Integrand size = 32, antiderivative size = 153 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^2} \, dx=\frac {2 A B (c+d x)}{(b c-a d) g^2 (a+b x)}-\frac {2 B^2 (c+d x)}{(b c-a d) g^2 (a+b x)}+\frac {2 B^2 (c+d x) \log \left (\frac {e (c+d x)}{a+b x}\right )}{(b c-a d) g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(b c-a d) g^2 (a+b x)} \] Output:
2*A*B*(d*x+c)/(-a*d+b*c)/g^2/(b*x+a)-2*B^2*(d*x+c)/(-a*d+b*c)/g^2/(b*x+a)+ 2*B^2*(d*x+c)*ln(e*(d*x+c)/(b*x+a))/(-a*d+b*c)/g^2/(b*x+a)-(d*x+c)*(A+B*ln (e*(d*x+c)/(b*x+a)))^2/(-a*d+b*c)/g^2/(b*x+a)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.46 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.05 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2+\frac {B \left (2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-2 (b c-a d) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-2 d (a+b x) \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+2 d (a+b x) \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+B d (a+b x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b c-a d}}{b g^2 (a+b x)} \] Input:
Integrate[(A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(a*g + b*g*x)^2,x]
Output:
-(((A + B*Log[(e*(c + d*x))/(a + b*x)])^2 + (B*(2*B*(b*c - a*d + d*(a + b* x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - 2*(b*c - a*d)*(A + B*Log[(e* (c + d*x))/(a + b*x)]) - 2*d*(a + b*x)*Log[a + b*x]*(A + B*Log[(e*(c + d*x ))/(a + b*x)]) + 2*d*(a + b*x)*Log[c + d*x]*(A + B*Log[(e*(c + d*x))/(a + b*x)]) - B*d*(a + b*x)*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/( b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + B*d*(a + b*x) *((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*Po lyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(b*c - a*d))/(b*g^2*(a + b*x)))
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.73, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2952, 2733, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{(a g+b g x)^2} \, dx\) |
| \(\Big \downarrow \) 2952 |
| \(\displaystyle -\frac {\int \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2d\frac {c+d x}{a+b x}}{g^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2733 |
| \(\displaystyle -\frac {\frac {(c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{a+b x}-2 B \int \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )d\frac {c+d x}{a+b x}}{g^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {(c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{a+b x}-2 B \left (\frac {A (c+d x)}{a+b x}+\frac {B (c+d x) \log \left (\frac {e (c+d x)}{a+b x}\right )}{a+b x}-\frac {B (c+d x)}{a+b x}\right )}{g^2 (b c-a d)}\) |
Input:
Int[(A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(a*g + b*g*x)^2,x]
Output:
-((((c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/(a + b*x) - 2*B*((A* (c + d*x))/(a + b*x) - (B*(c + d*x))/(a + b*x) + (B*(c + d*x)*Log[(e*(c + d*x))/(a + b*x)])/(a + b*x)))/((b*c - a*d)*g^2))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b *Log[c*x^n])^p, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( m + 1)*(g/d)^m Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, ( a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[ n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
Time = 1.04 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.18
| method | result | size |
| norman | \(\frac {\frac {\left (A^{2}-2 B A +2 B^{2}\right ) x}{g a}+\frac {B^{2} c \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2}}{g \left (d a -b c \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2}}{g \left (d a -b c \right )}+\frac {2 \left (A -B \right ) c B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{g \left (d a -b c \right )}+\frac {2 d \left (A -B \right ) B x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{g \left (d a -b c \right )}}{g \left (b x +a \right )}\) | \(180\) |
