Integrand size = 38, antiderivative size = 239 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {B (b c-a d)^3 g i^2 x}{12 b^2 d}+\frac {B (b c-a d)^2 g i^2 (c+d x)^2}{24 b d^2}-\frac {B (b c-a d) g i^2 (c+d x)^3}{12 d^2}+\frac {B (b c-a d)^4 g i^2 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac {(b c-a d) g i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^2}+\frac {b g i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^2}+\frac {B (b c-a d)^4 g i^2 \log (c+d x)}{12 b^3 d^2} \] Output:
1/12*B*(-a*d+b*c)^3*g*i^2*x/b^2/d+1/24*B*(-a*d+b*c)^2*g*i^2*(d*x+c)^2/b/d^ 2-1/12*B*(-a*d+b*c)*g*i^2*(d*x+c)^3/d^2+1/12*B*(-a*d+b*c)^4*g*i^2*ln((b*x+ a)/(d*x+c))/b^3/d^2-1/3*(-a*d+b*c)*g*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+ c)))/d^2+1/4*b*g*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^2+1/12*B*(-a* d+b*c)^4*g*i^2*ln(d*x+c)/b^3/d^2
Time = 0.21 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.90 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g i^2 \left (\frac {4 B (b c-a d)^2 \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )}{b^3}-\frac {B (b c-a d) \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )}{b^3}-8 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 b (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right )}{24 d^2} \] Input:
Integrate[(a*g + b*g*x)*(c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x) ]),x]
Output:
(g*i^2*((4*B*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]))/b^3 - (B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*b ^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b *x]))/b^3 - 8*(b*c - a*d)*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*b*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])))/(24*d^2)
Time = 0.42 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2962, 2782, 27, 86, 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.
| \(\displaystyle \int (a g+b g x) (c i+d i x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right ) \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle g i^2 (b c-a d)^4 \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-B \int -\frac {(c+d x) \left (b-\frac {4 d (a+b x)}{c+d x}\right )}{12 d^2 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {B \int \frac {(c+d x) \left (b-\frac {4 d (a+b x)}{c+d x}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{12 d^2}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (\frac {B \int \left (\frac {d}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {c+d x}{b^3 (a+b x)}\right )d\frac {a+b x}{c+d x}}{12 d^2}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{12 d^2}\right )\) |
Input:
Int[(a*g + b*g*x)*(c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
Output:
(b*c - a*d)^4*g*i^2*((b*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^2*(b - (d*(a + b*x))/(c + d*x))^4) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(3*d^2* (b - (d*(a + b*x))/(c + d*x))^3) + (B*(-(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))/(12*d^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 1.29 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.66
| method | result | size |
