Integrand size = 42, antiderivative size = 183 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {2 b B^2 (c+d x)}{(b c-a d)^2 g^2 i (a+b x)}-\frac {2 b B (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^2 i (a+b x)}-\frac {b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g^2 i (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^2 g^2 i} \] Output:
-2*b*B^2*(d*x+c)/(-a*d+b*c)^2/g^2/i/(b*x+a)-2*b*B*(d*x+c)*(A+B*ln(e*(b*x+a )/(d*x+c)))/(-a*d+b*c)^2/g^2/i/(b*x+a)-b*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c) ))^2/(-a*d+b*c)^2/g^2/i/(b*x+a)-1/3*d*(A+B*ln(e*(b*x+a)/(d*x+c)))^3/B/(-a* d+b*c)^2/g^2/i
Time = 0.70 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.02 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {3 \left (A^2+2 A B+2 B^2\right ) d (a+b x) \log (a+b x)+6 B (A+B) (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 B (a A d+A b d x+b B (c+d x)) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+B^2 d (a+b x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )+3 \left (A^2+2 A B+2 B^2\right ) (b c-a d-d (a+b x) \log (c+d x))}{3 (b c-a d)^2 g^2 i (a+b x)} \] Input:
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)^2*(c*i + d *i*x)),x]
Output:
-1/3*(3*(A^2 + 2*A*B + 2*B^2)*d*(a + b*x)*Log[a + b*x] + 6*B*(A + B)*(b*c - a*d)*Log[(e*(a + b*x))/(c + d*x)] + 3*B*(a*A*d + A*b*d*x + b*B*(c + d*x) )*Log[(e*(a + b*x))/(c + d*x)]^2 + B^2*d*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)]^3 + 3*(A^2 + 2*A*B + 2*B^2)*(b*c - a*d - d*(a + b*x)*Log[c + d*x]))/ ((b*c - a*d)^2*g^2*i*(a + b*x))
Time = 0.47 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.74, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {\int \frac {(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2}d\frac {a+b x}{c+d x}}{g^2 i (b c-a d)^2}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2}-\frac {d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x}\right )d\frac {a+b x}{c+d x}}{g^2 i (b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B}-\frac {b (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{a+b x}-\frac {2 b B (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {2 b B^2 (c+d x)}{a+b x}}{g^2 i (b c-a d)^2}\) |
Input:
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)^2*(c*i + d*i*x)) ,x]
Output:
((-2*b*B^2*(c + d*x))/(a + b*x) - (2*b*B*(c + d*x)*(A + B*Log[(e*(a + b*x) )/(c + d*x)]))/(a + b*x) - (b*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x) ])^2)/(a + b*x) - (d*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B))/((b*c - a*d)^2*g^2*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. \(399\) vs. \(2(181)=362\).
Time = 1.80 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.19
| method | result | size |
| norman | \(\frac {-\frac {\left (A^{2} a d +2 A B b c +2 B^{2} b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {B \left (A d a +B b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {B^{2} a d \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {b \left (A^{2} d +2 A B d +2 B^{2} d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {\left (A^{2}+2 A B +2 B^{2}\right ) b x}{g i a \left (d a -b c \right )}-\frac {b \,B^{2} d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {B b d \left (A +B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}}{g \left (b x +a \right )}\) | \(400\) |
| parts | \(\frac {A^{2} \left (\frac {d \ln \left (d x +c \right )}{\left (d a -b c \right )^{2}}+\frac {1}{\left (b x +a \right ) \left (d a -b c \right )}-\frac {d \ln \left (b x +a \right )}{\left (d a -b c \right )^{2}}\right )}{g^{2} i}-\frac {B^{2} \left (\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 \left (d a -b c \right )^{2}}-\frac {d b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{2}}\right )}{g^{2} i d}-\frac {2 A B \left (\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (d a -b c \right )^{2}}-\frac {d b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{2}}\right )}{g^{2} i d}\) | \(460\) |
