Integrand size = 32, antiderivative size = 152 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=-\frac {2 A B (a+b x)}{(b c-a d) i^2 (c+d x)}+\frac {2 B^2 (a+b x)}{(b c-a d) i^2 (c+d x)}-\frac {2 B^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d) i^2 (c+d x)}+\frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d) i^2 (c+d x)} \] Output:
-2*A*B*(b*x+a)/(-a*d+b*c)/i^2/(d*x+c)+2*B^2*(b*x+a)/(-a*d+b*c)/i^2/(d*x+c) -2*B^2*(b*x+a)*ln(e*(b*x+a)/(d*x+c))/(-a*d+b*c)/i^2/(d*x+c)+(b*x+a)*(A+B*l n(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)/i^2/(d*x+c)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.45 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.07 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\frac {-\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b (c+d x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-2 B (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-b B (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+b B (c+d x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b c-a d}}{d i^2 (c+d x)} \] Input:
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(c*i + d*i*x)^2,x]
Output:
(-(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(2*(b*c - a*d)*(A + B*Log[(e *(a + b*x))/(c + d*x)]) + 2*b*(c + d*x)*Log[a + b*x]*(A + B*Log[(e*(a + b* x))/(c + d*x)]) - 2*b*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 2*B*(b*c - a*d + b*(c + d*x)*Log[a + b*x] - b*(c + d*x)*Log[c + d*x]) - b*B*(c + d*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*B*(c + d*x)* ((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*Pol yLog[2, (b*(c + d*x))/(b*c - a*d)])))/(b*c - a*d))/(d*i^2*(c + d*x))
Time = 0.27 (sec) , antiderivative size = 111, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(c i+d i x)^2} \, dx\) |
| \(\Big \downarrow \) 2952 |
| \(\displaystyle \frac {\int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2d\frac {a+b x}{c+d x}}{i^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2733 |
| \(\displaystyle \frac {\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{c+d x}-2 B \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )d\frac {a+b x}{c+d x}}{i^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{c+d x}-2 B \left (\frac {A (a+b x)}{c+d x}+\frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x}-\frac {B (a+b x)}{c+d x}\right )}{i^2 (b c-a d)}\) |
Input:
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(c*i + d*i*x)^2,x]
Output:
(((a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(c + d*x) - 2*B*((A*(a + b*x))/(c + d*x) - (B*(a + b*x))/(c + d*x) + (B*(a + b*x)*Log[(e*(a + b* x))/(c + d*x)])/(c + d*x)))/((b*c - a*d)*i^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 = 1.20 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.20
| method | result | size |
| norman | \(\frac {\frac {\left (A^{2}-2 A B +2 B^{2}\right ) x}{i c}-\frac {B^{2} a \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{i \left (d a -b c \right )}-\frac {B^{2} b x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{\left (d a -b c \right ) i}-\frac {2 a B \left (A -B \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i \left (d a -b c \right )}-\frac {2 B b \left (A -B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i \left (d a -b c \right )}}{i \left (d x +c \right )}\) | \(182\) |
| parts | \(-\frac {A^{2}}{i^{2} \left (d x +c \right ) d}-\frac {B^{2} \left (\frac {\ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} e \left (b x +a \right )}{d x +c}-\frac {2 \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) e \left (b x +a \right )}{d x +c}+\frac {2 e \left (b x +a \right )}{d x +c}\right )}{i^{2} e \left (d a -b c \right )}-\frac {2 A B \left (\frac {\ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) e \left (b x +a \right )}{d x +c}-\frac {e \left (b x +a \right )}{d x +c}\right )}{i^{2} e \left (d a -b c \right )}\) | \(183\) |