| risch | \(\frac {g \,i^{2} B x \left (3 b \,d^{2} x^{3}+4 x^{2} a \,d^{2}+8 b c d \,x^{2}+12 a c d x +6 b \,c^{2} x +12 a \,c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{12}+\frac {i^{2} g b \,d^{2} A \,x^{4}}{4}+\frac {i^{2} g \,d^{2} A a \,x^{3}}{3}+\frac {2 i^{2} g b d A c \,x^{3}}{3}+\frac {i^{2} g \,d^{2} B a \,x^{3}}{12}-\frac {i^{2} g b d B c \,x^{3}}{12}+i^{2} g d A a c \,x^{2}+\frac {i^{2} g b A \,c^{2} x^{2}}{2}+\frac {i^{2} g \,d^{2} B \,a^{2} x^{2}}{24 b}+\frac {i^{2} g d B a c \,x^{2}}{6}-\frac {5 i^{2} g b B \,c^{2} x^{2}}{24}+i^{2} g A a \,c^{2} x +\frac {i^{2} g \,d^{2} B \ln \left (-b x -a \right ) a^{4}}{12 b^{3}}-\frac {i^{2} g d B \ln \left (-b x -a \right ) a^{3} c}{3 b^{2}}+\frac {i^{2} g B \ln \left (-b x -a \right ) a^{2} c^{2}}{2 b}-\frac {i^{2} g B \ln \left (d x +c \right ) a \,c^{3}}{3 d}+\frac {i^{2} g b B \ln \left (d x +c \right ) c^{4}}{12 d^{2}}-\frac {i^{2} g \,d^{2} B \,a^{3} x}{12 b^{2}}+\frac {i^{2} g d B \,a^{2} c x}{3 b}-\frac {i^{2} g B a \,c^{2} x}{6}-\frac {i^{2} g b B \,c^{3} x}{12 d}\) | \(396\) |
| parallelrisch | \(\frac {2 B \,a^{4} d^{4} g \,i^{2}+2 B \,b^{4} c^{4} g \,i^{2}-7 B \,a^{3} b c \,d^{3} g \,i^{2}-8 B \,a^{2} b^{2} c^{2} d^{2} g \,i^{2}+11 B a \,b^{3} c^{3} d g \,i^{2}+8 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} d^{4} g \,i^{2}+6 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} d^{4} g \,i^{2}+16 A \,x^{3} b^{4} c \,d^{3} g \,i^{2}+2 B \,x^{3} a \,b^{3} d^{4} g \,i^{2}-2 B \,x^{3} b^{4} c \,d^{3} g \,i^{2}+12 A \,x^{2} b^{4} c^{2} d^{2} g \,i^{2}-4 B x a \,b^{3} c^{2} d^{2} g \,i^{2}+8 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} c^{3} d g \,i^{2}-8 B \ln \left (b x +a \right ) a^{3} b c \,d^{3} g \,i^{2}-8 B \ln \left (b x +a \right ) a \,b^{3} c^{3} d g \,i^{2}+16 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c \,d^{3} g \,i^{2}+12 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c^{2} d^{2} g \,i^{2}+24 A \,x^{2} a \,b^{3} c \,d^{3} g \,i^{2}+24 A x a \,b^{3} c^{2} d^{2} g \,i^{2}+4 B \,x^{2} a \,b^{3} c \,d^{3} g \,i^{2}+12 B \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} g \,i^{2}-48 A \,a^{2} b^{2} c^{2} d^{2} g \,i^{2}-36 A a \,b^{3} c^{3} d g \,i^{2}+8 B x \,a^{2} b^{2} c \,d^{3} g \,i^{2}+B \,x^{2} a^{2} b^{2} d^{4} g \,i^{2}-5 B \,x^{2} b^{4} c^{2} d^{2} g \,i^{2}-2 B x \,a^{3} b \,d^{4} g \,i^{2}-2 B x \,b^{4} c^{3} d g \,i^{2}+8 A \,x^{3} a \,b^{3} d^{4} g \,i^{2}+6 A \,x^{4} b^{4} d^{4} g \,i^{2}-2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c^{4} g \,i^{2}+2 B \ln \left (b x +a \right ) a^{4} d^{4} g \,i^{2}+2 B \ln \left (b x +a \right ) b^{4} c^{4} g \,i^{2}+24 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} c \,d^{3} g \,i^{2}+24 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} c^{2} d^{2} g \,i^{2}}{24 b^{3} d^{2}}\) | \(737\) |
| parts | \(A g \,i^{2} \left (\frac {b \,d^{2} x^{4}}{4}+\frac {\left (a \,d^{2}+2 b c d \right ) x^{3}}{3}+\frac {\left (2 a c d +b \,c^{2}\right ) x^{2}}{2}+a \,c^{2} x \right )-\frac {B g \,i^{2} \left (d a -b c \right )^{3} e^{3} \left (\left (\frac {1}{3 b^{2} e^{2} d \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}-\frac {1}{6 b e d \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}+\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{3 b^{3} e^{3} d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (3 