| parallelrisch | \(-\frac {3 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{4} c^{2} d +3 A^{2} x \,a^{3} b \,c^{2} d -6 A B x \,a^{2} b^{2} c^{3}+6 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{3}+6 B^{2} x \,a^{3} b \,c^{2} d +3 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b \,c^{2} d +6 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d +B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} a^{4} c^{2} d +3 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b \,c^{3}-3 A^{2} x \,a^{2} b^{2} c^{3}+3 A^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} c^{2} d -6 B^{2} x \,a^{2} b^{2} c^{3}+6 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{3}+B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} a^{3} b \,c^{2} d +3 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{3} b \,c^{2} d +3 A^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d +6 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d +6 A B x \,a^{3} b \,c^{2} d}{3 i \,g^{2} \left (b x +a \right ) \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) a^{3} c^{2}}\) | \(473\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A^{2} b}{i \left (d a -b c \right )^{3} g^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{3} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (d a -b c \right )^{3} g^{2}}-\frac {2 d^{2} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{3} g^{2}}+\frac {d^{3} A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (d a -b c \right )^{3} g^{2}}-\frac {d^{2} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{3} g^{2}}+\frac {d^{3} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (d a -b c \right )^{3} g^{2}}\right )}{d^{2}}\) | \(527\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A^{2} b}{i \left (d a -b c \right )^{3} g^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{3} A^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (d a -b c \right )^{3} g^{2}}-\frac {2 d^{2} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{3} g^{2}}+\frac {d^{3} A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e i \left (d a -b c \right )^{3} g^{2}}-\frac {d^{2} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{3} g^{2}}+\frac {d^{3} B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e i \left (d a -b c \right )^{3} g^{2}}\right )}{d^{2}}\) | \(527\) |
| risch | \(\frac {A^{2} d \ln \left (d x +c \right )}{g^{2} i \left (d a -b c \right )^{2}}+\frac {A^{2}}{g^{2} i \left (b x +a \right ) \left (d a -b c \right )}-\frac {A^{2} d \ln \left (b x +a \right )}{g^{2} i \left (d a -b c \right )^{2}}-\frac {B^{2} d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 g^{2} i \left (d a -b c \right )^{2}}-\frac {B^{2} b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{2} i \left (d a -b c \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {2 B^{2} b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i \left (d a -b c \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {2 B^{2} b e}{g^{2} i \left (d a -b c \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {A B d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{2} i \left (d a -b c \right )^{2}}-\frac {2 A B b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i \left (d a -b c \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {2 A B b e}{g^{2} i \left (d a -b c \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}\) | \(556\) |
Input:
int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^2/(d*i*x+c*i),x,method=_RETU RNVERBOSE)
Output:
(-(A^2*a*d+2*A*B*b*c+2*B^2*b*c)/g/i/(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(e*(b*x+ a)/(d*x+c))-B*(A*a*d+B*b*c)/g/i/(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(e*(b*x+a)/( d*x+c))^2-1/3*B^2*a*d/g/i/(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(e*(b*x+a)/(d*x+c) )^3-1/g/i*b*(A^2*d+2*A*B*d+2*B^2*d)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*ln(e*(b* x+a)/(d*x+c))-(A^2+2*A*B+2*B^2)*b/g/i/a/(a*d-b*c)*x-1/3*b*B^2*d/g/i/(a^2*d ^2-2*a*b*c*d+b^2*c^2)*x*ln(e*(b*x+a)/(d*x+c))^3-B*b*d*(A+B)/g/i/(a^2*d^2-2 *a*b*c*d+b^2*c^2)*x*ln(e*(b*x+a)/(d*x+c))^2)/g/(b*x+a)
Time = 0.08 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.26 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=-\frac {{\left (B^{2} b d x + B^{2} a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b c - 3 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a d + 3 \, {\left (B^{2} b c + A B a d + {\left (A B + B^{2}\right )} b d x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, {\left (A^{2} a d + {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b d x + 2 \, {\left (A B + B^{2}\right )} b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{2} i x + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} g^{2} i\right )}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^2/(d*i*x+c*i),x, algo rithm="fricas")
Output:
-1/3*((B^2*b*d*x + B^2*a*d)*log((b*e*x + a*e)/(d*x + c))^3 + 3*(A^2 + 2*A* B + 2*B^2)*b*c - 3*(A^2 + 2*A*B + 2*B^2)*a*d + 3*(B^2*b*c + A*B*a*d + (A*B + B^2)*b*d*x)*log((b*e*x + a*e)/(d*x + c))^2 + 3*(A^2*a*d + (A^2 + 2*A*B + 2*B^2)*b*d*x + 2*(A*B + B^2)*b*c)*log((b*e*x + a*e)/(d*x + c)))/((b^3*c^ 2 - 2*a*b^2*c*d + a^2*b*d^2)*g^2*i*x + (a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2) *g^2*i)
Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (158) = 316\).
