3.66 \(\int \frac{A+B \log (e (\frac{a+b x}{c+d x})^n)}{(f+g x)^5} \, dx\)

Optimal. Leaf size=388 \[ -\frac{B n (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{4 (f+g x) (b f-a g)^3 (d f-c g)^3}-\frac{B n (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f) \left (-a^2 d^2 g^2+2 a b d^2 f g+b^2 \left (-\left (c^2 g^2-2 c d f g+2 d^2 f^2\right )\right )\right )}{4 (b f-a g)^4 (d f-c g)^4}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{4 g (f+g x)^4}+\frac{b^4 B n \log (a+b x)}{4 g (b f-a g)^4}-\frac{B n (b c-a d) (-a d g-b c g+2 b d f)}{8 (f+g x)^2 (b f-a g)^2 (d f-c g)^2}-\frac{B n (b c-a d)}{12 (f+g x)^3 (b f-a g) (d f-c g)}-\frac{B d^4 n \log (c+d x)}{4 g (d f-c g)^4} \]

[Out]

-(B*(b*c - a*d)*n)/(12*(b*f - a*g)*(d*f - c*g)*(f + g*x)^3) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*n)/(8*(
b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)^2) - (B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2
- 3*c*d*f*g + c^2*g^2))*n)/(4*(b*f - a*g)^3*(d*f - c*g)^3*(f + g*x)) + (b^4*B*n*Log[a + b*x])/(4*g*(b*f - a*g
)^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(4*g*(f + g*x)^4) - (B*d^4*n*Log[c + d*x])/(4*g*(d*f - c*g)^4) -
(B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2)
)*n*Log[f + g*x])/(4*(b*f - a*g)^4*(d*f - c*g)^4)

________________________________________________________________________________________

Rubi [A]  time = 0.713149, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 72} \[ -\frac{B n (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{4 (f+g x) (b f-a g)^3 (d f-c g)^3}-\frac{B n (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f) \left (-a^2 d^2 g^2+2 a b d^2 f g+b^2 \left (-\left (c^2 g^2-2 c d f g+2 d^2 f^2\right )\right )\right )}{4 (b f-a g)^4 (d f-c g)^4}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{4 g (f+g x)^4}+\frac{b^4 B n \log (a+b x)}{4 g (b f-a g)^4}-\frac{B n (b c-a d) (-a d g-b c g+2 b d f)}{8 (f+g x)^2 (b f-a g)^2 (d f-c g)^2}-\frac{B n (b c-a d)}{12 (f+g x)^3 (b f-a g) (d f-c g)}-\frac{B d^4 n \log (c+d x)}{4 g (d f-c g)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^5,x]

[Out]

-(B*(b*c - a*d)*n)/(12*(b*f - a*g)*(d*f - c*g)*(f + g*x)^3) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*n)/(8*(
b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)^2) - (B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2
- 3*c*d*f*g + c^2*g^2))*n)/(4*(b*f - a*g)^3*(d*f - c*g)^3*(f + g*x)) + (b^4*B*n*Log[a + b*x])/(4*g*(b*f - a*g
)^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(4*g*(f + g*x)^4) - (B*d^4*n*Log[c + d*x])/(4*g*(d*f - c*g)^4) -
(B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2)
)*n*Log[f + g*x])/(4*(b*f - a*g)^4*(d*f - c*g)^4)

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 12

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(f+g x)^5} \, dx &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 g (f+g x)^4}+\frac{(B n) \int \frac{b c-a d}{(a+b x) (c+d x) (f+g x)^4} \, dx}{4 g}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 g (f+g x)^4}+\frac{(B (b c-a d) n) \int \frac{1}{(a+b x) (c+d x) (f+g x)^4} \, dx}{4 g}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 g (f+g x)^4}+\frac{(B (b c-a d) n) \int \left (\frac{b^5}{(b c-a d) (b f-a g)^4 (a+b x)}-\frac{d^5}{(b c-a d) (-d f+c g)^4 (c+d x)}+\frac{g^2}{(b f-a g) (d f-c g) (f+g x)^4}-\frac{g^2 (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)^3}+\frac{g^2 \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)^2}+\frac{g^2 (2 b d f-b c g-a d g) \left (2 b^2 d^2 f^2-2 b^2 c d f g-2 a b d^2 f g+b^2 c^2 g^2+a^2 d^2 g^2\right )}{(b f-a g)^4 (d f-c g)^4 (f+g x)}\right ) \, dx}{4 g}\\ &=-\frac{B (b c-a d) n}{12 (b f-a g) (d f-c g) (f+g x)^3}-\frac{B (b c-a d) (2 b d f-b c g-a d g) n}{8 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac{B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n}{4 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac{b^4 B n \log (a+b x)}{4 g (b f-a g)^4}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 g (f+g x)^4}-\frac{B d^4 n \log (c+d x)}{4 g (d f-c g)^4}-\frac{B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n \log (f+g x)}{4 (b f-a g)^4 (d f-c g)^4}\\ \end{align*}

