Integrand size = 38, antiderivative size = 85 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {B i (c+d x)^2}{4 (b c-a d) g^3 (a+b x)^2}-\frac {i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d) g^3 (a+b x)^2} \] Output:
-1/4*B*i*(d*x+c)^2/(-a*d+b*c)/g^3/(b*x+a)^2-1/2*i*(d*x+c)^2*(A+B*ln(e*(b*x +a)/(d*x+c)))/(-a*d+b*c)/g^3/(b*x+a)^2
Leaf count is larger than twice the leaf count of optimal. \(208\) vs. \(2(85)=170\).
Time = 0.19 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.45 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=\frac {i \left (-\frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 (a+b x)^2}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (a+b x)}-\frac {B d \left (\frac {1}{a+b x}+\frac {d \log (a+b x)}{b c-a d}-\frac {d \log (c+d x)}{b c-a d}\right )}{b^2}-\frac {B \left (\frac {b c-a d}{(a+b x)^2}-\frac {2 d}{a+b x}-\frac {2 d^2 \log (a+b x)}{b c-a d}+\frac {2 d^2 \log (c+d x)}{b c-a d}\right )}{4 b^2}\right )}{g^3} \] Input:
Integrate[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g* x)^3,x]
Output:
(i*(-1/2*((b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*(a + b*x) ^2) - (d*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*(a + b*x)) - (B*d*((a + b*x)^(-1) + (d*Log[a + b*x])/(b*c - a*d) - (d*Log[c + d*x])/(b*c - a*d)) )/b^2 - (B*((b*c - a*d)/(a + b*x)^2 - (2*d)/(a + b*x) - (2*d^2*Log[a + b*x ])/(b*c - a*d) + (2*d^2*Log[c + d*x])/(b*c - a*d)))/(4*b^2)))/g^3
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2962, 2741}
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.
| \(\displaystyle \int \frac {(c i+d i x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a g+b g x)^3} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i \int \frac {(c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}}{g^3 (b c-a d)}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {i \left (-\frac {(c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {B (c+d x)^2}{4 (a+b x)^2}\right )}{g^3 (b c-a d)}\) |
Input:
Int[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^3,x ]
Output:
(i*(-1/4*(B*(c + d*x)^2)/(a + b*x)^2 - ((c + d*x)^2*(A + B*Log[(e*(a + b*x ))/(c + d*x)]))/(2*(a + b*x)^2)))/((b*c - a*d)*g^3)
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
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]
Time = 1.35 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.80
| method | result | size |
| parts | \(\frac {i A \left (-\frac {-d a +b c}{2 b^{2} \left (b x +a \right )^{2}}-\frac {d}{b^{2} \left (b x +a \right )}\right )}{g^{3}}-\frac {i B \,e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{g^{3} \left (d a -b c \right )}\) | \(153\) |
| norman | \(\frac {\frac {B c d i x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (d a -b c \right )}-\frac {2 A a d i +2 A b c i +B a d i +B b c i}{4 g \,b^{2}}-\frac {\left (2 A d i +B d i \right ) x}{2 g b}+\frac {B i \,c^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (d a -b c \right )}+\frac {B \,d^{2} i \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (d a -b c \right ) g}}{\left (b x +a \right )^{2} g^{2}}\) | \(170\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i \,d^{2} e A}{2 \left (d a -b c \right )^{2} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,d^{2} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{2} g^{3}}\right )}{d^{2}}\) | \(177\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i \,d^{2} e A}{2 \left (d a -b c \right )^{2} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,d^{2} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{2} g^{3}}\right )}{d^{2}}\) | \(177\) |