b^{2} e^{2}-3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d b e +d^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}\right )}{3 b^{3} e^{3} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{3}}\right ) d^{3} \left (d a -b c \right )+\left (-\frac {1}{4 b^{3} e^{3} d \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}+\frac {1}{8 b^{2} e^{2} d \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}-\frac {1}{12 b e d \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{3}}-\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{4 b^{4} e^{4} d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (d^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}-4 d^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} b e +6 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \,b^{2} e^{2}-4 b^{3} e^{3}\right )}{4 b^{4} e^{4} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{4}}\right ) b \,d^{3} e \left (d a -b c \right )\right )}{d^{4}}\) | \(821\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-A \,d^{2} e^{2} g \,i^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {1}{3 d^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}+\frac {b e}{4 d^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{4}}\right )-B \,d^{2} e^{2} g \,i^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {-\frac {1}{6 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}-\frac {1}{3 b^{2} e^{2} d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{3 b^{3} e^{3} d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (3 b^{2} e^{2}-3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d b e +d^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}\right )}{3 b^{3} e^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}}{d}-\frac {\left (\frac {1}{12 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}+\frac {1}{8 b^{2} e^{2} d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}+\frac {1}{4 b^{3} e^{3} d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{4 b^{4} e^{4} d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (4 b^{3} e^{3}-6 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \,b^{2} e^{2}+4 d^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} b e -d^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}\right )}{4 b^{4} e^{4} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{4}}\right ) b e}{d}\right )\right )}{d^{2}}\) | \(928\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-A \,d^{2} e^{2} g \,i^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {1}{3 d^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}+\frac {b e}{4 d^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{4}}\right )-B \,d^{2} e^{2} g \,i^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {-\frac {1}{6 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}-\frac {1}{3 b^{2} e^{2} d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{3 b^{3} e^{3} d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (3 b^{2} e^{2}-3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d b e +d^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}\right )}{3 b^{3} e^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}}{d}-\frac {\left (\frac {1}{12 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}+\frac {1}{8 