Mathematica [A]  time = 1.35501, size = 359, normalized size = 0.93 \[ \frac{B n (b c-a d) \left (-\frac{g \left (a^2 d^2 g^2+a b d g (c g-3 d f)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{(f+g x) (b f-a g)^3 (d f-c g)^3}-\frac{g \log (f+g x) (a d g+b c g-2 b d f) \left (a^2 d^2 g^2-2 a b d^2 f g+b^2 \left (c^2 g^2-2 c d f g+2 d^2 f^2\right )\right )}{(b f-a g)^4 (d f-c g)^4}+\frac{b^4 \log (a+b x)}{(b c-a d) (b f-a g)^4}-\frac{d^4 \log (c+d x)}{(b c-a d) (d f-c g)^4}+\frac{g (a d g+b c g-2 b d f)}{2 (f+g x)^2 (b f-a g)^2 (d f-c g)^2}-\frac{g}{3 (f+g x)^3 (b f-a g) (d f-c g)}\right )-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{(f+g x)^4}}{4 g} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^5,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.502, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{5}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^5,x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^5,x)

________________________________________________________________________________________

Maxima [B]  time = 1.96833, size = 2377, normalized size = 6.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^5,x, algorithm="maxima")

[Out]

1/24*(6*b^4*log(b*x + a)/(b^4*f^4*g - 4*a*b^3*f^3*g^2 + 6*a^2*b^2*f^2*g^3 - 4*a^3*b*f*g^4 + a^4*g^5) - 6*d^4*l
og(d*x + c)/(d^4*f^4*g - 4*c*d^3*f^3*g^2 + 6*c^2*d^2*f^2*g^3 - 4*c^3*d*f*g^4 + c^4*g^5) + 6*(4*(b^4*c*d^3 - a*
b^3*d^4)*f^3 - 6*(b^4*c^2*d^2 - a^2*b^2*d^4)*f^2*g + 4*(b^4*c^3*d - a^3*b*d^4)*f*g^2 - (b^4*c^4 - a^4*d^4)*g^3
)*log(g*x + f)/(b^4*d^4*f^8 + a^4*c^4*g^8 - 4*(b^4*c*d^3 + a*b^3*d^4)*f^7*g + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3
+ 3*a^2*b^2*d^4)*f^6*g^2 - 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*f^5*g^3 + (b^4*c^4 +
16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 + a^4*d^4)*f^4*g^4 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*
a^3*b*c^2*d^2 + a^4*c*d^3)*f^3*g^5 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*f^2*g^6 - 4*(a^3*b*c^4
+ a^4*c^3*d)*f*g^7) - (26*(b^3*c*d^2 - a*b^2*d^3)*f^4 - 31*(b^3*c^2*d - a^2*b*d^3)*f^3*g + (11*b^3*c^3 + 15*a*
b^2*c^2*d - 15*a^2*b*c*d^2 - 11*a^3*d^3)*f^2*g^2 - 7*(a*b^2*c^3 - a^3*c*d^2)*f*g^3 + 2*(a^2*b*c^3 - a^3*c^2*d)
*g^4 + 6*(3*(b^3*c*d^2 - a*b^2*d^3)*f^2*g^2 - 3*(b^3*c^2*d - a^2*b*d^3)*f*g^3 + (b^3*c^3 - a^3*d^3)*g^4)*x^2 +
3*(14*(b^3*c*d^2 - a*b^2*d^3)*f^3*g - 15*(b^3*c^2*d - a^2*b*d^3)*f^2*g^2 + (5*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2
*b*c*d^2 - 5*a^3*d^3)*f*g^3 - (a*b^2*c^3 - a^3*c*d^2)*g^4)*x)/(b^3*d^3*f^9 + a^3*c^3*f^3*g^6 - 3*(b^3*c*d^2 +
a*b^2*d^3)*f^8*g + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f^7*g^2 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^
2 + a^3*d^3)*f^6*g^3 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*f^5*g^4 - 3*(a^2*b*c^3 + a^3*c^2*d)*f^4*g^5 +
(b^3*d^3*f^6*g^3 + a^3*c^3*g^9 - 3*(b^3*c*d^2 + a*b^2*d^3)*f^5*g^4 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3
)*f^4*g^5 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*f^3*g^6 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c
*d^2)*f^2*g^7 - 3*(a^2*b*c^3 + a^3*c^2*d)*f*g^8)*x^3 + 3*(b^3*d^3*f^7*g^2 + a^3*c^3*f*g^8 - 3*(b^3*c*d^2 + a*b
^2*d^3)*f^6*g^3 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f^5*g^4 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2
+ a^3*d^3)*f^4*g^5 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*f^3*g^6 - 3*(a^2*b*c^3 + a^3*c^2*d)*f^2*g^7)*x
^2 + 3*(b^3*d^3*f^8*g + a^3*c^3*f^2*g^7 - 3*(b^3*c*d^2 + a*b^2*d^3)*f^7*g^2 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a
^2*b*d^3)*f^6*g^3 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*f^5*g^4 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d
+ a^3*c*d^2)*f^4*g^5 - 3*(a^2*b*c^3 + a^3*c^2*d)*f^3*g^6)*x))*B*n - 1/4*B*log(e*(b*x/(d*x + c) + a/(d*x + c))
^n)/(g^5*x^4 + 4*f*g^4*x^3 + 6*f^2*g^3*x^2 + 4*f^3*g^2*x + f^4*g) - 1/4*A/(g^5*x^4 + 4*f*g^4*x^3 + 6*f^2*g^3*x
^2 + 4*f^3*g^2*x + f^4*g)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^5,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(g*x+f)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 14.5389, size = 5863, normalized size = 15.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^5,x, algorithm="giac")

