Integrand size = 40, antiderivative size = 337 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=-\frac {B (b c-a d)^4 g^2 i^2 x}{30 b^2 d^2}-\frac {B (b c-a d)^3 g^2 i^2 (c+d x)^2}{60 b d^3}+\frac {B (b c-a d)^2 g^2 i^2 (c+d x)^3}{10 d^3}-\frac {b B (b c-a d) g^2 i^2 (c+d x)^4}{20 d^3}-\frac {B (b c-a d)^5 g^2 i^2 \log \left (\frac {a+b x}{c+d x}\right )}{30 b^3 d^3}+\frac {(b c-a d)^2 g^2 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^3}-\frac {b (b c-a d) g^2 i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3}+\frac {b^2 g^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^3}-\frac {B (b c-a d)^5 g^2 i^2 \log (c+d x)}{30 b^3 d^3} \]
-1/30*B*(-a*d+b*c)^4*g^2*i^2*x/b^2/d^2-1/60*B*(-a*d+b*c)^3*g^2*i^2*(d*x+c) ^2/b/d^3+1/10*B*(-a*d+b*c)^2*g^2*i^2*(d*x+c)^3/d^3-1/20*b*B*(-a*d+b*c)*g^2 *i^2*(d*x+c)^4/d^3-1/30*B*(-a*d+b*c)^5*g^2*i^2*ln((b*x+a)/(d*x+c))/b^3/d^3 +1/3*(-a*d+b*c)^2*g^2*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3-1/2*b* (-a*d+b*c)*g^2*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3+1/5*b^2*g^2*i ^2*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3-1/30*B*(-a*d+b*c)^5*g^2*i^2*l n(d*x+c)/b^3/d^3
Time = 0.16 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.07 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^2 i^2 \left (20 d^3 (b c-a d)^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+30 d^4 (b c-a d) (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+12 d^5 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+10 B (b c-a d)^3 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )-5 B (b c-a d)^2 \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+B (b c-a d) \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )\right )}{60 b^3 d^3} \]
(g^2*i^2*(20*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d *x)]) + 30*d^4*(b*c - a*d)*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)] ) + 12*d^5*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 10*B*(b*c - a*d)^3*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d* x]) - 5*B*(b*c - a*d)^2*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) + B*(b*c - a*d )*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(a + b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x])))/( 60*b^3*d^3)
Time = 0.49 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2962, 2782, 27, 1195, 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.
| \(\displaystyle \int (a g+b g x)^2 (c i+d i x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right ) \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle g^2 i^2 (b c-a d)^5 \int \frac {(a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle g^2 i^2 (b c-a d)^5 \left (-B \int \frac {(c+d x) \left (b^2-\frac {5 d (a+b x) b}{c+d x}+\frac {10 d^2 (a+b x)^2}{(c+d x)^2}\right )}{30 d^3 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}+\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle g^2 i^2 (b c-a d)^5 \left (-\frac {B \int \frac {(c+d x) \left (b^2-\frac {5 d (a+b x) b}{c+d x}+\frac {10 d^2 (a+b x)^2}{(c+d x)^2}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{30 d^3}+\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle g^2 i^2 (b c-a d)^5 \left (-\frac {B \int \left (\frac {d}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {9 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {6 b d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {c+d x}{b^3 (a+b x)}\right )d\frac {a+b x}{c+d x}}{30 d^3}+\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g^2 i^2 (b c-a d)^5 \left (\frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{30 d^3}\right )\) |
