B*x^2 + 3*(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)^2 + ((6*g^3*log(e) + 5*g^3)*b^3*c^2*d - 6*(3*g ^3*log(e) + 2*g^3)*a*b^2*c*d^2 + (18*g^3*log(e) + 7*g^3)*a^2*b*d^3)*B*x + (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*a*b^2*c^2*d*g^3 - 15*a^2*b*c*d^2*g^3 + 11*a^3*d^3*g^3)*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)*log(d*x + c))/(d^4*i)
Leaf count of result is larger than twice the leaf count of optimal. 3552 vs. \(2 (243) = 486\).
Time = 66.53 (sec) , antiderivative size = 3552, normalized size of antiderivative = 14.10 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\text {Too large to display} \]
-1/120*(6*(B*b^9*c^6*e^6*g^3 - 6*B*a*b^8*c^5*d*e^6*g^3 + 15*B*a^2*b^7*c^4* d^2*e^6*g^3 - 20*B*a^3*b^6*c^3*d^3*e^6*g^3 + 15*B*a^4*b^5*c^2*d^4*e^6*g^3 - 6*B*a^5*b^4*c*d^5*e^6*g^3 + B*a^6*b^3*d^6*e^6*g^3 - 5*(b*e*x + a*e)*B*b^ 8*c^6*d*e^5*g^3/(d*x + c) + 30*(b*e*x + a*e)*B*a*b^7*c^5*d^2*e^5*g^3/(d*x + c) - 75*(b*e*x + a*e)*B*a^2*b^6*c^4*d^3*e^5*g^3/(d*x + c) + 100*(b*e*x + a*e)*B*a^3*b^5*c^3*d^4*e^5*g^3/(d*x + c) - 75*(b*e*x + a*e)*B*a^4*b^4*c^2 *d^5*e^5*g^3/(d*x + c) + 30*(b*e*x + a*e)*B*a^5*b^3*c*d^6*e^5*g^3/(d*x + c ) - 5*(b*e*x + a*e)*B*a^6*b^2*d^7*e^5*g^3/(d*x + c) + 10*(b*e*x + a*e)^2*B *b^7*c^6*d^2*e^4*g^3/(d*x + c)^2 - 60*(b*e*x + a*e)^2*B*a*b^6*c^5*d^3*e^4* g^3/(d*x + c)^2 + 150*(b*e*x + a*e)^2*B*a^2*b^5*c^4*d^4*e^4*g^3/(d*x + c)^ 2 - 200*(b*e*x + a*e)^2*B*a^3*b^4*c^3*d^5*e^4*g^3/(d*x + c)^2 + 150*(b*e*x + a*e)^2*B*a^4*b^3*c^2*d^6*e^4*g^3/(d*x + c)^2 - 60*(b*e*x + a*e)^2*B*a^5 *b^2*c*d^7*e^4*g^3/(d*x + c)^2 + 10*(b*e*x + a*e)^2*B*a^6*b*d^8*e^4*g^3/(d *x + c)^2 - 10*(b*e*x + a*e)^3*B*b^6*c^6*d^3*e^3*g^3/(d*x + c)^3 + 60*(b*e *x + a*e)^3*B*a*b^5*c^5*d^4*e^3*g^3/(d*x + c)^3 - 150*(b*e*x + a*e)^3*B*a^ 2*b^4*c^4*d^5*e^3*g^3/(d*x + c)^3 + 200*(b*e*x + a*e)^3*B*a^3*b^3*c^3*d^6* e^3*g^3/(d*x + c)^3 - 150*(b*e*x + a*e)^3*B*a^4*b^2*c^2*d^7*e^3*g^3/(d*x + c)^3 + 60*(b*e*x + a*e)^3*B*a^5*b*c*d^8*e^3*g^3/(d*x + c)^3 - 10*(b*e*x + a*e)^3*B*a^6*d^9*e^3*g^3/(d*x + c)^3)*log((b*e*x + a*e)/(d*x + c))/(b^5*d ^4*e^5*i - 5*(b*e*x + a*e)*b^4*d^5*e^4*i/(d*x + c) + 10*(b*e*x + a*e)^2...
Timed out. \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^3\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{c\,i+d\,i\,x} \,d x \]