Integrand size = 30, antiderivative size = 144 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^3} \, dx=\frac {B}{4 d i^3 (c+d x)^2}+\frac {b B}{2 d (b c-a d) i^3 (c+d x)}+\frac {b^2 B \log (a+b x)}{2 d (b c-a d)^2 i^3}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 d i^3 (c+d x)^2}-\frac {b^2 B \log (c+d x)}{2 d (b c-a d)^2 i^3} \] Output:
1/4*B/d/i^3/(d*x+c)^2+1/2*b*B/d/(-a*d+b*c)/i^3/(d*x+c)+1/2*b^2*B*ln(b*x+a) /d/(-a*d+b*c)^2/i^3-1/2*(A+B*ln(e*(b*x+a)/(d*x+c)))/d/i^3/(d*x+c)^2-1/2*b^ 2*B*ln(d*x+c)/d/(-a*d+b*c)^2/i^3
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.77 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^3} \, dx=\frac {-2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+\frac {B \left ((b c-a d) (3 b c-a d+2 b d x)+2 b^2 (c+d x)^2 \log (a+b x)-2 b^2 (c+d x)^2 \log (c+d x)\right )}{(b c-a d)^2}}{4 d i^3 (c+d x)^2} \] Input:
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(c*i + d*i*x)^3,x]
Output:
(-2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + (B*((b*c - a*d)*(3*b*c - a*d + 2*b*d*x) + 2*b^2*(c + d*x)^2*Log[a + b*x] - 2*b^2*(c + d*x)^2*Log[c + d*x] ))/(b*c - a*d)^2)/(4*d*i^3*(c + d*x)^2)
Time = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2948, 27, 54, 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.
| \(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(c i+d i x)^3} \, dx\) |
| \(\Big \downarrow \) 2948 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {1}{i^2 (a+b x) (c+d x)^3}dx}{2 d i}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d i^3 (c+d x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {1}{(a+b x) (c+d x)^3}dx}{2 d i^3}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d i^3 (c+d x)^2}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {B (b c-a d) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d b^2}{(b c-a d)^3 (c+d x)}-\frac {d b}{(b c-a d)^2 (c+d x)^2}-\frac {d}{(b c-a d) (c+d x)^3}\right )dx}{2 d i^3}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d i^3 (c+d x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {B (b c-a d) \left (\frac {b^2 \log (a+b x)}{(b c-a d)^3}-\frac {b^2 \log (c+d x)}{(b c-a d)^3}+\frac {b}{(c+d x) (b c-a d)^2}+\frac {1}{2 (c+d x)^2 (b c-a d)}\right )}{2 d i^3}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d i^3 (c+d x)^2}\) |
Input:
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(c*i + d*i*x)^3,x]
Output:
-1/2*(A + B*Log[(e*(a + b*x))/(c + d*x)])/(d*i^3*(c + d*x)^2) + (B*(b*c - a*d)*(1/(2*(b*c - a*d)*(c + d*x)^2) + b/((b*c - a*d)^2*(c + d*x)) + (b^2*L og[a + b*x])/(b*c - a*d)^3 - (b^2*Log[c + d*x])/(b*c - a*d)^3))/(2*d*i^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*( (A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Simp[B*n*((b*c - a*d)/(g*(m + 1))) Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] / ; FreeQ[{a, b, c, d, e, f, g, A, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && !(EqQ[m, -2] && IntegerQ[n])
Time = 1.35 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.53
| method | result | size |
| parts | \(-\frac {A}{2 i^{3} \left (d x +c \right )^{2} d}-\frac {B d \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}-\frac {b e \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{d}\right )}{i^{3} \left (d a -b c \right )^{2} e^{2}}\) | \(221\) |
| norman | \(\frac {\frac {B c \,b^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) i}-\frac {2 A a \,d^{2}-2 A b c d -B a \,d^{2}+3 B b c d}{4 i \,d^{2} \left (d a -b c \right )}-\frac {B b x}{2 i \left (d a -b c \right )}-\frac {B a \left (d a -2 b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {b^{2} B d \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) i}}{i^{2} \left (d x +c \right )^{2}}\) | \(228\) |