(b*c - a*d)^5*g^2*i^2*((b^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^3*( b - (d*(a + b*x))/(c + d*x))^5) - (b*(A + B*Log[(e*(a + b*x))/(c + d*x)])) /(2*d^3*(b - (d*(a + b*x))/(c + d*x))^4) + (A + B*Log[(e*(a + b*x))/(c + d *x)])/(3*d^3*(b - (d*(a + b*x))/(c + d*x))^3) - (B*((3*b)/(2*(b - (d*(a + b*x))/(c + d*x))^4) - 3/(b - (d*(a + b*x))/(c + d*x))^3 + 1/(2*b*(b - (d*( a + b*x))/(c + d*x))^2) + 1/(b^2*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x)/(c + d*x)]/b^3 - Log[b - (d*(a + b*x))/(c + d*x)]/b^3))/(30*d^3))
3.1.11.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q _), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^q, x]}, Simp[(a + b*Log[c* x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[ {a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
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]
Time = 1.00 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.89
| method | result | size |
| risch | \(\frac {i^{2} g^{2} b^{2} d A c \,x^{4}}{2}+\frac {i^{2} g^{2} b \,d^{2} B a \,x^{4}}{20}-\frac {i^{2} g^{2} b^{2} d B c \,x^{4}}{20}+\frac {i^{2} g^{2} d^{2} A \,a^{2} x^{3}}{3}+\frac {i^{2} g^{2} b^{2} A \,c^{2} x^{3}}{3}+\frac {i^{2} g^{2} d^{2} B \,a^{2} x^{3}}{10}-\frac {i^{2} g^{2} b^{2} B \,c^{2} x^{3}}{10}+\frac {i^{2} g^{2} d^{2} B \,a^{3} x^{2}}{60 b}-\frac {i^{2} g^{2} b^{2} B \,c^{3} x^{2}}{60 d}-\frac {i^{2} g^{2} d^{2} B \,a^{4} x}{30 b^{2}}+\frac {i^{2} g^{2} b^{2} B \,c^{4} x}{30 d^{2}}+\frac {i^{2} g^{2} d^{2} B \ln \left (-b x -a \right ) a^{5}}{30 b^{3}}-\frac {i^{2} g^{2} b^{2} B \ln \left (d x +c \right ) c^{5}}{30 d^{3}}+\frac {i^{2} g^{2} B \ln \left (-b x -a \right ) a^{3} c^{2}}{3 b}+\frac {i^{2} g^{2} b \,d^{2} A a \,x^{4}}{2}-\frac {i^{2} g^{2} B \ln \left (d x +c \right ) a^{2} c^{3}}{3 d}+\frac {i^{2} g^{2} B x \left (6 b^{2} d^{2} x^{4}+15 a b \,d^{2} x^{3}+15 b^{2} c d \,x^{3}+10 x^{2} a^{2} d^{2}+40 a b c d \,x^{2}+10 b^{2} c^{2} x^{2}+30 a^{2} c d x +30 a b \,c^{2} x +30 c^{2} a^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{30}+\frac {4 i^{2} g^{2} b d A a c \,x^{3}}{3}+i^{2} g^{2} d A \,a^{2} c \,x^{2}+i^{2} g^{2} b A a \,c^{2} x^{2}+\frac {i^{2} g^{2} d B \,a^{2} c \,x^{2}}{4}-\frac {i^{2} g^{2} b B a \,c^{2} x^{2}}{4}+i^{2} g^{2} A \,a^{2} c^{2} x +\frac {i^{2} g^{2} d B \,a^{3} c x}{6 b}-\frac {i^{2} g^{2} b B a \,c^{3} x}{6 d}-\frac {i^{2} g^{2} d B \ln \left (-b x -a \right ) a^{4} c}{6 b^{2}}+\frac {i^{2} g^{2} b B \ln \left (d x +c \right ) a \,c^{4}}{6 d^{2}}+\frac {i^{2} g^{2} b^{2} d^{2} A \,x^{5}}{5}\) | \(636\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1140\) |
| parts | \(\text {Expression too large to display}\) | \(1409\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1500\) |
| default | \(\text {Expression too large to display}\) | \(1500\) |
