Integrand size = 42, antiderivative size = 343 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {4 b B^2 d (c+d x)}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 B^2 (c+d x)^2}{4 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {4 b B d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 B (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {2 b d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^3 g^3 i} \] Output:
4*b*B^2*d*(d*x+c)/(-a*d+b*c)^3/g^3/i/(b*x+a)-1/4*b^2*B^2*(d*x+c)^2/(-a*d+b *c)^3/g^3/i/(b*x+a)^2+4*b*B*d*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b* c)^3/g^3/i/(b*x+a)-1/2*b^2*B*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b *c)^3/g^3/i/(b*x+a)^2+2*b*d*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b* c)^3/g^3/i/(b*x+a)-1/2*b^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b *c)^3/g^3/i/(b*x+a)^2+1/3*d^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^3/B/(-a*d+b*c)^3 /g^3/i
Time = 1.08 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.93 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {-3 \left (2 A^2+2 A B+B^2\right ) (b c-a d)^2+6 \left (2 A^2+6 A B+7 B^2\right ) d (b c-a d) (a+b x)+6 \left (2 A^2+6 A B+7 B^2\right ) d^2 (a+b x)^2 \log (a+b x)-6 B (b c-a d) (-6 a A d-7 a B d+b B (c-6 d x)+2 A b (c-2 d x)) \log \left (\frac {e (a+b x)}{c+d x}\right )-6 B \left (-2 a^2 A d^2-4 a b d (A d x+B (c+d x))+b^2 \left (-2 A d^2 x^2+B \left (c^2-2 c d x-3 d^2 x^2\right )\right )\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+4 B^2 d^2 (a+b x)^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 \left (2 A^2+6 A B+7 B^2\right ) d^2 (a+b x)^2 \log (c+d x)}{12 (b c-a d)^3 g^3 i (a+b x)^2} \] Input:
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)^3*(c*i + d *i*x)),x]
Output:
(-3*(2*A^2 + 2*A*B + B^2)*(b*c - a*d)^2 + 6*(2*A^2 + 6*A*B + 7*B^2)*d*(b*c - a*d)*(a + b*x) + 6*(2*A^2 + 6*A*B + 7*B^2)*d^2*(a + b*x)^2*Log[a + b*x] - 6*B*(b*c - a*d)*(-6*a*A*d - 7*a*B*d + b*B*(c - 6*d*x) + 2*A*b*(c - 2*d* x))*Log[(e*(a + b*x))/(c + d*x)] - 6*B*(-2*a^2*A*d^2 - 4*a*b*d*(A*d*x + B* (c + d*x)) + b^2*(-2*A*d^2*x^2 + B*(c^2 - 2*c*d*x - 3*d^2*x^2)))*Log[(e*(a + b*x))/(c + d*x)]^2 + 4*B^2*d^2*(a + b*x)^2*Log[(e*(a + b*x))/(c + d*x)] ^3 - 6*(2*A^2 + 6*A*B + 7*B^2)*d^2*(a + b*x)^2*Log[c + d*x])/(12*(b*c - a* d)^3*g^3*i*(a + b*x)^2)
Time = 0.58 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 2795, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3}d\frac {a+b x}{c+d x}}{g^3 i (b c-a d)^3}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^3}{(a+b x)^3}-\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^2}{(a+b x)^2}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)}{a+b x}\right )d\frac {a+b x}{c+d x}}{g^3 i (b c-a d)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 (a+b x)^2}-\frac {b^2 B (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {d^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B}+\frac {2 b d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{a+b x}+\frac {4 b B d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {b^2 B^2 (c+d x)^2}{4 (a+b x)^2}+\frac {4 b B^2 d (c+d x)}{a+b x}}{g^3 i (b c-a d)^3}\) |
Input:
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)^3*(c*i + d*i*x)) ,x]
Output:
((4*b*B^2*d*(c + d*x))/(a + b*x) - (b^2*B^2*(c + d*x)^2)/(4*(a + b*x)^2) + (4*b*B*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) - (b^2 *B*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x)^2) + (2* b*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a + b*x) - (b^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(a + b*x)^2) + (d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B))/((b*c - a*d)^3*g^3*i)
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
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. \(766\) vs. \(2(335)=670\).