| parts | \(-\frac {A^{2}}{g^{2} \left (b x +a \right ) b}+\frac {B^{2} \left (\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} e \left (d x +c \right )}{b x +a}-\frac {2 \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) e \left (d x +c \right )}{b x +a}+\frac {2 e \left (d x +c \right )}{b x +a}\right )}{g^{2} e \left (d a -b c \right )}+\frac {2 B A \left (\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) e \left (d x +c \right )}{b x +a}-\frac {e \left (d x +c \right )}{b x +a}\right )}{g^{2} e \left (d a -b c \right )}\) | \(182\) |
| parallelrisch | \(-\frac {A^{2} a \,b^{2} d^{2}-A^{2} b^{3} c d +2 B^{2} a \,b^{2} d^{2}-2 B^{2} b^{3} c d -2 A B x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{3} d^{2}-2 A B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{3} c d +2 A B \,b^{3} c d -2 A B a \,b^{2} d^{2}-B^{2} x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} b^{3} d^{2}+2 B^{2} x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{3} d^{2}-B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} b^{3} c d +2 B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{3} c d}{g^{2} \left (b x +a \right ) b^{3} d \left (d a -b c \right )}\) | \(251\) |
| derivativedivides | \(\frac {e \left (d a -b c \right ) \left (\frac {b^{2} A^{2} \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{\left (d a -b c \right )^{2} e^{2} g^{2}}+\frac {2 b^{2} A B \left (\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}-\frac {d e}{b}\right )}{\left (d a -b c \right )^{2} e^{2} g^{2}}+\frac {b^{2} B^{2} \left (\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}-2 \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {2 e \left (d a -b c \right )}{b \left (b x +a \right )}+\frac {2 d e}{b}\right )}{\left (d a -b c \right )^{2} e^{2} g^{2}}\right )}{b^{2}}\) | \(346\) |
| default | \(\frac {e \left (d a -b c \right ) \left (\frac {b^{2} A^{2} \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{\left (d a -b c \right )^{2} e^{2} g^{2}}+\frac {2 b^{2} A B \left (\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}-\frac {d e}{b}\right )}{\left (d a -b c \right )^{2} e^{2} g^{2}}+\frac {b^{2} B^{2} \left (\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}-2 \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {2 e \left (d a -b c \right )}{b \left (b x +a \right )}+\frac {2 d e}{b}\right )}{\left (d a -b c \right )^{2} e^{2} g^{2}}\right )}{b^{2}}\) | \(346\) |
| risch | \(-\frac {A^{2}}{g^{2} \left (b x +a \right ) b}+\frac {B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} d x}{g^{2} \left (d a -b c \right ) \left (b x +a \right )}+\frac {B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} c}{g^{2} \left (d a -b c \right ) \left (b x +a \right )}-\frac {2 B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) d x}{g^{2} \left (d a -b c \right ) \left (b x +a \right )}-\frac {2 B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) c}{g^{2} \left (d a -b c \right ) \left (b x +a \right )}+\frac {2 B^{2} d x}{g^{2} \left (d a -b c \right ) \left (b x +a \right )}+\frac {2 B^{2} c}{g^{2} \left (d a -b c \right ) \left (b x +a \right )}+\frac {2 B A \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) d x}{g^{2} \left (d a -b c \right ) \left (b x +a \right )}+\frac {2 B A \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) c}{g^{2} \left (d a -b c \right ) \left (b x +a \right )}-\frac {2 B A d x}{g^{2} \left (d a -b c \right ) \left (b x +a \right )}-\frac {2 B A c}{g^{2} \left (d a -b c \right ) \left (b x +a \right )}\) | \(373\) |