b^{2} e^{2} d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}+\frac {1}{4 b^{3} e^{3} d \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{4 b^{4} e^{4} d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (4 b^{3} e^{3}-6 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \,b^{2} e^{2}+4 d^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} b e -d^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}\right )}{4 b^{4} e^{4} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )^{4}}\right ) b e}{d}\right )\right )}{d^{2}}\) | \(928\) |
Input:
int((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c))),x,method=_RETURN VERBOSE)
Output:
1/12*g*i^2*B*x*(3*b*d^2*x^3+4*a*d^2*x^2+8*b*c*d*x^2+12*a*c*d*x+6*b*c^2*x+1 2*a*c^2)*ln(e*(b*x+a)/(d*x+c))+1/4*i^2*g*b*d^2*A*x^4+1/3*i^2*g*d^2*A*a*x^3 +2/3*i^2*g*b*d*A*c*x^3+1/12*i^2*g*d^2*B*a*x^3-1/12*i^2*g*b*d*B*c*x^3+i^2*g *d*A*a*c*x^2+1/2*i^2*g*b*A*c^2*x^2+1/24*i^2*g/b*d^2*B*a^2*x^2+1/6*i^2*g*d* B*a*c*x^2-5/24*i^2*g*b*B*c^2*x^2+i^2*g*A*a*c^2*x+1/12*i^2*g/b^3*d^2*B*ln(- b*x-a)*a^4-1/3*i^2*g/b^2*d*B*ln(-b*x-a)*a^3*c+1/2*i^2*g/b*B*ln(-b*x-a)*a^2 *c^2-1/3*i^2*g/d*B*ln(d*x+c)*a*c^3+1/12*i^2*g*b/d^2*B*ln(d*x+c)*c^4-1/12*i ^2*g/b^2*d^2*B*a^3*x+1/3*i^2*g/b*d*B*a^2*c*x-1/6*i^2*g*B*a*c^2*x-1/12*i^2* g*b/d*B*c^3*x
Time = 0.14 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.54 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {6 \, A b^{4} d^{4} g i^{2} x^{4} + 2 \, {\left ({\left (8 \, A - B\right )} b^{4} c d^{3} + {\left (4 \, A + B\right )} a b^{3} d^{4}\right )} g i^{2} x^{3} + {\left ({\left (12 \, A - 5 \, B\right )} b^{4} c^{2} d^{2} + 4 \, {\left (6 \, A + B\right )} a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g i^{2} x^{2} - 2 \, {\left (B b^{4} c^{3} d - 2 \, {\left (6 \, A - B\right )} a b^{3} c^{2} d^{2} - 4 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g i^{2} x + 2 \, {\left (6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3} + B a^{4} d^{4}\right )} g i^{2} \log \left (b x + a\right ) + 2 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d\right )} g i^{2} \log \left (d x + c\right ) + 2 \, {\left (3 \, B b^{4} d^{4} g i^{2} x^{4} + 12 \, B a b^{3} c^{2} d^{2} g i^{2} x + 4 \, {\left (2 \, B b^{4} c d^{3} + B a b^{3} d^{4}\right )} g i^{2} x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} + 2 \, B a b^{3} c d^{3}\right )} g i^{2} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{24 \, b^{3} d^{2}} \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algori thm="fricas")
Output:
1/24*(6*A*b^4*d^4*g*i^2*x^4 + 2*((8*A - B)*b^4*c*d^3 + (4*A + B)*a*b^3*d^4 )*g*i^2*x^3 + ((12*A - 5*B)*b^4*c^2*d^2 + 4*(6*A + B)*a*b^3*c*d^3 + B*a^2* b^2*d^4)*g*i^2*x^2 - 2*(B*b^4*c^3*d - 2*(6*A - B)*a*b^3*c^2*d^2 - 4*B*a^2* b^2*c*d^3 + B*a^3*b*d^4)*g*i^2*x + 2*(6*B*a^2*b^2*c^2*d^2 - 4*B*a^3*b*c*d^ 3 + B*a^4*d^4)*g*i^2*log(b*x + a) + 2*(B*b^4*c^4 - 4*B*a*b^3*c^3*d)*g*i^2* log(d*x + c) + 2*(3*B*b^4*d^4*g*i^2*x^4 + 12*B*a*b^3*c^2*d^2*g*i^2*x + 4*( 2*B*b^4*c*d^3 + B*a*b^3*d^4)*g*i^2*x^3 + 6*(B*b^4*c^2*d^2 + 2*B*a*b^3*c*d^ 3)*g*i^2*x^2)*log((b*e*x + a*e)/(d*x + c)))/(b^3*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 850 vs. \(2 (221) = 442\).