| parallelrisch | \(-\frac {2 B^{2} a b \,d^{3}-2 B^{2} b^{2} c \,d^{2}+A^{2} a b \,d^{3}-A^{2} b^{2} c \,d^{2}-2 A B a b \,d^{3}+2 A B \,b^{2} c \,d^{2}+B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{2} d^{3}-2 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} d^{3}+B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a b \,d^{3}-2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a b \,d^{3}+2 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{2} d^{3}+2 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a b \,d^{3}}{i^{2} \left (d x +c \right ) b \,d^{3} \left (d a -b c \right )}\) | \(249\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\left (d a -b c \right )^{2} e^{2} i^{2}}+\frac {2 d^{2} A B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (d a -b c \right )^{2} e^{2} i^{2}}+\frac {d^{2} B^{2} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (d a -b c \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{\left (d a -b c \right )^{2} e^{2} i^{2}}\right )}{d^{2}}\) | \(341\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\left (d a -b c \right )^{2} e^{2} i^{2}}+\frac {2 d^{2} A B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (d a -b c \right )^{2} e^{2} i^{2}}+\frac {d^{2} B^{2} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (d a -b c \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{\left (d a -b c \right )^{2} e^{2} i^{2}}\right )}{d^{2}}\) | \(341\) |
| risch | \(-\frac {A^{2}}{i^{2} \left (d x +c \right ) d}-\frac {B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b x}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}-\frac {B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}+\frac {2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b x}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}+\frac {2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}-\frac {2 B^{2} b x}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}-\frac {2 B^{2} a}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}-\frac {2 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b x}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}-\frac {2 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}+\frac {2 A B b x}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}+\frac {2 A B a}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}\) | \(375\) |
| orering | \(-\frac {\left (d x +c \right ) \left (8 b^{2} d \,x^{2}+15 x a b d +b^{2} c x +7 a^{2} d +a b c \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (d i x +c i \right )^{2}}-\frac {\left (b x +a \right ) \left (d x +c \right )^{2} \left (7 b d x +6 d a +b c \right ) \left (\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{\left (d i x +c i \right )^{2} e \left (b x +a \right )}-\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2} d i}{\left (d i x +c i \right )^{3}}\right )}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) d}-\frac {\left (d x +c \right )^{3} \left (b x +a \right )^{2} \left (\frac {2 B^{2} \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right )^{2} \left (d x +c \right )^{2}}{e^{2} \left (b x +a \right )^{2} \left (d i x +c i \right )^{2}}-\frac {8 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right ) d i}{\left (d i x +c i \right )^{3} e \left (b x +a \right )}+\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) B \left (-\frac {2 e b d}{\left (d x +c \right )^{2}}+\frac {2 e \left (b x +a \right ) d^{2}}{\left (d x +c \right )^{3}}\right ) \left (d x +c \right )}{\left (d i x +c i \right )^{2} e \left (b x +a \right )}-\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right ) b}{\left (d i x +c i \right )^{2} e \left (b x +a \right )^{2}}+\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) d}{\left (d i x +c i \right )^{2} e \left (b x +a \right )}+\frac {6 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2} d^{2} i^{2}}{\left (d i x +c i \right )^{4}}\right )}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) d}\) | \(700\) |
Input:
int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i)^2,x,method=_RETURNVERBOSE)
Output:
((A^2-2*A*B+2*B^2)/i/c*x-B^2*a/i/(a*d-b*c)*ln(e*(b*x+a)/(d*x+c))^2-B^2*b/( a*d-b*c)/i*x*ln(e*(b*x+a)/(d*x+c))^2-2*a*B*(A-B)/i/(a*d-b*c)*ln(e*(b*x+a)/ (d*x+c))-2*B*b*(A-B)/i/(a*d-b*c)*x*ln(e*(b*x+a)/(d*x+c)))/i/(d*x+c)
Time = 0.08 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.02 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=-\frac {{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b c - {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a d - {\left (B^{2} b d x + B^{2} a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 2 \, {\left ({\left (A B - B^{2}\right )} b d x + {\left (A B - B^{2}\right )} a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} i^{2} x + {\left (b c^{2} d - a c d^{2}\right )} i^{2}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i)^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 *a*d)*log((b*e*x + a*e)/(d*x + c))^2 - 2*((A*B - B^2)*b*d*x + (A*B - B^2)* a*d)*log((b*e*x + a*e)/(d*x + c)))/((b*c*d^2 - a*d^3)*i^2*x + (b*c^2*d - a *c*d^2)*i^2)
Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (128) = 256\).