[Out]

-1/4*B*n*log((b*x + a)/(d*x + c))/(g^5*x^4 + 4*f*g^4*x^3 + 6*f^2*g^3*x^2 + 4*f^3*g^2*x + f^4*g) + 1/4*(4*B*b^4
*c*d^3*f^3*n - 4*B*a*b^3*d^4*f^3*n - 6*B*b^4*c^2*d^2*f^2*g*n + 6*B*a^2*b^2*d^4*f^2*g*n + 4*B*b^4*c^3*d*f*g^2*n
- 4*B*a^3*b*d^4*f*g^2*n - B*b^4*c^4*g^3*n + B*a^4*d^4*g^3*n)*log(g*x + f)/(b^4*d^4*f^8 - 4*b^4*c*d^3*f^7*g -
4*a*b^3*d^4*f^7*g + 6*b^4*c^2*d^2*f^6*g^2 + 16*a*b^3*c*d^3*f^6*g^2 + 6*a^2*b^2*d^4*f^6*g^2 - 4*b^4*c^3*d*f^5*g
^3 - 24*a*b^3*c^2*d^2*f^5*g^3 - 24*a^2*b^2*c*d^3*f^5*g^3 - 4*a^3*b*d^4*f^5*g^3 + b^4*c^4*f^4*g^4 + 16*a*b^3*c^
3*d*f^4*g^4 + 36*a^2*b^2*c^2*d^2*f^4*g^4 + 16*a^3*b*c*d^3*f^4*g^4 + a^4*d^4*f^4*g^4 - 4*a*b^3*c^4*f^3*g^5 - 24
*a^2*b^2*c^3*d*f^3*g^5 - 24*a^3*b*c^2*d^2*f^3*g^5 - 4*a^4*c*d^3*f^3*g^5 + 6*a^2*b^2*c^4*f^2*g^6 + 16*a^3*b*c^3
*d*f^2*g^6 + 6*a^4*c^2*d^2*f^2*g^6 - 4*a^3*b*c^4*f*g^7 - 4*a^4*c^3*d*f*g^7 + a^4*c^4*g^8) - 1/8*(4*B*b^4*c*d^3
*f^3*n - 4*B*a*b^3*d^4*f^3*n - 6*B*b^4*c^2*d^2*f^2*g*n + 6*B*a^2*b^2*d^4*f^2*g*n + 4*B*b^4*c^3*d*f*g^2*n - 4*B
*a^3*b*d^4*f*g^2*n - B*b^4*c^4*g^3*n + B*a^4*d^4*g^3*n)*log(abs(b*d*x^2 + b*c*x + a*d*x + a*c))/(b^4*d^4*f^8 -
4*b^4*c*d^3*f^7*g - 4*a*b^3*d^4*f^7*g + 6*b^4*c^2*d^2*f^6*g^2 + 16*a*b^3*c*d^3*f^6*g^2 + 6*a^2*b^2*d^4*f^6*g^
2 - 4*b^4*c^3*d*f^5*g^3 - 24*a*b^3*c^2*d^2*f^5*g^3 - 24*a^2*b^2*c*d^3*f^5*g^3 - 4*a^3*b*d^4*f^5*g^3 + b^4*c^4*
f^4*g^4 + 16*a*b^3*c^3*d*f^4*g^4 + 36*a^2*b^2*c^2*d^2*f^4*g^4 + 16*a^3*b*c*d^3*f^4*g^4 + a^4*d^4*f^4*g^4 - 4*a
*b^3*c^4*f^3*g^5 - 24*a^2*b^2*c^3*d*f^3*g^5 - 24*a^3*b*c^2*d^2*f^3*g^5 - 4*a^4*c*d^3*f^3*g^5 + 6*a^2*b^2*c^4*f
^2*g^6 + 16*a^3*b*c^3*d*f^2*g^6 + 6*a^4*c^2*d^2*f^2*g^6 - 4*a^3*b*c^4*f*g^7 - 4*a^4*c^3*d*f*g^7 + a^4*c^4*g^8)