1/2*i^2*g^2*b^2*d*A*c*x^4+1/20*i^2*g^2*b*d^2*B*a*x^4-1/20*i^2*g^2*b^2*d*B* c*x^4+1/3*i^2*g^2*d^2*A*a^2*x^3+1/3*i^2*g^2*b^2*A*c^2*x^3+1/10*i^2*g^2*d^2 *B*a^2*x^3-1/10*i^2*g^2*b^2*B*c^2*x^3+1/60*i^2*g^2/b*d^2*B*a^3*x^2-1/60*i^ 2*g^2*b^2/d*B*c^3*x^2-1/30*i^2*g^2/b^2*d^2*B*a^4*x+1/30*i^2*g^2*b^2/d^2*B* c^4*x+1/30*i^2*g^2/b^3*d^2*B*ln(-b*x-a)*a^5-1/30*i^2*g^2*b^2/d^3*B*ln(d*x+ c)*c^5+1/3*i^2*g^2/b*B*ln(-b*x-a)*a^3*c^2+1/2*i^2*g^2*b*d^2*A*a*x^4-1/3*i^ 2*g^2/d*B*ln(d*x+c)*a^2*c^3+1/30*i^2*g^2*B*x*(6*b^2*d^2*x^4+15*a*b*d^2*x^3 +15*b^2*c*d*x^3+10*a^2*d^2*x^2+40*a*b*c*d*x^2+10*b^2*c^2*x^2+30*a^2*c*d*x+ 30*a*b*c^2*x+30*a^2*c^2)*ln(e*(b*x+a)/(d*x+c))+4/3*i^2*g^2*b*d*A*a*c*x^3+i ^2*g^2*d*A*a^2*c*x^2+i^2*g^2*b*A*a*c^2*x^2+1/4*i^2*g^2*d*B*a^2*c*x^2-1/4*i ^2*g^2*b*B*a*c^2*x^2+i^2*g^2*A*a^2*c^2*x+1/6*i^2*g^2/b*d*B*a^3*c*x-1/6*i^2 *g^2*b/d*B*a*c^3*x-1/6*i^2*g^2/b^2*d*B*ln(-b*x-a)*a^4*c+1/6*i^2*g^2*b/d^2* B*ln(d*x+c)*a*c^4+1/5*i^2*g^2*b^2*d^2*A*x^5
Time = 0.42 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.58 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {12 \, A b^{5} d^{5} g^{2} i^{2} x^{5} + 3 \, {\left ({\left (10 \, A - B\right )} b^{5} c d^{4} + {\left (10 \, A + B\right )} a b^{4} d^{5}\right )} g^{2} i^{2} x^{4} + 2 \, {\left ({\left (10 \, A - 3 \, B\right )} b^{5} c^{2} d^{3} + 40 \, A a b^{4} c d^{4} + {\left (10 \, A + 3 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{2} i^{2} x^{3} - {\left (B b^{5} c^{3} d^{2} - 15 \, {\left (4 \, A - B\right )} a b^{4} c^{2} d^{3} - 15 \, {\left (4 \, A + B\right )} a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g^{2} i^{2} x^{2} + 2 \, {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 30 \, A a^{2} b^{3} c^{2} d^{3} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{2} i^{2} x + 2 \, {\left (10 \, B a^{3} b^{2} c^{2} d^{3} - 5 \, B a^{4} b c d^{4} + B a^{5} d^{5}\right )} g^{2} i^{2} \log \left (b x + a\right ) - 2 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2}\right )} g^{2} i^{2} \log \left (d x + c\right ) + 2 \, {\left (6 \, B b^{5} d^{5} g^{2} i^{2} x^{5} + 30 \, B a^{2} b^{3} c^{2} d^{3} g^{2} i^{2} x + 15 \, {\left (B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g^{2} i^{2} x^{4} + 10 \, {\left (B b^{5} c^{2} d^{3} + 4 \, B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{2} i^{2} x^{3} + 30 \, {\left (B a b^{4} c^{2} d^{3} + B a^{2} b^{3} c d^{4}\right )} g^{2} i^{2} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{60 \, b^{3} d^{3}} \]
1/60*(12*A*b^5*d^5*g^2*i^2*x^5 + 3*((10*A - B)*b^5*c*d^4 + (10*A + B)*a*b^ 4*d^5)*g^2*i^2*x^4 + 2*((10*A - 3*B)*b^5*c^2*d^3 + 40*A*a*b^4*c*d^4 + (10* A + 3*B)*a^2*b^3*d^5)*g^2*i^2*x^3 - (B*b^5*c^3*d^2 - 15*(4*A - B)*a*b^4*c^ 2*d^3 - 15*(4*A + B)*a^2*b^3*c*d^4 - B*a^3*b^2*d^5)*g^2*i^2*x^2 + 2*(B*b^5 *c^4*d - 5*B*a*b^4*c^3*d^2 + 30*A*a^2*b^3*c^2*d^3 + 5*B*a^3*b^2*c*d^4 - B* a^4*b*d^5)*g^2*i^2*x + 2*(10*B*a^3*b^2*c^2*d^3 - 5*B*a^4*b*c*d^4 + B*a^5*d ^5)*g^2*i^2*log(b*x + a) - 2*(B*b^5*c^5 - 5*B*a*b^4*c^4*d + 10*B*a^2*b^3*c ^3*d^2)*g^2*i^2*log(d*x + c) + 2*(6*B*b^5*d^5*g^2*i^2*x^5 + 30*B*a^2*b^3*c ^2*d^3*g^2*i^2*x + 15*(B*b^5*c*d^4 + B*a*b^4*d^5)*g^2*i^2*x^4 + 10*(B*b^5* c^2*d^3 + 4*B*a*b^4*c*d^4 + B*a^2*b^3*d^5)*g^2*i^2*x^3 + 30*(B*a*b^4*c^2*d ^3 + B*a^2*b^3*c*d^4)*g^2*i^2*x^2)*log((b*e*x + a*e)/(d*x + c)))/(b^3*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 1266 vs. \(2 (311) = 622\).