Time = 1.15 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.84 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\frac {2 B b \left (A - B\right ) \log {\left (x + \frac {2 A B a b d + 2 A B b^{2} c - 2 B^{2} a b d - 2 B^{2} b^{2} c - \frac {2 B a^{2} b d^{2} \left (A - B\right )}{a d - b c} + \frac {4 B a b^{2} c d \left (A - B\right )}{a d - b c} - \frac {2 B b^{3} c^{2} \left (A - B\right )}{a d - b c}}{4 A B b^{2} d - 4 B^{2} b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} - \frac {2 B b \left (A - B\right ) \log {\left (x + \frac {2 A B a b d + 2 A B b^{2} c - 2 B^{2} a b d - 2 B^{2} b^{2} c + \frac {2 B a^{2} b d^{2} \left (A - B\right )}{a d - b c} - \frac {4 B a b^{2} c d \left (A - B\right )}{a d - b c} + \frac {2 B b^{3} c^{2} \left (A - B\right )}{a d - b c}}{4 A B b^{2} d - 4 B^{2} b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} + \frac {\left (- 2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{c d i^{2} + d^{2} i^{2} x} + \frac {\left (- B^{2} a - B^{2} b x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a c d i^{2} + a d^{2} i^{2} x - b c^{2} i^{2} - b c d i^{2} x} + \frac {- A^{2} + 2 A B - 2 B^{2}}{c d i^{2} + d^{2} i^{2} x} \] Input:
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(d*i*x+c*i)**2,x)
Output:
2*B*b*(A - B)*log(x + (2*A*B*a*b*d + 2*A*B*b**2*c - 2*B**2*a*b*d - 2*B**2* b**2*c - 2*B*a**2*b*d**2*(A - B)/(a*d - b*c) + 4*B*a*b**2*c*d*(A - B)/(a*d - b*c) - 2*B*b**3*c**2*(A - B)/(a*d - b*c))/(4*A*B*b**2*d - 4*B**2*b**2*d ))/(d*i**2*(a*d - b*c)) - 2*B*b*(A - B)*log(x + (2*A*B*a*b*d + 2*A*B*b**2* c - 2*B**2*a*b*d - 2*B**2*b**2*c + 2*B*a**2*b*d**2*(A - B)/(a*d - b*c) - 4 *B*a*b**2*c*d*(A - B)/(a*d - b*c) + 2*B*b**3*c**2*(A - B)/(a*d - b*c))/(4* A*B*b**2*d - 4*B**2*b**2*d))/(d*i**2*(a*d - b*c)) + (-2*A*B + 2*B**2)*log( e*(a + b*x)/(c + d*x))/(c*d*i**2 + d**2*i**2*x) + (-B**2*a - B**2*b*x)*log (e*(a + b*x)/(c + d*x))**2/(a*c*d*i**2 + a*d**2*i**2*x - b*c**2*i**2 - b*c *d*i**2*x) + (-A**2 + 2*A*B - 2*B**2)/(c*d*i**2 + d**2*i**2*x)
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (152) = 304\).
Time = 0.06 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.74 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx={\left (2 \, {\left (\frac {1}{d^{2} i^{2} x + c d i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {{\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b c^{2} d i^{2} - a c d^{2} i^{2} + {\left (b c d^{2} i^{2} - a d^{3} i^{2}\right )} x}\right )} B^{2} - 2 \, A B {\left (\frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac {1}{d^{2} i^{2} x + c d i^{2}} - \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} + \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} - \frac {B^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2}}{d^{2} i^{2} x + c d i^{2}} - \frac {A^{2}}{d^{2} i^{2} x + c d i^{2}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i)^2,x, algorithm="maxim a")
Output:
(2*(1/(d^2*i^2*x + c*d*i^2) + b*log(b*x + a)/((b*c*d - a*d^2)*i^2) - b*log (d*x + c)/((b*c*d - a*d^2)*i^2))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - (( b*d*x + b*c)*log(b*x + a)^2 + (b*d*x + b*c)*log(d*x + c)^2 + 2*b*c - 2*a*d + 2*(b*d*x + b*c)*log(b*x + a) - 2*(b*d*x + b*c + (b*d*x + b*c)*log(b*x + a))*log(d*x + c))/(b*c^2*d*i^2 - a*c*d^2*i^2 + (b*c*d^2*i^2 - a*d^3*i^2)* x))*B^2 - 2*A*B*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^2*i^2*x + c*d*i^2 ) - 1/(d^2*i^2*x + c*d*i^2) - b*log(b*x + a)/((b*c*d - a*d^2)*i^2) + b*log (d*x + c)/((b*c*d - a*d^2)*i^2)) - B^2*log(b*e*x/(d*x + c) + a*e/(d*x + c) )^2/(d^2*i^2*x + c*d*i^2) - A^2/(d^2*i^2*x + c*d*i^2)