Time = 2.30 (sec) , antiderivative size = 850, normalized size of antiderivative = 3.56 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {A b d^{2} g i^{2} x^{4}}{4} + \frac {B a^{2} g i^{2} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right ) \log {\left (x + \frac {B a^{4} c d^{3} g i^{2} - 4 B a^{3} b c^{2} d^{2} g i^{2} + \frac {B a^{3} d^{2} g i^{2} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right )}{b} + 10 B a^{2} b^{2} c^{3} d g i^{2} - B a^{2} c d g i^{2} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right ) - B a b^{3} c^{4} g i^{2}}{B a^{4} d^{4} g i^{2} - 4 B a^{3} b c d^{3} g i^{2} + 6 B a^{2} b^{2} c^{2} d^{2} g i^{2} + 4 B a b^{3} c^{3} d g i^{2} - B b^{4} c^{4} g i^{2}} \right )}}{12 b^{3}} - \frac {B c^{3} g i^{2} \cdot \left (4 a d - b c\right ) \log {\left (x + \frac {B a^{4} c d^{3} g i^{2} - 4 B a^{3} b c^{2} d^{2} g i^{2} + 10 B a^{2} b^{2} c^{3} d g i^{2} - B a b^{3} c^{4} g i^{2} - B a b^{2} c^{3} g i^{2} \cdot \left (4 a d - b c\right ) + \frac {B b^{3} c^{4} g i^{2} \cdot \left (4 a d - b c\right )}{d}}{B a^{4} d^{4} g i^{2} - 4 B a^{3} b c d^{3} g i^{2} + 6 B a^{2} b^{2} c^{2} d^{2} g i^{2} + 4 B a b^{3} c^{3} d g i^{2} - B b^{4} c^{4} g i^{2}} \right )}}{12 d^{2}} + x^{3} \left (\frac {A a d^{2} g i^{2}}{3} + \frac {2 A b c d g i^{2}}{3} + \frac {B a d^{2} g i^{2}}{12} - \frac {B b c d g i^{2}}{12}\right ) + x^{2} \left (A a c d g i^{2} + \frac {A b c^{2} g i^{2}}{2} + \frac {B a^{2} d^{2} g i^{2}}{24 b} + \frac {B a c d g i^{2}}{6} - \frac {5 B b c^{2} g i^{2}}{24}\right ) + x \left (A a c^{2} g i^{2} - \frac {B a^{3} d^{2} g i^{2}}{12 b^{2}} + \frac {B a^{2} c d g i^{2}}{3 b} - \frac {B a c^{2} g i^{2}}{6} - \frac {B b c^{3} g i^{2}}{12 d}\right ) + \left (B a c^{2} g i^{2} x + B a c d g i^{2} x^{2} + \frac {B a d^{2} g i^{2} x^{3}}{3} + \frac {B b c^{2} g i^{2} x^{2}}{2} + \frac {2 B b c d g i^{2} x^{3}}{3} + \frac {B b d^{2} g i^{2} x^{4}}{4}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c))),x)
Output:
A*b*d**2*g*i**2*x**4/4 + B*a**2*g*i**2*(a**2*d**2 - 4*a*b*c*d + 6*b**2*c** 2)*log(x + (B*a**4*c*d**3*g*i**2 - 4*B*a**3*b*c**2*d**2*g*i**2 + B*a**3*d* *2*g*i**2*(a**2*d**2 - 4*a*b*c*d + 6*b**2*c**2)/b + 10*B*a**2*b**2*c**3*d* g*i**2 - B*a**2*c*d*g*i**2*(a**2*d**2 - 4*a*b*c*d + 6*b**2*c**2) - B*a*b** 3*c**4*g*i**2)/(B*a**4*d**4*g*i**2 - 4*B*a**3*b*c*d**3*g*i**2 + 6*B*a**2*b **2*c**2*d**2*g*i**2 + 4*B*a*b**3*c**3*d*g*i**2 - B*b**4*c**4*g*i**2))/(12 *b**3) - B*c**3*g*i**2*(4*a*d - b*c)*log(x + (B*a**4*c*d**3*g*i**2 - 4*B*a **3*b*c**2*d**2*g*i**2 + 10*B*a**2*b**2*c**3*d*g*i**2 - B*a*b**3*c**4*g*i* *2 - B*a*b**2*c**3*g*i**2*(4*a*d - b*c) + B*b**3*c**4*g*i**2*(4*a*d - b*c) /d)/(B*a**4*d**4*g*i**2 - 4*B*a**3*b*c*d**3*g*i**2 + 6*B*a**2*b**2*c**2*d* *2*g*i**2 + 4*B*a*b**3*c**3*d*g*i**2 - B*b**4*c**4*g*i**2))/(12*d**2) + x* *3*(A*a*d**2*g*i**2/3 + 2*A*b*c*d*g*i**2/3 + B*a*d**2*g*i**2/12 - B*b*c*d* g*i**2/12) + x**2*(A*a*c*d*g*i**2 + A*b*c**2*g*i**2/2 + B*a**2*d**2*g*i**2 /(24*b) + B*a*c*d*g*i**2/6 - 5*B*b*c**2*g*i**2/24) + x*(A*a*c**2*g*i**2 - B*a**3*d**2*g*i**2/(12*b**2) + B*a**2*c*d*g*i**2/(3*b) - B*a*c**2*g*i**2/6 - B*b*c**3*g*i**2/(12*d)) + (B*a*c**2*g*i**2*x + B*a*c*d*g*i**2*x**2 + B* a*d**2*g*i**2*x**3/3 + B*b*c**2*g*i**2*x**2/2 + 2*B*b*c*d*g*i**2*x**3/3 + B*b*d**2*g*i**2*x**4/4)*log(e*(a + b*x)/(c + d*x))
Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (225) = 450\).