- 1/24*(18*B*b^3*c*d^2*f^2*g^4*n*x^3 - 18*B*a*b^2*d^3*f^2*g^4*n*x^3 - 18*B*b^3*c^2*d*f*g^5*n*x^3 + 18*B*a^2*b
*d^3*f*g^5*n*x^3 + 6*B*b^3*c^3*g^6*n*x^3 - 6*B*a^3*d^3*g^6*n*x^3 + 60*B*b^3*c*d^2*f^3*g^3*n*x^2 - 60*B*a*b^2*d
^3*f^3*g^3*n*x^2 - 63*B*b^3*c^2*d*f^2*g^4*n*x^2 + 63*B*a^2*b*d^3*f^2*g^4*n*x^2 + 21*B*b^3*c^3*f*g^5*n*x^2 + 9*
B*a*b^2*c^2*d*f*g^5*n*x^2 - 9*B*a^2*b*c*d^2*f*g^5*n*x^2 - 21*B*a^3*d^3*f*g^5*n*x^2 - 3*B*a*b^2*c^3*g^6*n*x^2 +
3*B*a^3*c*d^2*g^6*n*x^2 + 68*B*b^3*c*d^2*f^4*g^2*n*x - 68*B*a*b^2*d^3*f^4*g^2*n*x - 76*B*b^3*c^2*d*f^3*g^3*n*
x + 76*B*a^2*b*d^3*f^3*g^3*n*x + 26*B*b^3*c^3*f^2*g^4*n*x + 24*B*a*b^2*c^2*d*f^2*g^4*n*x - 24*B*a^2*b*c*d^2*f^
2*g^4*n*x - 26*B*a^3*d^3*f^2*g^4*n*x - 10*B*a*b^2*c^3*f*g^5*n*x + 10*B*a^3*c*d^2*f*g^5*n*x + 2*B*a^2*b*c^3*g^6
*n*x - 2*B*a^3*c^2*d*g^6*n*x + 26*B*b^3*c*d^2*f^5*g*n - 26*B*a*b^2*d^3*f^5*g*n - 31*B*b^3*c^2*d*f^4*g^2*n + 31
*B*a^2*b*d^3*f^4*g^2*n + 11*B*b^3*c^3*f^3*g^3*n + 15*B*a*b^2*c^2*d*f^3*g^3*n - 15*B*a^2*b*c*d^2*f^3*g^3*n - 11
*B*a^3*d^3*f^3*g^3*n - 7*B*a*b^2*c^3*f^2*g^4*n + 7*B*a^3*c*d^2*f^2*g^4*n + 2*B*a^2*b*c^3*f*g^5*n - 2*B*a^3*c^2
*d*f*g^5*n + 6*A*b^3*d^3*f^6 + 6*B*b^3*d^3*f^6 - 18*A*b^3*c*d^2*f^5*g - 18*B*b^3*c*d^2*f^5*g - 18*A*a*b^2*d^3*
f^5*g - 18*B*a*b^2*d^3*f^5*g + 18*A*b^3*c^2*d*f^4*g^2 + 18*B*b^3*c^2*d*f^4*g^2 + 54*A*a*b^2*c*d^2*f^4*g^2 + 54
*B*a*b^2*c*d^2*f^4*g^2 + 18*A*a^2*b*d^3*f^4*g^2 + 18*B*a^2*b*d^3*f^4*g^2 - 6*A*b^3*c^3*f^3*g^3 - 6*B*b^3*c^3*f
^3*g^3 - 54*A*a*b^2*c^2*d*f^3*g^3 - 54*B*a*b^2*c^2*d*f^3*g^3 - 54*A*a^2*b*c*d^2*f^3*g^3 - 54*B*a^2*b*c*d^2*f^3
*g^3 - 6*A*a^3*d^3*f^3*g^3 - 6*B*a^3*d^3*f^3*g^3 + 18*A*a*b^2*c^3*f^2*g^4 + 18*B*a*b^2*c^3*f^2*g^4 + 54*A*a^2*