Time = 3.87 (sec) , antiderivative size = 1266, normalized size of antiderivative = 3.76 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {A b^{2} d^{2} g^{2} i^{2} x^{5}}{5} + \frac {B a^{3} g^{2} i^{2} \left (a^{2} d^{2} - 5 a b c d + 10 b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{5} c d^{4} g^{2} i^{2} - 5 B a^{4} b c^{2} d^{3} g^{2} i^{2} + \frac {B a^{4} d^{3} g^{2} i^{2} \left (a^{2} d^{2} - 5 a b c d + 10 b^{2} c^{2}\right )}{b} + 20 B a^{3} b^{2} c^{3} d^{2} g^{2} i^{2} - B a^{3} c d^{2} g^{2} i^{2} \left (a^{2} d^{2} - 5 a b c d + 10 b^{2} c^{2}\right ) - 5 B a^{2} b^{3} c^{4} d g^{2} i^{2} + B a b^{4} c^{5} g^{2} i^{2}}{B a^{5} d^{5} g^{2} i^{2} - 5 B a^{4} b c d^{4} g^{2} i^{2} + 10 B a^{3} b^{2} c^{2} d^{3} g^{2} i^{2} + 10 B a^{2} b^{3} c^{3} d^{2} g^{2} i^{2} - 5 B a b^{4} c^{4} d g^{2} i^{2} + B b^{5} c^{5} g^{2} i^{2}} \right )}}{30 b^{3}} - \frac {B c^{3} g^{2} i^{2} \cdot \left (10 a^{2} d^{2} - 5 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{5} c d^{4} g^{2} i^{2} - 5 B a^{4} b c^{2} d^{3} g^{2} i^{2} + 20 B a^{3} b^{2} c^{3} d^{2} g^{2} i^{2} - 5 B a^{2} b^{3} c^{4} d g^{2} i^{2} + B a b^{4} c^{5} g^{2} i^{2} - B a b^{2} c^{3} g^{2} i^{2} \cdot \left (10 a^{2} d^{2} - 5 a b c d + b^{2} c^{2}\right ) + \frac {B b^{3} c^{4} g^{2} i^{2} \cdot \left (10 a^{2} d^{2} - 5 a b c d + b^{2} c^{2}\right )}{d}}{B a^{5} d^{5} g^{2} i^{2} - 5 B a^{4} b c d^{4} g^{2} i^{2} + 10 B a^{3} b^{2} c^{2} d^{3} g^{2} i^{2} + 10 B a^{2} b^{3} c^{3} d^{2} g^{2} i^{2} - 5 B a b^{4} c^{4} d g^{2} i^{2} + B b^{5} c^{5} g^{2} i^{2}} \right )}}{30 d^{3}} + x^{4} \left (\frac {A a b d^{2} g^{2} i^{2}}{2} + \frac {A b^{2} c d g^{2} i^{2}}{2} + \frac {B a b d^{2} g^{2} i^{2}}{20} - \frac {B b^{2} c d g^{2} i^{2}}{20}\right ) + x^{3} \left (\frac {A a^{2} d^{2} g^{2} i^{2}}{3} + \frac {4 A a b c d g^{2} i^{2}}{3} + \frac {A b^{2} c^{2} g^{2} i^{2}}{3} + \frac {B a^{2} d^{2} g^{2} i^{2}}{10} - \frac {B b^{2} c^{2} g^{2} i^{2}}{10}\right ) + x^{2} \left (A a^{2} c d g^{2} i^{2} + A a b c^{2} g^{2} i^{2} + \frac {B a^{3} d^{2} g^{2} i^{2}}{60 b} + \frac {B a^{2} c d g^{2} i^{2}}{4} - \frac {B a b c^{2} g^{2} i^{2}}{4} - \frac {B