Time = 0.05 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.81 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {1}{4} \, A b d^{2} g i^{2} x^{4} + \frac {2}{3} \, A b c d g i^{2} x^{3} + \frac {1}{3} \, A a d^{2} g i^{2} x^{3} + \frac {1}{2} \, A b c^{2} g i^{2} x^{2} + A a c d g i^{2} x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a c^{2} g i^{2} + \frac {1}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B b c^{2} g i^{2} + {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a c d g i^{2} + \frac {1}{3} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b c d g i^{2} + \frac {1}{6} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a d^{2} g i^{2} + \frac {1}{24} \, {\left (6 \, x^{4} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b d^{2} g i^{2} + A a c^{2} g i^{2} x \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algori thm="maxima")
Output:
1/4*A*b*d^2*g*i^2*x^4 + 2/3*A*b*c*d*g*i^2*x^3 + 1/3*A*a*d^2*g*i^2*x^3 + 1/ 2*A*b*c^2*g*i^2*x^2 + A*a*c*d*g*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*c^2*g*i^2 + 1/2*(x^2*lo g(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*b*c^2*g*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*c*d*g*i^2 + 1/3*(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*b*c*d*g*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 *d^2*g*i^2 + 1/24*(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*d^2 *g*i^2 + A*a*c^2*g*i^2*x
Leaf count of result is larger than twice the leaf count of optimal. 1975 vs. \(2 (225) = 450\).
Time = 0.28 (sec) , antiderivative size = 1975, normalized size of antiderivative = 8.26 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:
integrate((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algori thm="giac")
Output:
-1/24*(2*(B*b^6*c^5*e^5*g*i^2 - 5*B*a*b^5*c^4*d*e^5*g*i^2 + 10*B*a^2*b^4*c ^3*d^2*e^5*g*i^2 - 10*B*a^3*b^3*c^2*d^3*e^5*g*i^2 + 5*B*a^4*b^2*c*d^4*e^5* g*i^2 - B*a^5*b*d^5*e^5*g*i^2 - 4*(b*e*x + a*e)*B*b^5*c^5*d*e^4*g*i^2/(d*x + c) + 20*(b*e*x + a*e)*B*a*b^4*c^4*d^2*e^4*g*i^2/(d*x + c) - 40*(b*e*x + a*e)*B*a^2*b^3*c^3*d^3*e^4*g*i^2/(d*x + c) + 40*(b*e*x + a*e)*B*a^3*b^2*c ^2*d^4*e^4*g*i^2/(d*x + c) - 20*(b*e*x + a*e)*B*a^4*b*c*d^5*e^4*g*i^2/(d*x + c) + 4*(b*e*x + a*e)*B*a^5*d^6*e^4*g*i^2/(d*x + c))*log((b*e*x + a*e)/( d*x + c))/(b^4*d^2*e^4 - 4*(b*e*x + a*e)*b^3*d^3*e^3/(d*x + c) + 6*(b*e*x + a*e)^2*b^2*d^4*e^2/(d*x + c)^2 - 4*(b*e*x + a*e)^3*b*d^5*e/(d*x + c)^3 + (b*e*x + a*e)^4*d^6/(d*x + c)^4) + (2*A*b^8*c^5*e^5*g*i^2 - B*b^8*c^5*e^5 *g*i^2 - 10*A*a*b^7*c^4*d*e^5*g*i^2 + 5*B*a*b^7*c^4*d*e^5*g*i^2 + 20*A*a^2 *b^6*c^3*d^2*e^5*g*i^2 - 10*B*a^2*b^6*c^3*d^2*e^5*g*i^2 - 20*A*a^3*b^5*c^2 *d^3*e^5*g*i^2 + 10*B*a^3*b^5*c^2*d^3*e^5*g*i^2 + 10*A*a^4*b^4*c*d^4*e^5*g *i^2 - 5*B*a^4*b^4*c*d^4*e^5*g*i^2 - 2*A*a^5*b^3*d^5*e^5*g*i^2 + B*a^5*b^3 *d^5*e^5*g*i^2 - 8*(b*e*x + a*e)*A*b^7*c^5*d*e^4*g*i^2/(d*x + c) + 6*(b*e* x + a*e)*B*b^7*c^5*d*e^4*g*i^2/(d*x + c) + 40*(b*e*x + a*e)*A*a*b^6*c^4*d^ 2*e^4*g*i^2/(d*x + c) - 30*(b*e*x + a*e)*B*a*b^6*c^4*d^2*e^4*g*i^2/(d*x + c) - 80*(b*e*x + a*e)*A*a^2*b^5*c^3*d^3*e^4*g*i^2/(d*x + c) + 60*(b*e*x + a*e)*B*a^2*b^5*c^3*d^3*e^4*g*i^2/(d*x + c) + 80*(b*e*x + a*e)*A*a^3*b^4*c^ 2*d^4*e^4*g*i^2/(d*x + c) - 60*(b*e*x + a*e)*B*a^3*b^4*c^2*d^4*e^4*g*i^...