Time = 2.23 (sec) , antiderivative size = 767, normalized size of antiderivative = 2.24
| method | result | size |
| parts | \(\frac {A^{2} \left (\frac {d^{2} \ln \left (d x +c \right )}{\left (d a -b c \right )^{3}}+\frac {1}{2 \left (d a -b c \right ) \left (b x +a \right )^{2}}+\frac {d}{\left (d a -b c \right )^{2} \left (b x +a \right )}-\frac {d^{2} \ln \left (b x +a \right )}{\left (d a -b c \right )^{3}}\right )}{g^{3} i}-\frac {B^{2} \left (\frac {d^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 \left (d a -b c \right )^{3}}-\frac {2 d^{2} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{3}}+\frac {d \,b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{3}}\right )}{g^{3} i d}-\frac {2 A B \left (\frac {d^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (d a -b c \right )^{3}}-\frac {2 d^{2} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{3}}+\frac {d \,b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{3}}\right )}{g^{3} i d}\) | \(767\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} e \,A^{2} b^{2}}{2 i \left (d a -b c \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 d^{3} A^{2} b}{i \left (d a -b c \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{4} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (d a -b c \right )^{4} g^{3}}+\frac {2 d^{2} e A B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (d a -b c \right )^{4} g^{3}}-\frac {4 d^{3} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{4} g^{3}}+\frac {d^{4} A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (d a -b c \right )^{4} g^{3}}+\frac {d^{2} e \,B^{2} b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (d a -b c \right )^{4} g^{3}}-\frac {2 d^{3} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{4} g^{3}}+\frac {d^{4} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (d a -b c \right )^{4} g^{3}}\right )}{d^{2}}\) | \(882\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} e \,A^{2} b^{2}}{2 i \left (d a -b c \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 d^{3} A^{2} b}{i \left (d a -b c \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{4} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (d a -b c \right )^{4} g^{3}}+\frac {2 d^{2} e A B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (d a -b c \right )^{4} g^{3}}-\frac {4 d^{3} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{4} g^{3}}+\frac {d^{4} A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (d a -b c \right )^{4} g^{3}}+\frac {d^{2} e \,B^{2} b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (d a -b c \right )^{4} g^{3}}-\frac {2 d^{3} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{4} g^{3}}+\frac {d^{4} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (d a -b c \right )^{4} g^{3}}\right )}{d^{2}}\) | \(882\) |
| norman | \(\frac {\frac {6 A^{2} a \,b^{2} d -2 A^{2} b^{3} c +14 A B a \,b^{2} d -2 A B \,b^{3} c +15 B^{2} a \,b^{2} d -B^{2} b^{3} c}{4 g i \,b^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {\left (2 A^{2} a^{2} d^{2}+8 A B a b c d -2 A B \,b^{2} c^{2}+8 B^{2} a b c d -B^{2} b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {B \left (2 A \,a^{2} d^{2}+4 B a b c d -B \,b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (2 A^{2} b^{2} d +6 A B \,b^{2} d +7 B^{2} b^{2} d \right ) x}{2 b \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) i g}-\frac {B^{2} a^{2} d^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b \left (2 A^{2} a \,d^{2}+4 A B a \,d^{2}+2 A B b c d +4 B^{2} a \,d^{2}+3 B^{2} b c d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{2} B^{2} d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\left (2 A^{2} d^{2}+6 A B \,d^{2}+7 B^{2} d^{2}\right ) b^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g i \left (d a -b c \right ) \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {2 b \,B^{2} a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b d B \left (2 A d a +2 B a d +B b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{2} B \,d^{2} \left (2 A +3 B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i g \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 (b x +a \right )^{2} g^{2}}\) | \(938\) |