b*c^2*d*f^2*g^4 + 54*B*a^2*b*c^2*d*f^2*g^4 + 18*A*a^3*c*d^2*f^2*g^4 + 18*B*a^3*c*d^2*f^2*g^4 - 18*A*a^2*b*c^3*
f*g^5 - 18*B*a^2*b*c^3*f*g^5 - 18*A*a^3*c^2*d*f*g^5 - 18*B*a^3*c^2*d*f*g^5 + 6*A*a^3*c^3*g^6 + 6*B*a^3*c^3*g^6
)/(b^3*d^3*f^6*g^5*x^4 - 3*b^3*c*d^2*f^5*g^6*x^4 - 3*a*b^2*d^3*f^5*g^6*x^4 + 3*b^3*c^2*d*f^4*g^7*x^4 + 9*a*b^2
*c*d^2*f^4*g^7*x^4 + 3*a^2*b*d^3*f^4*g^7*x^4 - b^3*c^3*f^3*g^8*x^4 - 9*a*b^2*c^2*d*f^3*g^8*x^4 - 9*a^2*b*c*d^2
*f^3*g^8*x^4 - a^3*d^3*f^3*g^8*x^4 + 3*a*b^2*c^3*f^2*g^9*x^4 + 9*a^2*b*c^2*d*f^2*g^9*x^4 + 3*a^3*c*d^2*f^2*g^9
*x^4 - 3*a^2*b*c^3*f*g^10*x^4 - 3*a^3*c^2*d*f*g^10*x^4 + a^3*c^3*g^11*x^4 + 4*b^3*d^3*f^7*g^4*x^3 - 12*b^3*c*d
^2*f^6*g^5*x^3 - 12*a*b^2*d^3*f^6*g^5*x^3 + 12*b^3*c^2*d*f^5*g^6*x^3 + 36*a*b^2*c*d^2*f^5*g^6*x^3 + 12*a^2*b*d
^3*f^5*g^6*x^3 - 4*b^3*c^3*f^4*g^7*x^3 - 36*a*b^2*c^2*d*f^4*g^7*x^3 - 36*a^2*b*c*d^2*f^4*g^7*x^3 - 4*a^3*d^3*f
^4*g^7*x^3 + 12*a*b^2*c^3*f^3*g^8*x^3 + 36*a^2*b*c^2*d*f^3*g^8*x^3 + 12*a^3*c*d^2*f^3*g^8*x^3 - 12*a^2*b*c^3*f
^2*g^9*x^3 - 12*a^3*c^2*d*f^2*g^9*x^3 + 4*a^3*c^3*f*g^10*x^3 + 6*b^3*d^3*f^8*g^3*x^2 - 18*b^3*c*d^2*f^7*g^4*x^
2 - 18*a*b^2*d^3*f^7*g^4*x^2 + 18*b^3*c^2*d*f^6*g^5*x^2 + 54*a*b^2*c*d^2*f^6*g^5*x^2 + 18*a^2*b*d^3*f^6*g^5*x^
2 - 6*b^3*c^3*f^5*g^6*x^2 - 54*a*b^2*c^2*d*f^5*g^6*x^2 - 54*a^2*b*c*d^2*f^5*g^6*x^2 - 6*a^3*d^3*f^5*g^6*x^2 +
18*a*b^2*c^3*f^4*g^7*x^2 + 54*a^2*b*c^2*d*f^4*g^7*x^2 + 18*a^3*c*d^2*f^4*g^7*x^2 - 18*a^2*b*c^3*f^3*g^8*x^2 -
18*a^3*c^2*d*f^3*g^8*x^2 + 6*a^3*c^3*f^2*g^9*x^2 + 4*b^3*d^3*f^9*g^2*x - 12*b^3*c*d^2*f^8*g^3*x - 12*a*b^2*d^3
*f^8*g^3*x + 12*b^3*c^2*d*f^7*g^4*x + 36*a*b^2*c*d^2*f^7*g^4*x + 12*a^2*b*d^3*f^7*g^4*x - 4*b^3*c^3*f^6*g^5*x