Time = 0.72 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.96 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=- \frac {B^{2} d \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a^{2} d^{2} g^{2} i - 6 a b c d g^{2} i + 3 b^{2} c^{2} g^{2} i} + \frac {\left (2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a^{2} d g^{2} i - a b c g^{2} i + a b d g^{2} i x - b^{2} c g^{2} i x} + \left (A^{2} + 2 A B + 2 B^{2}\right ) \left (\frac {d \log {\left (x + \frac {- \frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} - \frac {d \log {\left (x + \frac {\frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} + \frac {1}{a^{2} d g^{2} i - a b c g^{2} i + x \left (a b d g^{2} i - b^{2} c g^{2} i\right )}\right ) + \frac {\left (- A B a d - A B b d x - B^{2} b c - B^{2} b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{3} d^{2} g^{2} i - 2 a^{2} b c d g^{2} i + a^{2} b d^{2} g^{2} i x + a b^{2} c^{2} g^{2} i - 2 a b^{2} c d g^{2} i x + b^{3} c^{2} g^{2} i x} \] Input:
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**2/(d*i*x+c*i),x)
Output:
-B**2*d*log(e*(a + b*x)/(c + d*x))**3/(3*a**2*d**2*g**2*i - 6*a*b*c*d*g**2 *i + 3*b**2*c**2*g**2*i) + (2*A*B + 2*B**2)*log(e*(a + b*x)/(c + d*x))/(a* *2*d*g**2*i - a*b*c*g**2*i + a*b*d*g**2*i*x - b**2*c*g**2*i*x) + (A**2 + 2 *A*B + 2*B**2)*(d*log(x + (-a**3*d**4/(a*d - b*c)**2 + 3*a**2*b*c*d**3/(a* d - b*c)**2 - 3*a*b**2*c**2*d**2/(a*d - b*c)**2 + a*d**2 + b**3*c**3*d/(a* d - b*c)**2 + b*c*d)/(2*b*d**2))/(g**2*i*(a*d - b*c)**2) - d*log(x + (a**3 *d**4/(a*d - b*c)**2 - 3*a**2*b*c*d**3/(a*d - b*c)**2 + 3*a*b**2*c**2*d**2 /(a*d - b*c)**2 + a*d**2 - b**3*c**3*d/(a*d - b*c)**2 + b*c*d)/(2*b*d**2)) /(g**2*i*(a*d - b*c)**2) + 1/(a**2*d*g**2*i - a*b*c*g**2*i + x*(a*b*d*g**2 *i - b**2*c*g**2*i))) + (-A*B*a*d - A*B*b*d*x - B**2*b*c - B**2*b*d*x)*log (e*(a + b*x)/(c + d*x))**2/(a**3*d**2*g**2*i - 2*a**2*b*c*d*g**2*i + a**2* b*d**2*g**2*i*x + a*b**2*c**2*g**2*i - 2*a*b**2*c*d*g**2*i*x + b**3*c**2*g **2*i*x)
Leaf count of result is larger than twice the leaf count of optimal. 1008 vs. \(2 (181) = 362\).