| parallelrisch | \(-\frac {2 A \,a^{2} b^{2} d^{3} i +B \,a^{2} b^{2} d^{3} i -B \,b^{4} c^{2} d i -4 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c \,d^{2} i -4 A x \,b^{4} c \,d^{2} i -2 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} d^{3} i +4 A x a \,b^{3} d^{3} i -2 A \,c^{2} i \,b^{4} d -2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{4} c^{2} d i -2 B x \,b^{4} c \,d^{2} i +2 B x a \,b^{3} d^{3} i}{4 g^{3} \left (b x +a \right )^{2} b^{4} d \left (d a -b c \right )}\) | \(206\) |
| risch | \(-\frac {B i \left (2 b d x +d a +b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g^{3} \left (b x +a \right )^{2} b^{2}}-\frac {i \left (2 B \ln \left (d x +c \right ) b^{2} d^{2} x^{2}-2 B \ln \left (-b x -a \right ) b^{2} d^{2} x^{2}+4 B \ln \left (d x +c \right ) a b \,d^{2} x -4 B \ln \left (-b x -a \right ) a b \,d^{2} x +4 A a b \,d^{2} x -4 A \,b^{2} c d x +2 B \ln \left (d x +c \right ) a^{2} d^{2}-2 B \,a^{2} \ln \left (-b x -a \right ) d^{2}+2 B a b \,d^{2} x -2 B \,b^{2} c d x +2 A \,a^{2} d^{2}-2 A \,b^{2} c^{2}+B \,a^{2} d^{2}-B \,b^{2} c^{2}\right )}{4 g^{3} \left (b x +a \right )^{2} b^{2} \left (d a -b c \right )}\) | \(249\) |
| orering | \(-\frac {\left (b x +a \right ) \left (2 d^{2} b^{2} x^{3}-3 x^{2} a b \,d^{2}+3 x^{2} b^{2} c d -6 x a c d b +a^{2} c d -5 a b \,c^{2}\right ) \left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{4 a \left (d a -b c \right ) b \left (d x +c \right ) \left (b g x +a g \right )^{3}}+\frac {\left (-b d \,x^{2}+a c \right ) \left (b x +a \right )^{2} \left (\frac {d i \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (b g x +a g \right )^{3}}+\frac {\left (d i x +c i \right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (b g x +a g \right )^{3}}-\frac {3 \left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) b g}{\left (b g x +a g \right )^{4}}\right )}{4 a \left (d a -b c \right ) b}\) | \(293\) |
Input:
int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x,method=_RETURN VERBOSE)
Output:
i*A/g^3*(-1/2*(-a*d+b*c)/b^2/(b*x+a)^2-d/b^2/(b*x+a))-i*B/g^3/(a*d-b*c)*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)
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (81) = 162\).
Time = 0.08 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.08 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {2 \, {\left ({\left (2 \, A + B\right )} b^{2} c d - {\left (2 \, A + B\right )} a b d^{2}\right )} i x + {\left ({\left (2 \, A + B\right )} b^{2} c^{2} - {\left (2 \, A + B\right )} a^{2} d^{2}\right )} i + 2 \, {\left (B b^{2} d^{2} i x^{2} + 2 \, B b^{2} c d i x + B b^{2} c^{2} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algori thm="fricas")
Output:
-1/4*(2*((2*A + B)*b^2*c*d - (2*A + B)*a*b*d^2)*i*x + ((2*A + B)*b^2*c^2 - (2*A + B)*a^2*d^2)*i + 2*(B*b^2*d^2*i*x^2 + 2*B*b^2*c*d*i*x + B*b^2*c^2*i )*log((b*e*x + a*e)/(d*x + c)))/((b^5*c - a*b^4*d)*g^3*x^2 + 2*(a*b^4*c - a^2*b^3*d)*g^3*x + (a^2*b^3*c - a^3*b^2*d)*g^3)
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (73) = 146\).