| risch | \(\frac {A^{2} d^{2} \ln \left (d x +c \right )}{g^{3} i \left (d a -b c \right )^{3}}+\frac {A^{2}}{2 g^{3} i \left (d a -b c \right ) \left (b x +a \right )^{2}}+\frac {A^{2} d}{g^{3} i \left (d a -b c \right )^{2} \left (b x +a \right )}-\frac {A^{2} d^{2} \ln \left (b x +a \right )}{g^{3} i \left (d a -b c \right )^{3}}-\frac {B^{2} d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 g^{3} i \left (d a -b c \right )^{3}}-\frac {2 B^{2} d b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {4 B^{2} d b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {4 B^{2} d b e}{g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}+\frac {B^{2} b^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}+\frac {B^{2} b^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}+\frac {B^{2} b^{2} e^{2}}{4 g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}-\frac {A B \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{3} i \left (d a -b c \right )^{3}}-\frac {4 A B d b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {4 A B d b e}{g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}+\frac {A B \,b^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}+\frac {A B \,b^{2} e^{2}}{2 g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}\) | \(986\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1122\) |
Input:
int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3/(d*i*x+c*i),x,method=_RETU RNVERBOSE)
Output:
A^2/g^3/i*(d^2/(a*d-b*c)^3*ln(d*x+c)+1/2/(a*d-b*c)/(b*x+a)^2+d/(a*d-b*c)^2 /(b*x+a)-d^2/(a*d-b*c)^3*ln(b*x+a))-B^2/g^3/i/d*(1/3*d^3/(a*d-b*c)^3*ln(b* e/d+(a*d-b*c)*e/d/(d*x+c))^3-2*d^2/(a*d-b*c)^3*b*e*(-1/(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))^2-2/(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))-2/(b*e/d+(a*d-b*c)*e/d/(d*x+c)))+d/(a*d -b*c)^3*b^2*e^2*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e /d/(d*x+c))^2-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/( d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2))-2*A*B/g^3/i/d*(1/2*d^3/(a*d- b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2*d^2/(a*d-b*c)^3*b*e*(-1/(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))-1/(b*e/d+(a*d-b*c)* e/d/(d*x+c)))+d/(a*d-b*c)^3*b^2*e^2*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2* ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2))
Time = 0.10 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.57 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {3 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - 24 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b c d + 3 \, {\left (6 \, A^{2} + 14 \, A B + 15 \, B^{2}\right )} a^{2} d^{2} - 4 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x + B^{2} a^{2} d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} - 6 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} d^{2} x^{2} - B^{2} b^{2} c^{2} + 4 \, B^{2} a b c d + 2 \, A B a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} c d + 2 \, {\left (A B + B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 6 \, {\left ({\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} a b d^{2}\right )} x - 6 \, {\left ({\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, A^{2} a^{2} d^{2} - {\left (2 \, A B + B^{2}\right )} b^{2} c^{2} + 8 \, {\left (A B + B^{2}\right )} a b c d + 2 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} c d + 2 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{12 \, {\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x + {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3/(d*i*x+c*i),x, algo rithm="fricas")
Output:
-1/12*(3*(2*A^2 + 2*A*B + B^2)*b^2*c^2 - 24*(A^2 + 2*A*B + 2*B^2)*a*b*c*d + 3*(6*A^2 + 14*A*B + 15*B^2)*a^2*d^2 - 4*(B^2*b^2*d^2*x^2 + 2*B^2*a*b*d^2 *x + B^2*a^2*d^2)*log((b*e*x + a*e)/(d*x + c))^3 - 6*((2*A*B + 3*B^2)*b^2* d^2*x^2 - B^2*b^2*c^2 + 4*B^2*a*b*c*d + 2*A*B*a^2*d^2 + 2*(B^2*b^2*c*d + 2 *(A*B + B^2)*a*b*d^2)*x)*log((b*e*x + a*e)/(d*x + c))^2 - 6*((2*A^2 + 6*A* B + 7*B^2)*b^2*c*d - (2*A^2 + 6*A*B + 7*B^2)*a*b*d^2)*x - 6*((2*A^2 + 6*A* B + 7*B^2)*b^2*d^2*x^2 + 2*A^2*a^2*d^2 - (2*A*B + B^2)*b^2*c^2 + 8*(A*B + B^2)*a*b*c*d + 2*((2*A*B + 3*B^2)*b^2*c*d + 2*(A^2 + 2*A*B + 2*B^2)*a*b*d^ 2)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3* c*d^2 - a^3*b^2*d^3)*g^3*i*x^2 + 2*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^ 2*c*d^2 - a^4*b*d^3)*g^3*i*x + (a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c* d^2 - a^5*d^3)*g^3*i)
Leaf count of result is larger than twice the leaf count of optimal. 1488 vs. \(2 (303) = 606\).