- 36*a*b^2*c^2*d*f^6*g^5*x - 36*a^2*b*c*d^2*f^6*g^5*x - 4*a^3*d^3*f^6*g^5*x + 12*a*b^2*c^3*f^5*g^6*x + 36*a^2*
b*c^2*d*f^5*g^6*x + 12*a^3*c*d^2*f^5*g^6*x - 12*a^2*b*c^3*f^4*g^7*x - 12*a^3*c^2*d*f^4*g^7*x + 4*a^3*c^3*f^3*g
^8*x + b^3*d^3*f^10*g - 3*b^3*c*d^2*f^9*g^2 - 3*a*b^2*d^3*f^9*g^2 + 3*b^3*c^2*d*f^8*g^3 + 9*a*b^2*c*d^2*f^8*g^
3 + 3*a^2*b*d^3*f^8*g^3 - b^3*c^3*f^7*g^4 - 9*a*b^2*c^2*d*f^7*g^4 - 9*a^2*b*c*d^2*f^7*g^4 - a^3*d^3*f^7*g^4 +
3*a*b^2*c^3*f^6*g^5 + 9*a^2*b*c^2*d*f^6*g^5 + 3*a^3*c*d^2*f^6*g^5 - 3*a^2*b*c^3*f^5*g^6 - 3*a^3*c^2*d*f^5*g^6
+ a^3*c^3*f^4*g^7) + 1/8*(2*B*b^5*c*d^4*f^4*n - 2*B*a*b^4*d^5*f^4*n - 4*B*b^5*c^2*d^3*f^3*g*n + 4*B*a^2*b^3*d^
5*f^3*g*n + 6*B*b^5*c^3*d^2*f^2*g^2*n - 6*B*a*b^4*c^2*d^3*f^2*g^2*n + 6*B*a^2*b^3*c*d^4*f^2*g^2*n - 6*B*a^3*b^
2*d^5*f^2*g^2*n - 4*B*b^5*c^4*d*f*g^3*n + 4*B*a*b^4*c^3*d^2*f*g^3*n - 4*B*a^3*b^2*c*d^4*f*g^3*n + 4*B*a^4*b*d^
5*f*g^3*n + B*b^5*c^5*g^4*n - B*a*b^4*c^4*d*g^4*n + B*a^4*b*c*d^4*g^4*n - B*a^5*d^5*g^4*n)*log(abs((2*b*d*x +
b*c + a*d - abs(-b*c + a*d))/(2*b*d*x + b*c + a*d + abs(-b*c + a*d))))/((b^4*d^4*f^8*g - 4*b^4*c*d^3*f^7*g^2 -
4*a*b^3*d^4*f^7*g^2 + 6*b^4*c^2*d^2*f^6*g^3 + 16*a*b^3*c*d^3*f^6*g^3 + 6*a^2*b^2*d^4*f^6*g^3 - 4*b^4*c^3*d*f^
5*g^4 - 24*a*b^3*c^2*d^2*f^5*g^4 - 24*a^2*b^2*c*d^3*f^5*g^4 - 4*a^3*b*d^4*f^5*g^4 + b^4*c^4*f^4*g^5 + 16*a*b^3
*c^3*d*f^4*g^5 + 36*a^2*b^2*c^2*d^2*f^4*g^5 + 16*a^3*b*c*d^3*f^4*g^5 + a^4*d^4*f^4*g^5 - 4*a*b^3*c^4*f^3*g^6 -
24*a^2*b^2*c^3*d*f^3*g^6 - 24*a^3*b*c^2*d^2*f^3*g^6 - 4*a^4*c*d^3*f^3*g^6 + 6*a^2*b^2*c^4*f^2*g^7 + 16*a^3*b*
c^3*d*f^2*g^7 + 6*a^4*c^2*d^2*f^2*g^7 - 4*a^3*b*c^4*f*g^8 - 4*a^4*c^3*d*f*g^8 + a^4*c^4*g^9)*abs(-b*c + a*d))