Time = 0.22 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.14 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx={\left (\frac {{\left (b e x + a e\right )} B^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{{\left (d x + c\right )} i^{2}} + \frac {2 \, {\left (b e x + a e\right )} {\left (A B - B^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (d x + c\right )} i^{2}} + \frac {{\left (b e x + a e\right )} {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )}}{{\left (d x + c\right )} i^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i)^2,x, algorithm="giac" )
Output:
((b*e*x + a*e)*B^2*log((b*e*x + a*e)/(d*x + c))^2/((d*x + c)*i^2) + 2*(b*e *x + a*e)*(A*B - B^2)*log((b*e*x + a*e)/(d*x + c))/((d*x + c)*i^2) + (b*e* x + a*e)*(A^2 - 2*A*B + 2*B^2)/((d*x + c)*i^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 = 27.36 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.46 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {2\,B^2}{b\,d^2\,i^2}-\frac {2\,A\,B}{b\,d^2\,i^2}\right )}{\frac {x}{b}+\frac {c}{b\,d}}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B^2}{d^2\,i^2\,\left (x+\frac {c}{d}\right )}+\frac {B^2\,b}{d\,i^2\,\left (a\,d-b\,c\right )}\right )-\frac {A^2-2\,A\,B+2\,B^2}{x\,d^2\,i^2+c\,d\,i^2}+\frac {B\,b\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {a\,d^2\,i^2+b\,c\,d\,i^2}{d\,i^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A-B\right )\,4{}\mathrm {i}}{d\,i^2\,\left (a\,d-b\,c\right )} \] Input:
int((A + B*log((e*(a + b*x))/(c + d*x)))^2/(c*i + d*i*x)^2,x)
Output:
(log((e*(a + b*x))/(c + d*x))*((2*B^2)/(b*d^2*i^2) - (2*A*B)/(b*d^2*i^2))) /(x/b + c/(b*d)) - log((e*(a + b*x))/(c + d*x))^2*(B^2/(d^2*i^2*(x + c/d)) + (B^2*b)/(d*i^2*(a*d - b*c))) - (A^2 + 2*B^2 - 2*A*B)/(d^2*i^2*x + c*d*i ^2) + (B*b*atan(((2*b*d*x + (a*d^2*i^2 + b*c*d*i^2)/(d*i^2))*1i)/(a*d - b* c))*(A - B)*4i)/(d*i^2*(a*d - b*c))
Time = 0.20 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.18 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\frac {2 \,\mathrm {log}\left (b x +a \right ) a^{2} b c +2 \,\mathrm {log}\left (b x +a \right ) a^{2} b d x -2 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c -2 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} d x -2 \,\mathrm {log}\left (d x +c \right ) a^{2} b c -2 \,\mathrm {log}\left (d x +c \right ) a^{2} b d x +2 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c +2 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} d x +\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a \,b^{2} c +\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} b^{3} c x -2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b d x +2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{2} c x +2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{2} d x -2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{3} c x -a^{3} d x +a^{2} b c x +2 a^{2} b d x -2 a \,b^{2} c x -2 a \,b^{2} d x +2 b^{3} c x}{c \left (a \,d^{2} x -b c d x +a c d -b \,c^{2}\right )} \] Input:
int((A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i)^2,x)
Output:
(2*log(a + b*x)*a**2*b*c + 2*log(a + b*x)*a**2*b*d*x - 2*log(a + b*x)*a*b* *2*c - 2*log(a + b*x)*a*b**2*d*x - 2*log(c + d*x)*a**2*b*c - 2*log(c + d*x )*a**2*b*d*x + 2*log(c + d*x)*a*b**2*c + 2*log(c + d*x)*a*b**2*d*x + log(( a*e + b*e*x)/(c + d*x))**2*a*b**2*c + log((a*e + b*e*x)/(c + d*x))**2*b**3 *c*x - 2*log((a*e + b*e*x)/(c + d*x))*a**2*b*d*x + 2*log((a*e + b*e*x)/(c + d*x))*a*b**2*c*x + 2*log((a*e + b*e*x)/(c + d*x))*a*b**2*d*x - 2*log((a* e + b*e*x)/(c + d*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)/(c*(a*c*d + a*d**2*x - b*c**2 - b* c*d*x))