Time = 26.63 (sec) , antiderivative size = 636, normalized size of antiderivative = 2.66 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=x^3\,\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{12}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{36}\right )-x^2\,\left (\frac {\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{24\,b\,d}-\frac {g\,i^2\,\left (3\,A\,a^2\,d^2+9\,A\,b^2\,c^2+B\,a^2\,d^2-2\,B\,b^2\,c^2+18\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\right )}{6\,b}+\frac {A\,a\,c\,d\,g\,i^2}{2}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,a\,c^2\,g\,i^2\,x+\frac {B\,c\,g\,i^2\,x^2\,\left (2\,a\,d+b\,c\right )}{2}+\frac {B\,d\,g\,i^2\,x^3\,\left (a\,d+2\,b\,c\right )}{3}+\frac {B\,b\,d^2\,g\,i^2\,x^4}{4}\right )+x\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{12\,b\,d}-\frac {g\,i^2\,\left (3\,A\,a^2\,d^2+9\,A\,b^2\,c^2+B\,a^2\,d^2-2\,B\,b^2\,c^2+18\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\right )}{3\,b}+A\,a\,c\,d\,g\,i^2\right )}{12\,b\,d}-\frac {a\,c\,\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{4}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{b\,d}+\frac {c\,g\,i^2\,\left (6\,A\,a^2\,d^2+2\,A\,b^2\,c^2+2\,B\,a^2\,d^2-B\,b^2\,c^2+12\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\right )}{2\,b\,d}\right )+\frac {\ln \left (a+b\,x\right )\,\left (B\,g\,a^4\,d^2\,i^2-4\,B\,g\,a^3\,b\,c\,d\,i^2+6\,B\,g\,a^2\,b^2\,c^2\,i^2\right )}{12\,b^3}+\frac {\ln \left (c+d\,x\right )\,\left (B\,b\,c^4\,g\,i^2-4\,B\,a\,c^3\,d\,g\,i^2\right )}{12\,d^2}+\frac {A\,b\,d^2\,g\,i^2\,x^4}{4} \] Input:
int((a*g + b*g*x)*(c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))),x)
Output:
x^3*((d*g*i^2*(8*A*a*d + 12*A*b*c + B*a*d - B*b*c))/12 - (A*d*g*i^2*(12*a* d + 12*b*c))/36) - x^2*((((d*g*i^2*(8*A*a*d + 12*A*b*c + B*a*d - B*b*c))/4 - (A*d*g*i^2*(12*a*d + 12*b*c))/12)*(12*a*d + 12*b*c))/(24*b*d) - (g*i^2* (3*A*a^2*d^2 + 9*A*b^2*c^2 + B*a^2*d^2 - 2*B*b^2*c^2 + 18*A*a*b*c*d + B*a* b*c*d))/(6*b) + (A*a*c*d*g*i^2)/2) + log((e*(a + b*x))/(c + d*x))*(B*a*c^2 *g*i^2*x + (B*c*g*i^2*x^2*(2*a*d + b*c))/2 + (B*d*g*i^2*x^3*(a*d + 2*b*c)) /3 + (B*b*d^2*g*i^2*x^4)/4) + x*(((12*a*d + 12*b*c)*((((d*g*i^2*(8*A*a*d + 12*A*b*c + B*a*d - B*b*c))/4 - (A*d*g*i^2*(12*a*d + 12*b*c))/12)*(12*a*d + 12*b*c))/(12*b*d) - (g*i^2*(3*A*a^2*d^2 + 9*A*b^2*c^2 + B*a^2*d^2 - 2*B* b^2*c^2 + 18*A*a*b*c*d + B*a*b*c*d))/(3*b) + A*a*c*d*g*i^2))/(12*b*d) - (a *c*((d*g*i^2*(8*A*a*d + 12*A*b*c + B*a*d - B*b*c))/4 - (A*d*g*i^2*(12*a*d + 12*b*c))/12))/(b*d) + (c*g*i^2*(6*A*a^2*d^2 + 2*A*b^2*c^2 + 2*B*a^2*d^2 - B*b^2*c^2 + 12*A*a*b*c*d - B*a*b*c*d))/(2*b*d)) + (log(a + b*x)*(B*a^4*d ^2*g*i^2 + 6*B*a^2*b^2*c^2*g*i^2 - 4*B*a^3*b*c*d*g*i^2))/(12*b^3) + (log(c + d*x)*(B*b*c^4*g*i^2 - 4*B*a*c^3*d*g*i^2))/(12*d^2) + (A*b*d^2*g*i^2*x^4 )/4