| parallelrisch | \(-\frac {2 A \,a^{2} b \,d^{5}+2 A \,b^{3} c^{2} d^{3}-B \,a^{2} b \,d^{5}-3 B \,b^{3} c^{2} d^{3}-4 A a \,b^{2} c \,d^{4}+4 B a \,b^{2} c \,d^{4}-2 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{5}+2 B x a \,b^{2} d^{5}-2 B x \,b^{3} c \,d^{4}+2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b \,d^{5}-4 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c \,d^{4}-4 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} c \,d^{4}}{4 i^{3} \left (d x +c \right )^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b \,d^{4}}\) | \(235\) |
| orering | \(-\frac {\left (d x +c \right ) \left (-8 b^{2} d \,x^{2}-3 x a b d -13 b^{2} c x +5 a^{2} d -13 a b c \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{4 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (d i x +c i \right )^{3}}-\frac {\left (-2 b d x +d a -3 b c \right ) \left (b x +a \right ) \left (d x +c \right )^{2} \left (\frac {B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (d i x +c i \right )^{3}}-\frac {3 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) d i}{\left (d i x +c i \right )^{4}}\right )}{4 d \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}\) | \(236\) |
| risch | \(-\frac {B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 d \,i^{3} \left (d x +c \right )^{2}}-\frac {2 B \ln \left (d x +c \right ) b^{2} d^{2} x^{2}-2 B \ln \left (-b x -a \right ) b^{2} d^{2} x^{2}+4 B \ln \left (d x +c \right ) b^{2} c d x -4 B \ln \left (-b x -a \right ) b^{2} c d x +2 B \ln \left (d x +c \right ) b^{2} c^{2}-2 B \ln \left (-b x -a \right ) b^{2} c^{2}+2 B a b \,d^{2} x -2 B \,b^{2} c d x +2 A \,a^{2} d^{2}-4 A a b c d +2 A \,b^{2} c^{2}-B \,a^{2} d^{2}+4 B a b c d -3 B \,b^{2} c^{2}}{4 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) i^{3} \left (d x +c \right )^{2} d}\) | \(245\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A b \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\left (d a -b c \right )^{3} e^{2} i^{3}}+\frac {d^{3} A \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (d a -b c \right )^{3} e^{3} i^{3}}-\frac {d^{2} B b \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (d a -b c \right )^{3} e^{2} i^{3}}+\frac {d^{3} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (d a -b c \right )^{3} e^{3} i^{3}}\right )}{d^{2}}\) | \(337\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A b \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\left (d a -b c \right )^{3} e^{2} i^{3}}+\frac {d^{3} A \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (d a -b c \right )^{3} e^{3} i^{3}}-\frac {d^{2} B b \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (d a -b c \right )^{3} e^{2} i^{3}}+\frac {d^{3} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (d a -b c \right )^{3} e^{3} i^{3}}\right )}{d^{2}}\) | \(337\) |
Input:
int((A+B*ln(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*A/i^3/(d*x+c)^2/d-B/i^3*d/(a*d-b*c)^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-b*e/d*((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))-( a*d-b*c)*e/d/(d*x+c)-b*e/d))
Time = 0.08 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.53 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^3} \, dx=-\frac {{\left (2 \, A - 3 \, B\right )} b^{2} c^{2} - 4 \, {\left (A - B\right )} a b c d + {\left (2 \, A - B\right )} a^{2} d^{2} - 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x - 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B b^{2} c d x + 2 \, B a b c d - B a^{2} d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} i^{3} x^{2} + 2 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} i^{3} x + {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} i^{3}\right )}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^3,x, algorithm="fricas" )
Output:
-1/4*((2*A - 3*B)*b^2*c^2 - 4*(A - B)*a*b*c*d + (2*A - B)*a^2*d^2 - 2*(B*b ^2*c*d - B*a*b*d^2)*x - 2*(B*b^2*d^2*x^2 + 2*B*b^2*c*d*x + 2*B*a*b*c*d - B *a^2*d^2)*log((b*e*x + a*e)/(d*x + c)))/((b^2*c^2*d^3 - 2*a*b*c*d^4 + a^2* d^5)*i^3*x^2 + 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*i^3*x + (b^2*c^ 4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3)*i^3)
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (122) = 244\).