Time = 2.30 (sec) , antiderivative size = 384, normalized size of antiderivative = 4.52 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=- \frac {B d^{2} i \log {\left (x + \frac {- \frac {B a^{2} d^{4} i}{a d - b c} + \frac {2 B a b c d^{3} i}{a d - b c} + B a d^{3} i - \frac {B b^{2} c^{2} d^{2} i}{a d - b c} + B b c d^{2} i}{2 B b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {B d^{2} i \log {\left (x + \frac {\frac {B a^{2} d^{4} i}{a d - b c} - \frac {2 B a b c d^{3} i}{a d - b c} + B a d^{3} i + \frac {B b^{2} c^{2} d^{2} i}{a d - b c} + B b c d^{2} i}{2 B b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {- 2 A a d i - 2 A b c i - B a d i - B b c i + x \left (- 4 A b d i - 2 B b d i\right )}{4 a^{2} b^{2} g^{3} + 8 a b^{3} g^{3} x + 4 b^{4} g^{3} x^{2}} + \frac {\left (- B a d i - B b c i - 2 B b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} b^{2} g^{3} + 4 a b^{3} g^{3} x + 2 b^{4} g^{3} x^{2}} \] Input:
integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**3,x)
Output:
-B*d**2*i*log(x + (-B*a**2*d**4*i/(a*d - b*c) + 2*B*a*b*c*d**3*i/(a*d - b* c) + B*a*d**3*i - B*b**2*c**2*d**2*i/(a*d - b*c) + B*b*c*d**2*i)/(2*B*b*d* *3*i))/(2*b**2*g**3*(a*d - b*c)) + B*d**2*i*log(x + (B*a**2*d**4*i/(a*d - b*c) - 2*B*a*b*c*d**3*i/(a*d - b*c) + B*a*d**3*i + B*b**2*c**2*d**2*i/(a*d - b*c) + B*b*c*d**2*i)/(2*B*b*d**3*i))/(2*b**2*g**3*(a*d - b*c)) + (-2*A* a*d*i - 2*A*b*c*i - B*a*d*i - B*b*c*i + x*(-4*A*b*d*i - 2*B*b*d*i))/(4*a** 2*b**2*g**3 + 8*a*b**3*g**3*x + 4*b**4*g**3*x**2) + (-B*a*d*i - B*b*c*i - 2*B*b*d*i*x)*log(e*(a + b*x)/(c + d*x))/(2*a**2*b**2*g**3 + 4*a*b**3*g**3* x + 2*b**4*g**3*x**2)
Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (81) = 162\).
Time = 0.05 (sec) , antiderivative size = 570, normalized size of antiderivative = 6.71 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {1}{4} \, B d i {\left (\frac {2 \, {\left (2 \, b x + a\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}} + \frac {3 \, a b c - a^{2} d + 2 \, {\left (2 \, b^{2} c - a b d\right )} x}{{\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}} + \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (b x + a\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}} - \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}}\right )} + \frac {1}{4} \, B c i {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {{\left (2 \, b x + a\right )} A d i}{2 \, {\left (b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}\right )}} - \frac {A c i}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algori thm="maxima")
Output:
-1/4*B*d*i*(2*(2*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^4*g^3*x^ 2 + 2*a*b^3*g^3*x + a^2*b^2*g^3) + (3*a*b*c - a^2*d + 2*(2*b^2*c - a*b*d)* x)/((b^5*c - a*b^4*d)*g^3*x^2 + 2*(a*b^4*c - a^2*b^3*d)*g^3*x + (a^2*b^3*c - a^3*b^2*d)*g^3) + 2*(2*b*c*d - a*d^2)*log(b*x + a)/((b^4*c^2 - 2*a*b^3* c*d + a^2*b^2*d^2)*g^3) - 2*(2*b*c*d - a*d^2)*log(d*x + c)/((b^4*c^2 - 2*a *b^3*c*d + a^2*b^2*d^2)*g^3)) + 1/4*B*c*i*((2*b*d*x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - a^2*b^2*d)*g^3*x + (a^2*b^2*c - a^3*b*d )*g^3) - 2*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^3*g^3*x^2 + 2*a*b^2*g^3 *x + a^2*b*g^3) + 2*d^2*log(b*x + a)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)* g^3) - 2*d^2*log(d*x + c)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3)) - 1/2 *(2*b*x + a)*A*d*i/(b^4*g^3*x^2 + 2*a*b^3*g^3*x + a^2*b^2*g^3) - 1/2*A*c*i /(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3)