Time = 4.17 (sec) , antiderivative size = 1488, normalized size of antiderivative = 4.34 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=\text {Too large to display} \] Input:
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**3/(d*i*x+c*i),x)
Output:
-B**2*d**2*log(e*(a + b*x)/(c + d*x))**3/(3*a**3*d**3*g**3*i - 9*a**2*b*c* d**2*g**3*i + 9*a*b**2*c**2*d*g**3*i - 3*b**3*c**3*g**3*i) + d**2*(2*A**2 + 6*A*B + 7*B**2)*log(x + (2*A**2*a*d**3 + 2*A**2*b*c*d**2 + 6*A*B*a*d**3 + 6*A*B*b*c*d**2 + 7*B**2*a*d**3 + 7*B**2*b*c*d**2 - a**4*d**6*(2*A**2 + 6 *A*B + 7*B**2)/(a*d - b*c)**3 + 4*a**3*b*c*d**5*(2*A**2 + 6*A*B + 7*B**2)/ (a*d - b*c)**3 - 6*a**2*b**2*c**2*d**4*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b* c)**3 + 4*a*b**3*c**3*d**3*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3 - b**4 *c**4*d**2*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3)/(4*A**2*b*d**3 + 12*A *B*b*d**3 + 14*B**2*b*d**3))/(2*g**3*i*(a*d - b*c)**3) - d**2*(2*A**2 + 6* A*B + 7*B**2)*log(x + (2*A**2*a*d**3 + 2*A**2*b*c*d**2 + 6*A*B*a*d**3 + 6* A*B*b*c*d**2 + 7*B**2*a*d**3 + 7*B**2*b*c*d**2 + a**4*d**6*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3 - 4*a**3*b*c*d**5*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3 + 6*a**2*b**2*c**2*d**4*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)** 3 - 4*a*b**3*c**3*d**3*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3 + b**4*c** 4*d**2*(2*A**2 + 6*A*B + 7*B**2)/(a*d - b*c)**3)/(4*A**2*b*d**3 + 12*A*B*b *d**3 + 14*B**2*b*d**3))/(2*g**3*i*(a*d - b*c)**3) + (6*A*B*a*d - 2*A*B*b* c + 4*A*B*b*d*x + 7*B**2*a*d - B**2*b*c + 6*B**2*b*d*x)*log(e*(a + b*x)/(c + d*x))/(2*a**4*d**2*g**3*i - 4*a**3*b*c*d*g**3*i + 4*a**3*b*d**2*g**3*i* x + 2*a**2*b**2*c**2*g**3*i - 8*a**2*b**2*c*d*g**3*i*x + 2*a**2*b**2*d**2* g**3*i*x**2 + 4*a*b**3*c**2*g**3*i*x - 4*a*b**3*c*d*g**3*i*x**2 + 2*b**...
Leaf count of result is larger than twice the leaf count of optimal. 2115 vs. \(2 (335) = 670\).