Time = 1.10 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.93 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^3} \, dx=- \frac {B b^{2} \log {\left (x + \frac {- \frac {B a^{3} b^{2} d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b^{3} c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{4} c^{2} d}{\left (a d - b c\right )^{2}} + B a b^{2} d + \frac {B b^{5} c^{3}}{\left (a d - b c\right )^{2}} + B b^{3} c}{2 B b^{3} d} \right )}}{2 d i^{3} \left (a d - b c\right )^{2}} + \frac {B b^{2} \log {\left (x + \frac {\frac {B a^{3} b^{2} d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b^{3} c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{4} c^{2} d}{\left (a d - b c\right )^{2}} + B a b^{2} d - \frac {B b^{5} c^{3}}{\left (a d - b c\right )^{2}} + B b^{3} c}{2 B b^{3} d} \right )}}{2 d i^{3} \left (a d - b c\right )^{2}} - \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 c^{2} d i^{3} + 4 c d^{2} i^{3} x + 2 d^{3} i^{3} x^{2}} + \frac {- 2 A a d + 2 A b c + B a d - 3 B b c - 2 B b d x}{4 a c^{2} d^{2} i^{3} - 4 b c^{3} d i^{3} + x^{2} \cdot \left (4 a d^{4} i^{3} - 4 b c d^{3} i^{3}\right ) + x \left (8 a c d^{3} i^{3} - 8 b c^{2} d^{2} i^{3}\right )} \] Input:
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)**3,x)
Output:
-B*b**2*log(x + (-B*a**3*b**2*d**3/(a*d - b*c)**2 + 3*B*a**2*b**3*c*d**2/( a*d - b*c)**2 - 3*B*a*b**4*c**2*d/(a*d - b*c)**2 + B*a*b**2*d + B*b**5*c** 3/(a*d - b*c)**2 + B*b**3*c)/(2*B*b**3*d))/(2*d*i**3*(a*d - b*c)**2) + B*b **2*log(x + (B*a**3*b**2*d**3/(a*d - b*c)**2 - 3*B*a**2*b**3*c*d**2/(a*d - b*c)**2 + 3*B*a*b**4*c**2*d/(a*d - b*c)**2 + B*a*b**2*d - B*b**5*c**3/(a* d - b*c)**2 + B*b**3*c)/(2*B*b**3*d))/(2*d*i**3*(a*d - b*c)**2) - B*log(e* (a + b*x)/(c + d*x))/(2*c**2*d*i**3 + 4*c*d**2*i**3*x + 2*d**3*i**3*x**2) + (-2*A*a*d + 2*A*b*c + B*a*d - 3*B*b*c - 2*B*b*d*x)/(4*a*c**2*d**2*i**3 - 4*b*c**3*d*i**3 + x**2*(4*a*d**4*i**3 - 4*b*c*d**3*i**3) + x*(8*a*c*d**3* i**3 - 8*b*c**2*d**2*i**3))
Time = 0.05 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.77 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^3} \, dx=\frac {1}{4} \, B {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b c d^{3} - a d^{4}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} i^{3} x + {\left (b c^{3} d - a c^{2} d^{2}\right )} i^{3}} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}} + \frac {2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}} - \frac {2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} i^{3}}\right )} - \frac {A}{2 \, {\left (d^{3} i^{3} x^{2} + 2 \, c d^{2} i^{3} x + c^{2} d i^{3}\right )}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^3,x, algorithm="maxima" )
Output:
1/4*B*((2*b*d*x + 3*b*c - a*d)/((b*c*d^3 - a*d^4)*i^3*x^2 + 2*(b*c^2*d^2 - a*c*d^3)*i^3*x + (b*c^3*d - a*c^2*d^2)*i^3) - 2*log(b*e*x/(d*x + c) + a*e /(d*x + c))/(d^3*i^3*x^2 + 2*c*d^2*i^3*x + c^2*d*i^3) + 2*b^2*log(b*x + a) /((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3) - 2*b^2*log(d*x + c)/((b^2*c^2* d - 2*a*b*c*d^2 + a^2*d^3)*i^3)) - 1/2*A/(d^3*i^3*x^2 + 2*c*d^2*i^3*x + c^ 2*d*i^3)
Time = 0.21 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.64 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^3} \, dx=\frac {1}{4} \, {\left (2 \, {\left (\frac {2 \, {\left (b e x + a e\right )} B b}{{\left (b c i^{3} - a d i^{3}\right )} {\left (d x + c\right )}} - \frac {{\left (b e x + a e\right )}^{2} B d}{{\left (b c e i^{3} - a d e i^{3}\right )} {\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right ) - \frac {{\left (b e x + a e\right )}^{2} {\left (2 \, A d - B d\right )}}{{\left (b c e i^{3} - a d e i^{3}\right )} {\left (d x + c\right )}^{2}} + \frac {4 \, {\left (b e x + a e\right )} {\left (A b - B b\right )}}{{\left (b c i^{3} - a d i^{3}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^3,x, algorithm="giac")
Output:
1/4*(2*(2*(b*e*x + a*e)*B*b/((b*c*i^3 - a*d*i^3)*(d*x + c)) - (b*e*x + a*e )^2*B*d/((b*c*e*i^3 - a*d*e*i^3)*(d*x + c)^2))*log((b*e*x + a*e)/(d*x + c) ) - (b*e*x + a*e)^2*(2*A*d - B*d)/((b*c*e*i^3 - a*d*e*i^3)*(d*x + c)^2) + 4*(b*e*x + a*e)*(A*b - B*b)/((b*c*i^3 - a*d*i^3)*(d*x + c)))*(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 = 27.25 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.44 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^3} \, dx=\frac {B\,b^2\,\mathrm {atanh}\left (\frac {2\,a^2\,d^3\,i^3-2\,b^2\,c^2\,d\,i^3}{2\,d\,i^3\,{\left (a\,d-b\,c\right )}^2}+\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{d\,i^3\,{\left (a\,d-b\,c\right )}^2}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,d^2\,i^3\,\left (2\,c\,x+d\,x^2+\frac {c^2}{d}\right )}-\frac {\frac {2\,A\,a\,d-2\,A\,b\,c-B\,a\,d+3\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b\,d\,x}{a\,d-b\,c}}{2\,c^2\,d\,i^3+4\,c\,d^2\,i^3\,x+2\,d^3\,i^3\,x^2} \] Input:
int((A + B*log((e*(a + b*x))/(c + d*x)))/(c*i + d*i*x)^3,x)
Output:
(B*b^2*atanh((2*a^2*d^3*i^3 - 2*b^2*c^2*d*i^3)/(2*d*i^3*(a*d - b*c)^2) + ( 2*b*d*x)/(a*d - b*c)))/(d*i^3*(a*d - b*c)^2) - (B*log((e*(a + b*x))/(c + d *x)))/(2*d^2*i^3*(2*c*x + d*x^2 + c^2/d)) - ((2*A*a*d - 2*A*b*c - B*a*d + 3*B*b*c)/(2*(a*d - b*c)) + (B*b*d*x)/(a*d - b*c))/(2*c^2*d*i^3 + 2*d^3*i^3 *x^2 + 4*c*d^2*i^3*x)