Time = 0.10 (sec) , antiderivative size = 1008, normalized size of antiderivative = 5.51 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx =\text {Too large to display} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^2/(d*i*x+c*i),x, algo rithm="maxima")
Output:
-B^2*(1/((b^2*c - a*b*d)*g^2*i*x + (a*b*c - a^2*d)*g^2*i) + d*log(b*x + a) /((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*g^2*i) - d*log(d*x + c)/((b^2*c^2 - 2*a* b*c*d + a^2*d^2)*g^2*i))*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2 - 2*A*B*(1 /((b^2*c - a*b*d)*g^2*i*x + (a*b*c - a^2*d)*g^2*i) + d*log(b*x + a)/((b^2* c^2 - 2*a*b*c*d + a^2*d^2)*g^2*i) - d*log(d*x + c)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*g^2*i))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 1/3*B^2*(3*((b*d* x + a*d)*log(b*x + a)^2 + (b*d*x + a*d)*log(d*x + c)^2 - 2*b*c + 2*a*d - 2 *(b*d*x + a*d)*log(b*x + a) + 2*(b*d*x + a*d - (b*d*x + a*d)*log(b*x + a)) *log(d*x + c))*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(a*b^2*c^2*g^2*i - 2*a ^2*b*c*d*g^2*i + a^3*d^2*g^2*i + (b^3*c^2*g^2*i - 2*a*b^2*c*d*g^2*i + a^2* b*d^2*g^2*i)*x) - ((b*d*x + a*d)*log(b*x + a)^3 - (b*d*x + a*d)*log(d*x + c)^3 - 3*(b*d*x + a*d)*log(b*x + a)^2 - 3*(b*d*x + a*d - (b*d*x + a*d)*log (b*x + a))*log(d*x + c)^2 + 6*b*c - 6*a*d + 6*(b*d*x + a*d)*log(b*x + a) - 3*(2*b*d*x + (b*d*x + a*d)*log(b*x + a)^2 + 2*a*d - 2*(b*d*x + a*d)*log(b *x + a))*log(d*x + c))/(a*b^2*c^2*g^2*i - 2*a^2*b*c*d*g^2*i + a^3*d^2*g^2* i + (b^3*c^2*g^2*i - 2*a*b^2*c*d*g^2*i + a^2*b*d^2*g^2*i)*x)) - A^2*(1/((b ^2*c - a*b*d)*g^2*i*x + (a*b*c - a^2*d)*g^2*i) + d*log(b*x + a)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*g^2*i) - d*log(d*x + c)/((b^2*c^2 - 2*a*b*c*d + a^2 *d^2)*g^2*i)) + ((b*d*x + a*d)*log(b*x + a)^2 + (b*d*x + a*d)*log(d*x + c) ^2 - 2*b*c + 2*a*d - 2*(b*d*x + a*d)*log(b*x + a) + 2*(b*d*x + a*d - (b...
\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )}} \,d x } \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^2/(d*i*x+c*i),x, algo rithm="giac")
Output:
integrate((B*log((b*x + a)*e/(d*x + c)) + A)^2/((b*g*x + a*g)^2*(d*i*x + c *i)), x)
Time = 28.04 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.29 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=\frac {A^2+2\,A\,B+2\,B^2}{\left (a\,d-b\,c\right )\,\left (a\,g^2\,i+b\,g^2\,i\,x\right )}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B\,d\,\left (A+B\right )}{g^2\,i\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,\left (a\,d-b\,c\right )}{b\,d\,g^2\,i\,\left (\frac {x}{d}+\frac {a}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {B^2\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g^2\,i\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {2\,B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,d-b\,c\right )\,\left (A+B\right )}{b\,d\,g^2\,i\,\left (\frac {x}{d}+\frac {a}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,\mathrm {atan}\left (\frac {d\,\left (2\,b\,d\,x+\frac {a^2\,d^2\,g^2\,i-b^2\,c^2\,g^2\,i}{g^2\,i\,\left (a\,d-b\,c\right )}\right )\,\left (A^2+2\,A\,B+2\,B^2\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (d\,A^2+2\,d\,A\,B+2\,d\,B^2\right )}\right )\,\left (A^2+2\,A\,B+2\,B^2\right )\,2{}\mathrm {i}}{g^2\,i\,{\left (a\,d-b\,c\right )}^2} \] Input:
int((A + B*log((e*(a + b*x))/(c + d*x)))^2/((a*g + b*g*x)^2*(c*i + d*i*x)) ,x)
Output:
(A^2 + 2*B^2 + 2*A*B)/((a*d - b*c)*(a*g^2*i + b*g^2*i*x)) - log((e*(a + b* x))/(c + d*x))^2*((B*d*(A + B))/(g^2*i*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (B^2*(a*d - b*c))/(b*d*g^2*i*(x/d + a/(b*d))*(a^2*d^2 + b^2*c^2 - 2*a*b*c* d))) - (B^2*d*log((e*(a + b*x))/(c + d*x))^3)/(3*g^2*i*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (d*atan((d*(2*b*d*x + (a^2*d^2*g^2*i - b^2*c^2*g^2*i)/(g^2 *i*(a*d - b*c)))*(A^2 + 2*B^2 + 2*A*B)*1i)/((a*d - b*c)*(A^2*d + 2*B^2*d + 2*A*B*d)))*(A^2 + 2*B^2 + 2*A*B)*2i)/(g^2*i*(a*d - b*c)^2) + (2*B*log((e* (a + b*x))/(c + d*x))*(a*d - b*c)*(A + B))/(b*d*g^2*i*(x/d + a/(b*d))*(a^2 *d^2 + b^2*c^2 - 2*a*b*c*d))
Time = 0.21 (sec) , antiderivative size = 537, normalized size of antiderivative = 2.93 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)} \, dx=\frac {i \left (3 \,\mathrm {log}\left (b x +a \right ) a^{4} d -3 \,\mathrm {log}\left (d x +c \right ) a^{4} d -6 b^{4} c x +3 \,\mathrm {log}\left (b x +a \right ) a^{3} b d x +6 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} c x -3 \,\mathrm {log}\left (d x +c \right ) a^{3} b d x -6 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c x +3 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a^{2} b^{2} d x +3 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a \,b^{3} d x +6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{2} d x -6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} c x +6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} d x +\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{3} a \,b^{3} d x +\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{3} a^{2} b^{2} d +6 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} c +6 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} c +6 \,\mathrm {log}\left (b x +a \right ) b^{4} c x -6 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c -6 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c -6 \,\mathrm {log}\left (d x +c \right ) b^{4} c x +3 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a^{3} b d +3 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a \,b^{3} c -6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{4} c x +3 a^{3} b d x -3 a^{2} b^{2} c x +6 a^{2} b^{2} d x -6 a \,b^{3} c x +6 a \,b^{3} d x \right )}{3 a \,g^{2} \left (a^{2} b \,d^{2} x -2 a \,b^{2} c d x +b^{3} c^{2} x +a^{3} d^{2}-2 a^{2} b c d +a \,b^{2} c^{2}\right )} \] Input:
int((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^2/(d*i*x+c*i),x)
Output:
(i*(3*log(a + b*x)*a**4*d + 3*log(a + b*x)*a**3*b*d*x + 6*log(a + b*x)*a** 2*b**2*c + 6*log(a + b*x)*a*b**3*c*x + 6*log(a + b*x)*a*b**3*c + 6*log(a + b*x)*b**4*c*x - 3*log(c + d*x)*a**4*d - 3*log(c + d*x)*a**3*b*d*x - 6*log (c + d*x)*a**2*b**2*c - 6*log(c + d*x)*a*b**3*c*x - 6*log(c + d*x)*a*b**3* c - 6*log(c + d*x)*b**4*c*x + log((a*e + b*e*x)/(c + d*x))**3*a**2*b**2*d + log((a*e + b*e*x)/(c + d*x))**3*a*b**3*d*x + 3*log((a*e + b*e*x)/(c + d* x))**2*a**3*b*d + 3*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**2*d*x + 3*log( (a*e + b*e*x)/(c + d*x))**2*a*b**3*c + 3*log((a*e + b*e*x)/(c + d*x))**2*a *b**3*d*x + 6*log((a*e + b*e*x)/(c + d*x))*a**2*b**2*d*x - 6*log((a*e + b* e*x)/(c + d*x))*a*b**3*c*x + 6*log((a*e + b*e*x)/(c + d*x))*a*b**3*d*x - 6 *log((a*e + b*e*x)/(c + d*x))*b**4*c*x + 3*a**3*b*d*x - 3*a**2*b**2*c*x + 6*a**2*b**2*d*x - 6*a*b**3*c*x + 6*a*b**3*d*x - 6*b**4*c*x))/(3*a*g**2*(a* *3*d**2 - 2*a**2*b*c*d + a**2*b*d**2*x + a*b**2*c**2 - 2*a*b**2*c*d*x + b* *3*c**2*x))