b^{2} c^{3} g^{2} i^{2}}{60 d}\right ) + x \left (A a^{2} c^{2} g^{2} i^{2} - \frac {B a^{4} d^{2} g^{2} i^{2}}{30 b^{2}} + \frac {B a^{3} c d g^{2} i^{2}}{6 b} - \frac {B a b c^{3} g^{2} i^{2}}{6 d} + \frac {B b^{2} c^{4} g^{2} i^{2}}{30 d^{2}}\right ) + \left (B a^{2} c^{2} g^{2} i^{2} x + B a^{2} c d g^{2} i^{2} x^{2} + \frac {B a^{2} d^{2} g^{2} i^{2} x^{3}}{3} + B a b c^{2} g^{2} i^{2} x^{2} + \frac {4 B a b c d g^{2} i^{2} x^{3}}{3} + \frac {B a b d^{2} g^{2} i^{2} x^{4}}{2} + \frac {B b^{2} c^{2} g^{2} i^{2} x^{3}}{3} + \frac {B b^{2} c d g^{2} i^{2} x^{4}}{2} + \frac {B b^{2} d^{2} g^{2} i^{2} x^{5}}{5}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]
A*b**2*d**2*g**2*i**2*x**5/5 + B*a**3*g**2*i**2*(a**2*d**2 - 5*a*b*c*d + 1 0*b**2*c**2)*log(x + (B*a**5*c*d**4*g**2*i**2 - 5*B*a**4*b*c**2*d**3*g**2* i**2 + B*a**4*d**3*g**2*i**2*(a**2*d**2 - 5*a*b*c*d + 10*b**2*c**2)/b + 20 *B*a**3*b**2*c**3*d**2*g**2*i**2 - B*a**3*c*d**2*g**2*i**2*(a**2*d**2 - 5* a*b*c*d + 10*b**2*c**2) - 5*B*a**2*b**3*c**4*d*g**2*i**2 + B*a*b**4*c**5*g **2*i**2)/(B*a**5*d**5*g**2*i**2 - 5*B*a**4*b*c*d**4*g**2*i**2 + 10*B*a**3 *b**2*c**2*d**3*g**2*i**2 + 10*B*a**2*b**3*c**3*d**2*g**2*i**2 - 5*B*a*b** 4*c**4*d*g**2*i**2 + B*b**5*c**5*g**2*i**2))/(30*b**3) - B*c**3*g**2*i**2* (10*a**2*d**2 - 5*a*b*c*d + b**2*c**2)*log(x + (B*a**5*c*d**4*g**2*i**2 - 5*B*a**4*b*c**2*d**3*g**2*i**2 + 20*B*a**3*b**2*c**3*d**2*g**2*i**2 - 5*B* a**2*b**3*c**4*d*g**2*i**2 + B*a*b**4*c**5*g**2*i**2 - B*a*b**2*c**3*g**2* i**2*(10*a**2*d**2 - 5*a*b*c*d + b**2*c**2) + B*b**3*c**4*g**2*i**2*(10*a* *2*d**2 - 5*a*b*c*d + b**2*c**2)/d)/(B*a**5*d**5*g**2*i**2 - 5*B*a**4*b*c* d**4*g**2*i**2 + 10*B*a**3*b**2*c**2*d**3*g**2*i**2 + 10*B*a**2*b**3*c**3* d**2*g**2*i**2 - 5*B*a*b**4*c**4*d*g**2*i**2 + B*b**5*c**5*g**2*i**2))/(30 *d**3) + x**4*(A*a*b*d**2*g**2*i**2/2 + A*b**2*c*d*g**2*i**2/2 + B*a*b*d** 2*g**2*i**2/20 - B*b**2*c*d*g**2*i**2/20) + x**3*(A*a**2*d**2*g**2*i**2/3 + 4*A*a*b*c*d*g**2*i**2/3 + A*b**2*c**2*g**2*i**2/3 + B*a**2*d**2*g**2*i** 2/10 - B*b**2*c**2*g**2*i**2/10) + x**2*(A*a**2*c*d*g**2*i**2 + A*a*b*c**2 *g**2*i**2 + B*a**3*d**2*g**2*i**2/(60*b) + B*a**2*c*d*g**2*i**2/4 - B*...
