3.7 \(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^3} \, dx\)

Optimal. Leaf size=85 \[ -\frac {i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B i (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \]

[Out]

-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

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Rubi [B]  time = 0.28, antiderivative size = 191, normalized size of antiderivative = 2.25, number of steps used = 10, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2528, 2525, 12, 44} \[ -\frac {d i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (a+b x)}-\frac {i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^3 (a+b x)^2}-\frac {B d^2 i \log (a+b x)}{2 b^2 g^3 (b c-a d)}+\frac {B d^2 i \log (c+d x)}{2 b^2 g^3 (b c-a d)}-\frac {B i (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {B d i}{2 b^2 g^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^3,x]

[Out]

-(B*(b*c - a*d)*i)/(4*b^2*g^3*(a + b*x)^2) - (B*d*i)/(2*b^2*g^3*(a + b*x)) - (B*d^2*i*Log[a + b*x])/(2*b^2*(b*
c - a*d)*g^3) - ((b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^2*g^3*(a + b*x)^2) - (d*i*(A + B*Log
[(e*(a + b*x))/(c + d*x)]))/(b^2*g^3*(a + b*x)) + (B*d^2*i*Log[c + d*x])/(2*b^2*(b*c - a*d)*g^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {(7 c+7 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac {7 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^3 (a+b x)^3}+\frac {7 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^3 (a+b x)^2}\right ) \, dx\\ &=\frac {(7 d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac {(7 (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac {7 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {7 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac {(7 B d) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {(7 B (b c-a d)) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac {7 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {7 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac {(7 B d (b c-a d)) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac {\left (7 B (b c-a d)^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac {7 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {7 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac {(7 B d (b c-a d)) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^3}+\frac {\left (7 B (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^2 g^3}\\ &=-\frac {7 B (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac {7 B d}{2 b^2 g^3 (a+b x)}-\frac {7 B d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac {7 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {7 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}+\frac {7 B d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}\\ \end {align*}

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Mathematica [B]  time = 0.15, size = 208, normalized size = 2.45 \[ \frac {i \left (-\frac {d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 (a+b x)}-\frac {(b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 (a+b x)^2}-\frac {B \left (-\frac {2 d^2 \log (a+b x)}{b c-a d}+\frac {2 d^2 \log (c+d x)}{b c-a d}+\frac {b c-a d}{(a+b x)^2}-\frac {2 d}{a+b x}\right )}{4 b^2}-\frac {B d \left (\frac {d \log (a+b x)}{b c-a d}-\frac {d \log (c+d x)}{b c-a d}+\frac {1}{a+b x}\right )}{b^2}\right )}{g^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^3,x]

[Out]

(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

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fricas [B]  time = 0.76, size = 177, normalized size = 2.08 \[ -\frac {2 \, {\left ({\left (2 \, A + B\right )} b^{2} c d - {\left (2 \, A + B\right )} a b d^{2}\right )} i x + {\left ({\left (2 \, A + B\right )} b^{2} c^{2} - {\left (2 \, A + B\right )} a^{2} d^{2}\right )} i + 2 \, {\left (B b^{2} d^{2} i x^{2} + 2 \, B b^{2} c d i x + B b^{2} c^{2} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algorithm="fricas")

[Out]

-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)

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giac [A]  time = 1.11, size = 117, normalized size = 1.38 \[ -\frac {{\left (2 \, B i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) + 2 \, A i e^{3} + B i e^{3}\right )} {\left (d x + c\right )}^{2} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{4 \, {\left (b x e + a e\right )}^{2} g^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algorithm="giac")

[Out]

-1/4*(2*B*i*e^3*log((b*x*e + a*e)/(d*x + c)) + 2*A*i*e^3 + B*i*e^3)*(d*x + c)^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)))/((b*x*e + a*e)^2*g^3)

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maple [B]  time = 0.05, size = 394, normalized size = 4.64 \[ \frac {B a d \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}-\frac {B b c \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}+\frac {A a d \,e^{2} i}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}-\frac {A b c \,e^{2} i}{2 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}+\frac {B a d \,e^{2} i}{4 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}}-\frac {B b c \,e^{2} i}{4 \left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)*(B*ln((b*x+a)/(d*x+c)*e)+A)/(b*g*x+a*g)^3,x)

[Out]

1/2*d*e^2*i/(a*d-b*c)^2/g^3*A/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*a-1/2*e^2*i/(a*d-b*c)^2/g^3*A/(1/(d*x+
c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*b*c+1/2*d*e^2*i/(a*d-b*c)^2/g^3*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*
ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a-1/2*e^2*i/(a*d-b*c)^2/g^3*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d
*e+(a*d-b*c)/(d*x+c)/d*e)*b*c+1/4*d*e^2*i/(a*d-b*c)^2/g^3*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*a-1/4*e^
2*i/(a*d-b*c)^2/g^3*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*b*c

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maxima [B]  time = 1.32, size = 570, normalized size = 6.71 \[ -\frac {1}{4} \, B d i {\left (\frac {2 \, {\left (2 \, b x + a\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}} + \frac {3 \, a b c - a^{2} d + 2 \, {\left (2 \, b^{2} c - a b d\right )} x}{{\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}} + \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (b x + a\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}} - \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}}\right )} + \frac {1}{4} \, B c i {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {{\left (2 \, b x + a\right )} A d i}{2 \, {\left (b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}\right )}} - \frac {A c i}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algorithm="maxima")

[Out]

-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)

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mupad [B]  time = 5.58, size = 197, normalized size = 2.32 \[ -\frac {x\,\left (2\,A\,b\,d\,i+B\,b\,d\,i\right )+A\,a\,d\,i+A\,b\,c\,i+\frac {B\,a\,d\,i}{2}+\frac {B\,b\,c\,i}{2}}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,c\,i}{2\,b^2\,g^3}+\frac {B\,a\,d\,i}{2\,b^3\,g^3}+\frac {B\,d\,i\,x}{b^2\,g^3}\right )}{2\,a\,x+b\,x^2+\frac {a^2}{b}}-\frac {B\,d^2\,i\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^3,x)

[Out]

- (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))

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sympy [B]  time = 6.25, size = 384, normalized size = 4.52 \[ - \frac {B d^{2} i \log {\left (x + \frac {- \frac {B a^{2} d^{4} i}{a d - b c} + \frac {2 B a b c d^{3} i}{a d - b c} + B a d^{3} i - \frac {B b^{2} c^{2} d^{2} i}{a d - b c} + B b c d^{2} i}{2 B b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {B d^{2} i \log {\left (x + \frac {\frac {B a^{2} d^{4} i}{a d - b c} - \frac {2 B a b c d^{3} i}{a d - b c} + B a d^{3} i + \frac {B b^{2} c^{2} d^{2} i}{a d - b c} + B b c d^{2} i}{2 B b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac {- 2 A a d i - 2 A b c i - B a d i - B b c i + x \left (- 4 A b d i - 2 B b d i\right )}{4 a^{2} b^{2} g^{3} + 8 a b^{3} g^{3} x + 4 b^{4} g^{3} x^{2}} + \frac {\left (- B a d i - B b c i - 2 B b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} b^{2} g^{3} + 4 a b^{3} g^{3} x + 2 b^{4} g^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**3,x)

[Out]

-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)

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