Time = 0.22 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.58 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (d x + c\right )}^{2} B e^{3} i \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{2} g^{3}} + \frac {{\left (2 \, A e^{3} i + B e^{3} i\right )} {\left (d x + c\right )}^{2}}{{\left (b e x + a e\right )}^{2} g^{3}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algori thm="giac")
Output:
-1/4*(2*(d*x + c)^2*B*e^3*i*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^2* g^3) + (2*A*e^3*i + B*e^3*i)*(d*x + c)^2/((b*e*x + a*e)^2*g^3))*(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 = 27.16 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.32 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {x\,\left (2\,A\,b\,d\,i+B\,b\,d\,i\right )+A\,a\,d\,i+A\,b\,c\,i+\frac {B\,a\,d\,i}{2}+\frac {B\,b\,c\,i}{2}}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,c\,i}{2\,b^2\,g^3}+\frac {B\,a\,d\,i}{2\,b^3\,g^3}+\frac {B\,d\,i\,x}{b^2\,g^3}\right )}{2\,a\,x+b\,x^2+\frac {a^2}{b}}-\frac {B\,d^2\,i\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \] Input:
int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^3,x )
Output:
- (x*(2*A*b*d*i + B*b*d*i) + A*a*d*i + A*b*c*i + (B*a*d*i)/2 + (B*b*c*i)/2 )/(2*a^2*b^2*g^3 + 2*b^4*g^3*x^2 + 4*a*b^3*g^3*x) - (log((e*(a + b*x))/(c + d*x))*((B*c*i)/(2*b^2*g^3) + (B*a*d*i)/(2*b^3*g^3) + (B*d*i*x)/(b^2*g^3) ))/(2*a*x + b*x^2 + a^2/b) - (B*d^2*i*atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*1i)/(b^2*g^3*(a*d - b*c))
Time = 0.20 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.94 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=\frac {i \left (2 \,\mathrm {log}\left (b x +a \right ) a^{2} b c d +4 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c d x +2 \,\mathrm {log}\left (b x +a \right ) b^{3} c d \,x^{2}-2 \,\mathrm {log}\left (d x +c \right ) a^{2} b c d -4 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c d x -2 \,\mathrm {log}\left (d x +c \right ) b^{3} c d \,x^{2}-2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b c d +2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{2} c^{2}+2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{2} d^{2} x^{2}-2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{3} c d \,x^{2}-2 a^{3} c d +2 a^{2} b \,c^{2}-a^{2} b c d +2 a^{2} b \,d^{2} x^{2}+a \,b^{2} c^{2}-2 a \,b^{2} c d \,x^{2}+a \,b^{2} d^{2} x^{2}-b^{3} c d \,x^{2}\right )}{4 a b \,g^{3} \left (a \,b^{2} d \,x^{2}-b^{3} c \,x^{2}+2 a^{2} b d x -2 a \,b^{2} c x +a^{3} d -a^{2} b c \right )} \] Input:
int((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x)
Output:
(i*(2*log(a + b*x)*a**2*b*c*d + 4*log(a + b*x)*a*b**2*c*d*x + 2*log(a + b* x)*b**3*c*d*x**2 - 2*log(c + d*x)*a**2*b*c*d - 4*log(c + d*x)*a*b**2*c*d*x - 2*log(c + d*x)*b**3*c*d*x**2 - 2*log((a*e + b*e*x)/(c + d*x))*a**2*b*c* d + 2*log((a*e + b*e*x)/(c + d*x))*a*b**2*c**2 + 2*log((a*e + b*e*x)/(c + d*x))*a*b**2*d**2*x**2 - 2*log((a*e + b*e*x)/(c + d*x))*b**3*c*d*x**2 - 2* a**3*c*d + 2*a**2*b*c**2 - a**2*b*c*d + 2*a**2*b*d**2*x**2 + a*b**2*c**2 - 2*a*b**2*c*d*x**2 + a*b**2*d**2*x**2 - b**3*c*d*x**2))/(4*a*b*g**3*(a**3* d - a**2*b*c + 2*a**2*b*d*x - 2*a*b**2*c*x + a*b**2*d*x**2 - b**3*c*x**2))