Time = 0.20 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.28 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g \left (2 a^{3} b \,d^{4} x -a^{2} b^{2} d^{4} x^{2}-2 a \,b^{3} d^{4} x^{3}+2 b^{4} c^{3} d x +5 b^{4} c^{2} d^{2} x^{2}+2 b^{4} c \,d^{3} x^{3}-24 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} c^{2} d^{2} x +8 \,\mathrm {log}\left (d x +c \right ) a^{3} b c \,d^{3}-12 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c^{2} d^{2}+8 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c^{3} d -8 a^{2} b^{2} c \,d^{3} x +4 a \,b^{3} c^{2} d^{2} x -4 a \,b^{3} c \,d^{3} x^{2}-8 a^{2} b^{2} d^{4} x^{3}-6 a \,b^{3} d^{4} x^{4}-12 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{2} c^{2} d^{2}-8 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} d^{4} x^{3}-16 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{4} c \,d^{3} x^{3}-24 a^{2} b^{2} c \,d^{3} x^{2}+8 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{3} b c \,d^{3}-12 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{4} c^{2} d^{2} x^{2}-24 a^{2} b^{2} c^{2} d^{2} x -12 a \,b^{3} c^{2} d^{2} x^{2}-16 a \,b^{3} c \,d^{3} x^{3}-24 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} c \,d^{3} x^{2}-6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{4} d^{4} x^{4}-2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{4} d^{4}-2 \,\mathrm {log}\left (d x +c \right ) a^{4} d^{4}-2 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{4}\right )}{24 b^{2} d^{2}} \] Input:
int((b*g*x+a*g)*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x)
Output:
(g*( - 2*log(c + d*x)*a**4*d**4 + 8*log(c + d*x)*a**3*b*c*d**3 - 12*log(c + d*x)*a**2*b**2*c**2*d**2 + 8*log(c + d*x)*a*b**3*c**3*d - 2*log(c + d*x) *b**4*c**4 - 2*log((a*e + b*e*x)/(c + d*x))*a**4*d**4 + 8*log((a*e + b*e*x )/(c + d*x))*a**3*b*c*d**3 - 12*log((a*e + b*e*x)/(c + d*x))*a**2*b**2*c** 2*d**2 - 24*log((a*e + b*e*x)/(c + d*x))*a*b**3*c**2*d**2*x - 24*log((a*e + b*e*x)/(c + d*x))*a*b**3*c*d**3*x**2 - 8*log((a*e + b*e*x)/(c + d*x))*a* b**3*d**4*x**3 - 12*log((a*e + b*e*x)/(c + d*x))*b**4*c**2*d**2*x**2 - 16* log((a*e + b*e*x)/(c + d*x))*b**4*c*d**3*x**3 - 6*log((a*e + b*e*x)/(c + d *x))*b**4*d**4*x**4 + 2*a**3*b*d**4*x - 24*a**2*b**2*c**2*d**2*x - 24*a**2 *b**2*c*d**3*x**2 - 8*a**2*b**2*c*d**3*x - 8*a**2*b**2*d**4*x**3 - a**2*b* *2*d**4*x**2 - 12*a*b**3*c**2*d**2*x**2 + 4*a*b**3*c**2*d**2*x - 16*a*b**3 *c*d**3*x**3 - 4*a*b**3*c*d**3*x**2 - 6*a*b**3*d**4*x**4 - 2*a*b**3*d**4*x **3 + 2*b**4*c**3*d*x + 5*b**4*c**2*d**2*x**2 + 2*b**4*c*d**3*x**3))/(24*b **2*d**2)