Leaf count of result is larger than twice the leaf count of optimal. 1200 vs. \(2 (319) = 638\).
Time = 0.23 (sec) , antiderivative size = 1200, normalized size of antiderivative = 3.56 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]
1/5*A*b^2*d^2*g^2*i^2*x^5 + 1/2*A*b^2*c*d*g^2*i^2*x^4 + 1/2*A*a*b*d^2*g^2* i^2*x^4 + 1/3*A*b^2*c^2*g^2*i^2*x^3 + 4/3*A*a*b*c*d*g^2*i^2*x^3 + 1/3*A*a^ 2*d^2*g^2*i^2*x^3 + A*a*b*c^2*g^2*i^2*x^2 + A*a^2*c*d*g^2*i^2*x^2 + (x*log (b*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*B *a^2*c^2*g^2*i^2 + (x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a*b*c^2*g^2*i^2 + 1/6*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^ 2)*x)/(b^2*d^2))*B*b^2*c^2*g^2*i^2 + (x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))* B*a^2*c*d*g^2*i^2 + 2/3*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^ 3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2 *(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a*b*c*d*g^2*i^2 + 1/12*(6*x^4*log(b*e *x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c )/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6 *(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*b^2*c*d*g^2*i^2 + 1/6*(2*x^3*log(b*e* x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c) /d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a^ 2*d^2*g^2*i^2 + 1/12*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*l og(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x...
Leaf count of result is larger than twice the leaf count of optimal. 3144 vs. \(2 (319) = 638\).
Time = 0.52 (sec) , antiderivative size = 3144, normalized size of antiderivative = 9.33 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]
1/60*(2*(B*b^8*c^6*e^6*g^2*i^2 - 6*B*a*b^7*c^5*d*e^6*g^2*i^2 + 15*B*a^2*b^ 6*c^4*d^2*e^6*g^2*i^2 - 20*B*a^3*b^5*c^3*d^3*e^6*g^2*i^2 + 15*B*a^4*b^4*c^ 2*d^4*e^6*g^2*i^2 - 6*B*a^5*b^3*c*d^5*e^6*g^2*i^2 + B*a^6*b^2*d^6*e^6*g^2* i^2 - 5*(b*e*x + a*e)*B*b^7*c^6*d*e^5*g^2*i^2/(d*x + c) + 30*(b*e*x + a*e) *B*a*b^6*c^5*d^2*e^5*g^2*i^2/(d*x + c) - 75*(b*e*x + a*e)*B*a^2*b^5*c^4*d^ 3*e^5*g^2*i^2/(d*x + c) + 100*(b*e*x + a*e)*B*a^3*b^4*c^3*d^4*e^5*g^2*i^2/ (d*x + c) - 75*(b*e*x + a*e)*B*a^4*b^3*c^2*d^5*e^5*g^2*i^2/(d*x + c) + 30* (b*e*x + a*e)*B*a^5*b^2*c*d^6*e^5*g^2*i^2/(d*x + c) - 5*(b*e*x + a*e)*B*a^ 6*b*d^7*e^5*g^2*i^2/(d*x + c) + 10*(b*e*x + a*e)^2*B*b^6*c^6*d^2*e^4*g^2*i ^2/(d*x + c)^2 - 60*(b*e*x + a*e)^2*B*a*b^5*c^5*d^3*e^4*g^2*i^2/(d*x + c)^ 2 + 150*(b*e*x + a*e)^2*B*a^2*b^4*c^4*d^4*e^4*g^2*i^2/(d*x + c)^2 - 200*(b *e*x + a*e)^2*B*a^3*b^3*c^3*d^5*e^4*g^2*i^2/(d*x + c)^2 + 150*(b*e*x + a*e )^2*B*a^4*b^2*c^2*d^6*e^4*g^2*i^2/(d*x + c)^2 - 60*(b*e*x + a*e)^2*B*a^5*b *c*d^7*e^4*g^2*i^2/(d*x + c)^2 + 10*(b*e*x + a*e)^2*B*a^6*d^8*e^4*g^2*i^2/ (d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/(b^5*d^3*e^5 - 5*(b*e*x + a*e)*b ^4*d^4*e^4/(d*x + c) + 10*(b*e*x + a*e)^2*b^3*d^5*e^3/(d*x + c)^2 - 10*(b* e*x + a*e)^3*b^2*d^6*e^2/(d*x + c)^3 + 5*(b*e*x + a*e)^4*b*d^7*e/(d*x + c) ^4 - (b*e*x + a*e)^5*d^8/(d*x + c)^5) + (2*A*b^10*c^6*e^6*g^2*i^2 - 12*A*a *b^9*c^5*d*e^6*g^2*i^2 + 30*A*a^2*b^8*c^4*d^2*e^6*g^2*i^2 - 40*A*a^3*b^7*c ^3*d^3*e^6*g^2*i^2 + 30*A*a^4*b^6*c^2*d^4*e^6*g^2*i^2 - 12*A*a^5*b^5*c*...
Time = 1.93 (sec) , antiderivative size = 1287, normalized size of antiderivative = 3.82 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]
log((e*(a + b*x))/(c + d*x))*((B*g^2*i^2*x^3*(a^2*d^2 + b^2*c^2 + 4*a*b*c* d))/3 + B*a^2*c^2*g^2*i^2*x + (B*b^2*d^2*g^2*i^2*x^5)/5 + B*a*c*g^2*i^2*x^ 2*(a*d + b*c) + (B*b*d*g^2*i^2*x^4*(a*d + b*c))/2) - x^3*(((30*a*d + 30*b* c)*((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d - B*b*c))/5 - (A*b*d*g^2*i^2 *(30*a*d + 30*b*c))/30))/(90*b*d) - (g^2*i^2*(6*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 18*A*a*b*c*d))/6 + (A*a*b*c*d*g^2*i^2)/3) + x*((a* c*(((30*a*d + 30*b*c)*((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d - B*b*c)) /5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30))/(30*b*d) - (g^2*i^2*(6*A*a^2*d ^2 + 6*A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 18*A*a*b*c*d))/2 + A*a*b*c*d*g^ 2*i^2))/(b*d) - ((30*a*d + 30*b*c)*(((30*a*d + 30*b*c)*(((30*a*d + 30*b*c) *((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d - B*b*c))/5 - (A*b*d*g^2*i^2*( 30*a*d + 30*b*c))/30))/(30*b*d) - (g^2*i^2*(6*A*a^2*d^2 + 6*A*b^2*c^2 + B* a^2*d^2 - B*b^2*c^2 + 18*A*a*b*c*d))/2 + A*a*b*c*d*g^2*i^2))/(30*b*d) + (g ^2*i^2*(3*A*a^3*d^3 + 3*A*b^3*c^3 + B*a^3*d^3 - B*b^3*c^3 + 27*A*a*b^2*c^2 *d + 27*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(3*b*d) - (a*c *((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d - B*b*c))/5 - (A*b*d*g^2*i^2*( 30*a*d + 30*b*c))/30))/(b*d)))/(30*b*d) + (a*c*g^2*i^2*(3*A*a^2*d^2 + 3*A* b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 9*A*a*b*c*d))/(b*d)) + x^2*(((30*a*d + 3 0*b*c)*(((30*a*d + 30*b*c)*((b*d*g^2*i^2*(15*A*a*d + 15*A*b*c + B*a*d - B* b*c))/5 - (A*b*d*g^2*i^2*(30*a*d + 30*b*c))/30))/(30*b*d) - (g^2*i^2*(6...