Time = 0.20 (sec) , antiderivative size = 577, normalized size of antiderivative = 4.01 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c i+d i x)^3} \, dx=\frac {i \left (-3 a \,b^{2} c^{3} d +a \,b^{2} c \,d^{3} x^{2}-4 \,\mathrm {log}\left (b x +a \right ) a^{2} b c \,d^{3} x +8 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c^{2} d^{2} x +4 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c \,d^{3} x^{2}+4 \,\mathrm {log}\left (d x +c \right ) a^{2} b c \,d^{3} x -8 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c^{2} d^{2} x -4 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c \,d^{3} x^{2}+4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b c \,d^{3} x -2 \,\mathrm {log}\left (b x +a \right ) a^{2} b \,c^{2} d^{2}-2 \,\mathrm {log}\left (b x +a \right ) a^{2} b \,d^{4} x^{2}+4 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c^{3} d +2 \,\mathrm {log}\left (d x +c \right ) a^{2} b \,c^{2} d^{2}+2 \,\mathrm {log}\left (d x +c \right ) a^{2} b \,d^{4} x^{2}-2 a^{3} c^{2} d^{2}-2 a \,b^{2} c^{4}-8 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{2} c^{2} d^{2} x -4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{2} c \,d^{3} x^{2}-4 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c^{3} d +2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b \,d^{4} x^{2}+4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{3} c^{3} d x +2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{3} c^{2} d^{2} x^{2}+a^{2} b \,c^{2} d^{2}-b^{3} c^{2} d^{2} x^{2}+2 b^{3} c^{4}+4 a^{2} b \,c^{3} d \right )}{4 c^{2} d \left (a^{2} d^{4} x^{2}-2 a b c \,d^{3} x^{2}+b^{2} c^{2} d^{2} x^{2}+2 a^{2} c \,d^{3} x -4 a b \,c^{2} d^{2} x +2 b^{2} c^{3} d x +a^{2} c^{2} d^{2}-2 a b \,c^{3} d +b^{2} c^{4}\right )} \] Input:
int((A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^3,x)
Output:
(i*( - 2*log(a + b*x)*a**2*b*c**2*d**2 - 4*log(a + b*x)*a**2*b*c*d**3*x - 2*log(a + b*x)*a**2*b*d**4*x**2 + 4*log(a + b*x)*a*b**2*c**3*d + 8*log(a + b*x)*a*b**2*c**2*d**2*x + 4*log(a + b*x)*a*b**2*c*d**3*x**2 + 2*log(c + d *x)*a**2*b*c**2*d**2 + 4*log(c + d*x)*a**2*b*c*d**3*x + 2*log(c + d*x)*a** 2*b*d**4*x**2 - 4*log(c + d*x)*a*b**2*c**3*d - 8*log(c + d*x)*a*b**2*c**2* d**2*x - 4*log(c + d*x)*a*b**2*c*d**3*x**2 + 4*log((a*e + b*e*x)/(c + d*x) )*a**2*b*c*d**3*x + 2*log((a*e + b*e*x)/(c + d*x))*a**2*b*d**4*x**2 - 8*lo g((a*e + b*e*x)/(c + d*x))*a*b**2*c**2*d**2*x - 4*log((a*e + b*e*x)/(c + d *x))*a*b**2*c*d**3*x**2 + 4*log((a*e + b*e*x)/(c + d*x))*b**3*c**3*d*x + 2 *log((a*e + b*e*x)/(c + d*x))*b**3*c**2*d**2*x**2 - 2*a**3*c**2*d**2 + 4*a **2*b*c**3*d + a**2*b*c**2*d**2 - 2*a*b**2*c**4 - 3*a*b**2*c**3*d + a*b**2 *c*d**3*x**2 + 2*b**3*c**4 - b**3*c**2*d**2*x**2))/(4*c**2*d*(a**2*c**2*d* *2 + 2*a**2*c*d**3*x + a**2*d**4*x**2 - 2*a*b*c**3*d - 4*a*b*c**2*d**2*x - 2*a*b*c*d**3*x**2 + b**2*c**4 + 2*b**2*c**3*d*x + b**2*c**2*d**2*x**2))