| orering | \(-\frac {\left (b x +a \right ) \left (8 b \,d^{2} x^{2}+a \,d^{2} x +15 b c d x +a c d +7 b \,c^{2}\right ) \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )^{2}}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (b g x +a g \right )^{2}}-\frac {\left (b x +a \right )^{2} \left (d x +c \right ) \left (7 b d x +d a +6 b c \right ) \left (\frac {2 \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right ) B \left (\frac {e d}{b x +a}-\frac {e \left (d x +c \right ) b}{\left (b x +a \right )^{2}}\right ) \left (b x +a \right )}{\left (b g x +a g \right )^{2} e \left (d x +c \right )}-\frac {2 \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )^{2} b g}{\left (b g x +a g \right )^{3}}\right )}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b}-\frac {\left (d x +c \right )^{2} \left (b x +a \right )^{3} \left (\frac {2 B^{2} \left (\frac {e d}{b x +a}-\frac {e \left (d x +c \right ) b}{\left (b x +a \right )^{2}}\right )^{2} \left (b x +a \right )^{2}}{e^{2} \left (d x +c \right )^{2} \left (b g x +a g \right )^{2}}-\frac {8 \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right ) B \left (\frac {e d}{b x +a}-\frac {e \left (d x +c \right ) b}{\left (b x +a \right )^{2}}\right ) \left (b x +a \right ) b g}{\left (b g x +a g \right )^{3} e \left (d x +c \right )}+\frac {2 \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right ) B \left (-\frac {2 e d b}{\left (b x +a \right )^{2}}+\frac {2 e \left (d x +c \right ) b^{2}}{\left (b x +a \right )^{3}}\right ) \left (b x +a \right )}{\left (b g x +a g \right )^{2} e \left (d x +c \right )}-\frac {2 \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right ) B \left (\frac {e d}{b x +a}-\frac {e \left (d x +c \right ) b}{\left (b x +a \right )^{2}}\right ) \left (b x +a \right ) d}{\left (b g x +a g \right )^{2} e \left (d x +c \right )^{2}}+\frac {2 \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right ) B \left (\frac {e d}{b x +a}-\frac {e \left (d x +c \right ) b}{\left (b x +a \right )^{2}}\right ) b}{\left (b g x +a g \right )^{2} e \left (d x +c \right )}+\frac {6 \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )^{2} b^{2} g^{2}}{\left (b g x +a g \right )^{4}}\right )}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b}\) | \(700\) |
Input:
int((A+B*ln(e*(d*x+c)/(b*x+a)))^2/(b*g*x+a*g)^2,x,method=_RETURNVERBOSE)
Output:
((A^2-2*A*B+2*B^2)/g/a*x+B^2*c/g/(a*d-b*c)*ln(e*(d*x+c)/(b*x+a))^2+B^2*d/g /(a*d-b*c)*x*ln(e*(d*x+c)/(b*x+a))^2+2*(A-B)*c*B/g/(a*d-b*c)*ln(e*(d*x+c)/ (b*x+a))+2*d*(A-B)*B/g/(a*d-b*c)*x*ln(e*(d*x+c)/(b*x+a)))/g/(b*x+a)
Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.01 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b c - {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a d + {\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {d e x + c e}{b x + a}\right )^{2} + 2 \, {\left ({\left (A B - B^{2}\right )} b d x + {\left (A B - B^{2}\right )} b c\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \] Input:
integrate((A+B*log(e*(d*x+c)/(b*x+a)))^2/(b*g*x+a*g)^2,x, algorithm="frica s")
Output:
-((A^2 - 2*A*B + 2*B^2)*b*c - (A^2 - 2*A*B + 2*B^2)*a*d + (B^2*b*d*x + B^2 *b*c)*log((d*e*x + c*e)/(b*x + a))^2 + 2*((A*B - B^2)*b*d*x + (A*B - B^2)* b*c)*log((d*e*x + c*e)/(b*x + a)))/((b^3*c - a*b^2*d)*g^2*x + (a*b^2*c - a ^2*b*d)*g^2)
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (128) = 256\).