Time = 0.19 (sec) , antiderivative size = 2115, normalized size of antiderivative = 6.17 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=\text {Too large to display} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3/(d*i*x+c*i),x, algo rithm="maxima")
Output:
1/2*B^2*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^ 3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2* c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^3*c^3 - 3 *a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i))*log(b*e*x/(d*x + c) + a*e/ (d*x + c))^2 + A*B*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2* b^2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + ( a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/(( b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 1/12*B^2*(6*(b^2*c^2 - 8*a*b*c*d + 7*a^2*d^2 + 2*( b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*a *b*d^2*x + a^2*d^2)*log(d*x + c)^2 - 6*(b^2*c*d - a*b*d^2)*x - 6*(b^2*d^2* x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 2*(3*b^2*d^2*x^2 + 6*a*b*d^2*x + 3*a^2*d^2 - 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a))*log(d *x + c))*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(a^2*b^3*c^3*g^3*i - 3*a^3*b ^2*c^2*d*g^3*i + 3*a^4*b*c*d^2*g^3*i - a^5*d^3*g^3*i + (b^5*c^3*g^3*i - 3* a*b^4*c^2*d*g^3*i + 3*a^2*b^3*c*d^2*g^3*i - a^3*b^2*d^3*g^3*i)*x^2 + 2*(a* b^4*c^3*g^3*i - 3*a^2*b^3*c^2*d*g^3*i + 3*a^3*b^2*c*d^2*g^3*i - a^4*b*d^3* g^3*i)*x) + (3*b^2*c^2 - 48*a*b*c*d + 45*a^2*d^2 - 4*(b^2*d^2*x^2 + 2*a...
Time = 57.21 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.62 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (d x + c\right )}^{2} B^{2} e^{3} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{{\left (b e x + a e\right )}^{2} g^{3} i} + \frac {2 \, {\left (2 \, A B e^{3} + B^{2} e^{3}\right )} {\left (d x + c\right )}^{2} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{2} g^{3} i} + \frac {{\left (2 \, A^{2} e^{3} + 2 \, A B e^{3} + B^{2} e^{3}\right )} {\left (d x + c\right )}^{2}}{{\left (b e x + a e\right )}^{2} g^{3} i}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3/(d*i*x+c*i),x, algo rithm="giac")
Output:
-1/4*(2*(d*x + c)^2*B^2*e^3*log((b*e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^ 2*g^3*i) + 2*(2*A*B*e^3 + B^2*e^3)*(d*x + c)^2*log((b*e*x + a*e)/(d*x + c) )/((b*e*x + a*e)^2*g^3*i) + (2*A^2*e^3 + 2*A*B*e^3 + B^2*e^3)*(d*x + c)^2/ ((b*e*x + a*e)^2*g^3*i))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))^2
Time = 29.85 (sec) , antiderivative size = 981, normalized size of antiderivative = 2.86 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx =\text {Too large to display} \] Input:
int((A + B*log((e*(a + b*x))/(c + d*x)))^2/((a*g + b*g*x)^3*(c*i + d*i*x)) ,x)
Output:
log((e*(a + b*x))/(c + d*x))^2*(((B^2*d^2*((2*a^2*d^2 + b^2*c^2 - 3*a*b*c* d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)))/(g^3*i*(a^3*d^3 - b^3*c^3 + 3*a *b^2*c^2*d - 3*a^2*b*c*d^2)) + (B^2*x*(a*d - b*c))/(g^3*i*(a^3*d^3 - b^3*c ^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/((b*x^2)/d + a^2/(b*d) + (2*a*x)/d) - (B*d^2*(2*A + 3*B))/(2*g^3*i*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2* b*c*d^2))) - ((6*A^2*a*d - 2*A^2*b*c + 15*B^2*a*d - B^2*b*c + 14*A*B*a*d - 2*A*B*b*c)/(2*(a*d - b*c)) + (x*(2*A^2*b*d + 7*B^2*b*d + 6*A*B*b*d))/(a*d - b*c))/(x^2*(2*b^3*c*g^3*i - 2*a*b^2*d*g^3*i) + x*(4*a*b^2*c*g^3*i - 4*a ^2*b*d*g^3*i) - 2*a^3*d*g^3*i + 2*a^2*b*c*g^3*i) + (log((e*(a + b*x))/(c + d*x))*((B*d^2*((2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b* c))/(2*b*d^2))*(2*A + 3*B))/(g^3*i*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3* a^2*b*c*d^2)) - B^2/(b*d*g^3*i*(a*d - b*c)) + (B*x*(2*A + 3*B)*(a*d - b*c) )/(g^3*i*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((b*x^2)/d + a^2/(b*d) + (2*a*x)/d) - (B^2*d^2*log((e*(a + b*x))/(c + d*x))^3)/(3*g^ 3*i*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (d^2*atan((d^2* (A^2 + (7*B^2)/2 + 3*A*B)*(2*a^3*d^3*g^3*i + 2*b^3*c^3*g^3*i - 2*a*b^2*c^2 *d*g^3*i - 2*a^2*b*c*d^2*g^3*i)*1i)/(g^3*i*(a*d - b*c)^3*(2*A^2*d^2 + 7*B^ 2*d^2 + 6*A*B*d^2)) + (b*d^3*x*(a^2*d^2*g^3*i + b^2*c^2*g^3*i - 2*a*b*c*d* g^3*i)*(A^2 + (7*B^2)/2 + 3*A*B)*4i)/(g^3*i*(a*d - b*c)^3*(2*A^2*d^2 + 7*B ^2*d^2 + 6*A*B*d^2)))*(A^2 + (7*B^2)/2 + 3*A*B)*2i)/(g^3*i*(a*d - b*c)^...