Time = 1.17 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.81 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^2} \, dx=\frac {2 B d \left (A - B\right ) \log {\left (x + \frac {2 A B a d^{2} + 2 A B b c d - 2 B^{2} a d^{2} - 2 B^{2} b c d - \frac {2 B a^{2} d^{3} \left (A - B\right )}{a d - b c} + \frac {4 B a b c d^{2} \left (A - B\right )}{a d - b c} - \frac {2 B b^{2} c^{2} d \left (A - B\right )}{a d - b c}}{4 A B b d^{2} - 4 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} - \frac {2 B d \left (A - B\right ) \log {\left (x + \frac {2 A B a d^{2} + 2 A B b c d - 2 B^{2} a d^{2} - 2 B^{2} b c d + \frac {2 B a^{2} d^{3} \left (A - B\right )}{a d - b c} - \frac {4 B a b c d^{2} \left (A - B\right )}{a d - b c} + \frac {2 B b^{2} c^{2} d \left (A - B\right )}{a d - b c}}{4 A B b d^{2} - 4 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {\left (- 2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}}{a b g^{2} + b^{2} g^{2} x} + \frac {\left (B^{2} c + B^{2} d x\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}^{2}}{a^{2} d g^{2} - a b c g^{2} + a b d g^{2} x - b^{2} c g^{2} x} + \frac {- A^{2} + 2 A B - 2 B^{2}}{a b g^{2} + b^{2} g^{2} x} \] Input:
integrate((A+B*ln(e*(d*x+c)/(b*x+a)))**2/(b*g*x+a*g)**2,x)
Output:
2*B*d*(A - B)*log(x + (2*A*B*a*d**2 + 2*A*B*b*c*d - 2*B**2*a*d**2 - 2*B**2 *b*c*d - 2*B*a**2*d**3*(A - B)/(a*d - b*c) + 4*B*a*b*c*d**2*(A - B)/(a*d - b*c) - 2*B*b**2*c**2*d*(A - B)/(a*d - b*c))/(4*A*B*b*d**2 - 4*B**2*b*d**2 ))/(b*g**2*(a*d - b*c)) - 2*B*d*(A - B)*log(x + (2*A*B*a*d**2 + 2*A*B*b*c* d - 2*B**2*a*d**2 - 2*B**2*b*c*d + 2*B*a**2*d**3*(A - B)/(a*d - b*c) - 4*B *a*b*c*d**2*(A - B)/(a*d - b*c) + 2*B*b**2*c**2*d*(A - B)/(a*d - b*c))/(4* A*B*b*d**2 - 4*B**2*b*d**2))/(b*g**2*(a*d - b*c)) + (-2*A*B + 2*B**2)*log( e*(c + d*x)/(a + b*x))/(a*b*g**2 + b**2*g**2*x) + (B**2*c + B**2*d*x)*log( e*(c + d*x)/(a + b*x))**2/(a**2*d*g**2 - a*b*c*g**2 + a*b*d*g**2*x - b**2* c*g**2*x) + (-A**2 + 2*A*B - 2*B**2)/(a*b*g**2 + b**2*g**2*x)
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (153) = 306\).
Time = 0.06 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.72 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^2} \, dx={\left (2 \, {\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) + \frac {{\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c g^{2} - a^{2} b d g^{2} + {\left (b^{3} c g^{2} - a b^{2} d g^{2}\right )} x}\right )} B^{2} - 2 \, A B {\left (\frac {\log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right )}{b^{2} g^{2} x + a b g^{2}} - \frac {1}{b^{2} g^{2} x + a b g^{2}} - \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} + \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {B^{2} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right )^{2}}{b^{2} g^{2} x + a b g^{2}} - \frac {A^{2}}{b^{2} g^{2} x + a b g^{2}} \] Input:
integrate((A+B*log(e*(d*x+c)/(b*x+a)))^2/(b*g*x+a*g)^2,x, algorithm="maxim a")
Output:
(2*(1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*log (d*x + c)/((b^2*c - a*b*d)*g^2))*log(d*e*x/(b*x + a) + c*e/(b*x + a)) + (( 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))/(a*b^2*c*g^2 - a^2*b*d*g^2 + (b^3*c*g^2 - a*b^2*d*g^2)* x))*B^2 - 2*A*B*(log(d*e*x/(b*x + a) + c*e/(b*x + a))/(b^2*g^2*x + a*b*g^2 ) - 1/(b^2*g^2*x + a*b*g^2) - d*log(b*x + a)/((b^2*c - a*b*d)*g^2) + d*log (d*x + c)/((b^2*c - a*b*d)*g^2)) - B^2*log(d*e*x/(b*x + a) + c*e/(b*x + a) )^2/(b^2*g^2*x + a*b*g^2) - A^2/(b^2*g^2*x + a*b*g^2)