Time = 0.21 (sec) , antiderivative size = 1426, normalized size of antiderivative = 4.16 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx =\text {Too large to display} \] Input:
int((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3/(d*i*x+c*i),x)
Output:
(i*(12*log(a + b*x)*a**5*d**2 + 24*log(a + b*x)*a**4*b*d**2*x + 24*log(a + b*x)*a**4*b*d**2 + 12*log(a + b*x)*a**3*b**2*c*d + 12*log(a + b*x)*a**3*b **2*d**2*x**2 + 48*log(a + b*x)*a**3*b**2*d**2*x + 24*log(a + b*x)*a**3*b* *2*d**2 + 24*log(a + b*x)*a**2*b**3*c*d*x + 18*log(a + b*x)*a**2*b**3*c*d + 24*log(a + b*x)*a**2*b**3*d**2*x**2 + 48*log(a + b*x)*a**2*b**3*d**2*x + 12*log(a + b*x)*a*b**4*c*d*x**2 + 36*log(a + b*x)*a*b**4*c*d*x + 24*log(a + b*x)*a*b**4*d**2*x**2 + 18*log(a + b*x)*b**5*c*d*x**2 - 12*log(c + d*x) *a**5*d**2 - 24*log(c + d*x)*a**4*b*d**2*x - 24*log(c + d*x)*a**4*b*d**2 - 12*log(c + d*x)*a**3*b**2*c*d - 12*log(c + d*x)*a**3*b**2*d**2*x**2 - 48* log(c + d*x)*a**3*b**2*d**2*x - 24*log(c + d*x)*a**3*b**2*d**2 - 24*log(c + d*x)*a**2*b**3*c*d*x - 18*log(c + d*x)*a**2*b**3*c*d - 24*log(c + d*x)*a **2*b**3*d**2*x**2 - 48*log(c + d*x)*a**2*b**3*d**2*x - 12*log(c + d*x)*a* b**4*c*d*x**2 - 36*log(c + d*x)*a*b**4*c*d*x - 24*log(c + d*x)*a*b**4*d**2 *x**2 - 18*log(c + d*x)*b**5*c*d*x**2 + 4*log((a*e + b*e*x)/(c + d*x))**3* a**3*b**2*d**2 + 8*log((a*e + b*e*x)/(c + d*x))**3*a**2*b**3*d**2*x + 4*lo g((a*e + b*e*x)/(c + d*x))**3*a*b**4*d**2*x**2 + 12*log((a*e + b*e*x)/(c + d*x))**2*a**4*b*d**2 + 24*log((a*e + b*e*x)/(c + d*x))**2*a**3*b**2*d**2* x + 24*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**3*c*d + 12*log((a*e + b*e*x )/(c + d*x))**2*a**2*b**3*d**2*x**2 + 24*log((a*e + b*e*x)/(c + d*x))**2*a **2*b**3*d**2*x - 6*log((a*e + b*e*x)/(c + d*x))**2*a*b**4*c**2 + 12*lo...