Time = 0.25 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.14 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^2} \, dx=-{\left (\frac {{\left (d e x + c e\right )} B^{2} \log \left (\frac {d e x + c e}{b x + a}\right )^{2}}{{\left (b x + a\right )} g^{2}} + \frac {2 \, {\left (d e x + c e\right )} {\left (A B - B^{2}\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{{\left (b x + a\right )} g^{2}} + \frac {{\left (d e x + c e\right )} {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )}}{{\left (b x + a\right )} g^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((A+B*log(e*(d*x+c)/(b*x+a)))^2/(b*g*x+a*g)^2,x, algorithm="giac" )
Output:
-((d*e*x + c*e)*B^2*log((d*e*x + c*e)/(b*x + a))^2/((b*x + a)*g^2) + 2*(d* e*x + c*e)*(A*B - B^2)*log((d*e*x + c*e)/(b*x + a))/((b*x + a)*g^2) + (d*e *x + c*e)*(A^2 - 2*A*B + 2*B^2)/((b*x + a)*g^2))*(b*c/((b*c*e - a*d*e)*(b* c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
Time = 27.47 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.46 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^2} \, dx=\frac {\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\,\left (\frac {2\,B^2}{b^2\,d\,g^2}-\frac {2\,A\,B}{b^2\,d\,g^2}\right )}{\frac {x}{d}+\frac {a}{b\,d}}-{\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )}^2\,\left (\frac {B^2}{b^2\,g^2\,\left (x+\frac {a}{b}\right )}-\frac {B^2\,d}{b\,g^2\,\left (a\,d-b\,c\right )}\right )-\frac {A^2-2\,A\,B+2\,B^2}{x\,b^2\,g^2+a\,b\,g^2}+\frac {B\,d\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {c\,b^2\,g^2+a\,d\,b\,g^2}{b\,g^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A-B\right )\,4{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \] Input:
int((A + B*log((e*(c + d*x))/(a + b*x)))^2/(a*g + b*g*x)^2,x)
Output:
(log((e*(c + d*x))/(a + b*x))*((2*B^2)/(b^2*d*g^2) - (2*A*B)/(b^2*d*g^2))) /(x/d + a/(b*d)) - log((e*(c + d*x))/(a + b*x))^2*(B^2/(b^2*g^2*(x + a/b)) - (B^2*d)/(b*g^2*(a*d - b*c))) - (A^2 + 2*B^2 - 2*A*B)/(b^2*g^2*x + a*b*g ^2) + (B*d*atan(((2*b*d*x + (b^2*c*g^2 + a*b*d*g^2)/(b*g^2))*1i)/(a*d - b* c))*(A - B)*4i)/(b*g^2*(a*d - b*c))
Time = 0.17 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.18 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^2} \, dx=\frac {-2 \,\mathrm {log}\left (b x +a \right ) a^{2} b c -2 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c x +2 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c +2 \,\mathrm {log}\left (b x +a \right ) b^{3} c x +2 \,\mathrm {log}\left (d x +c \right ) a^{2} b c +2 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c x -2 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c -2 \,\mathrm {log}\left (d x +c \right ) b^{3} c x +\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right )^{2} a \,b^{2} c +\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right )^{2} a \,b^{2} d x +2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{2} b d x -2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{2} c x -2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{2} d x +2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) b^{3} c x +a^{3} d x -a^{2} b c x -2 a^{2} b d x +2 a \,b^{2} c x +2 a \,b^{2} d x -2 b^{3} c x}{a \,g^{2} \left (a b d x -b^{2} c x +a^{2} d -a b c \right )} \] Input:
int((A+B*log(e*(d*x+c)/(b*x+a)))^2/(b*g*x+a*g)^2,x)
Output:
( - 2*log(a + b*x)*a**2*b*c - 2*log(a + b*x)*a*b**2*c*x + 2*log(a + b*x)*a *b**2*c + 2*log(a + b*x)*b**3*c*x + 2*log(c + d*x)*a**2*b*c + 2*log(c + d* x)*a*b**2*c*x - 2*log(c + d*x)*a*b**2*c - 2*log(c + d*x)*b**3*c*x + log((c *e + d*e*x)/(a + b*x))**2*a*b**2*c + log((c*e + d*e*x)/(a + b*x))**2*a*b** 2*d*x + 2*log((c*e + d*e*x)/(a + b*x))*a**2*b*d*x - 2*log((c*e + d*e*x)/(a + b*x))*a*b**2*c*x - 2*log((c*e + d*e*x)/(a + b*x))*a*b**2*d*x + 2*log((c *e + d*e*x)/(a + b*x))*b**3*c*x + a**3*d*x - a**2*b*c*x - 2*a**2*b*d*x + 2 *a*b**2*c*x + 2*a*b**2*d*x - 2*b**3*c*x)/(a*g**2*(a**2*d - a*b*c + a*b*d*x - b**2*c*x))