3.34 \(\int \frac{\log (e (f (a+b x)^p (c+d x)^q)^r)}{(g+h x)^5} \, dx\)
Optimal. Leaf size=318 \[ \frac{b^3 p r}{4 h (g+h x) (b g-a h)^3}+\frac{b^2 p r}{8 h (g+h x)^2 (b g-a h)^2}+\frac{b^4 p r \log (a+b x)}{4 h (b g-a h)^4}-\frac{b^4 p r \log (g+h x)}{4 h (b g-a h)^4}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h (g+h x)^4}+\frac{b p r}{12 h (g+h x)^3 (b g-a h)}+\frac{d^3 q r}{4 h (g+h x) (d g-c h)^3}+\frac{d^2 q r}{8 h (g+h x)^2 (d g-c h)^2}+\frac{d^4 q r \log (c+d x)}{4 h (d g-c h)^4}-\frac{d^4 q r \log (g+h x)}{4 h (d g-c h)^4}+\frac{d q r}{12 h (g+h x)^3 (d g-c h)} \]
[Out]
(b*p*r)/(12*h*(b*g - a*h)*(g + h*x)^3) + (d*q*r)/(12*h*(d*g - c*h)*(g + h*x)^3) + (b^2*p*r)/(8*h*(b*g - a*h)^2
*(g + h*x)^2) + (d^2*q*r)/(8*h*(d*g - c*h)^2*(g + h*x)^2) + (b^3*p*r)/(4*h*(b*g - a*h)^3*(g + h*x)) + (d^3*q*r
)/(4*h*(d*g - c*h)^3*(g + h*x)) + (b^4*p*r*Log[a + b*x])/(4*h*(b*g - a*h)^4) + (d^4*q*r*Log[c + d*x])/(4*h*(d*
g - c*h)^4) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(4*h*(g + h*x)^4) - (b^4*p*r*Log[g + h*x])/(4*h*(b*g - a*h)
^4) - (d^4*q*r*Log[g + h*x])/(4*h*(d*g - c*h)^4)
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Rubi [A] time = 0.194835, antiderivative size = 318, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used =
{2495, 44} \[ \frac{b^3 p r}{4 h (g+h x) (b g-a h)^3}+\frac{b^2 p r}{8 h (g+h x)^2 (b g-a h)^2}+\frac{b^4 p r \log (a+b x)}{4 h (b g-a h)^4}-\frac{b^4 p r \log (g+h x)}{4 h (b g-a h)^4}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h (g+h x)^4}+\frac{b p r}{12 h (g+h x)^3 (b g-a h)}+\frac{d^3 q r}{4 h (g+h x) (d g-c h)^3}+\frac{d^2 q r}{8 h (g+h x)^2 (d g-c h)^2}+\frac{d^4 q r \log (c+d x)}{4 h (d g-c h)^4}-\frac{d^4 q r \log (g+h x)}{4 h (d g-c h)^4}+\frac{d q r}{12 h (g+h x)^3 (d g-c h)} \]
Antiderivative was successfully verified.
[In]
Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^5,x]
[Out]
(b*p*r)/(12*h*(b*g - a*h)*(g + h*x)^3) + (d*q*r)/(12*h*(d*g - c*h)*(g + h*x)^3) + (b^2*p*r)/(8*h*(b*g - a*h)^2
*(g + h*x)^2) + (d^2*q*r)/(8*h*(d*g - c*h)^2*(g + h*x)^2) + (b^3*p*r)/(4*h*(b*g - a*h)^3*(g + h*x)) + (d^3*q*r
)/(4*h*(d*g - c*h)^3*(g + h*x)) + (b^4*p*r*Log[a + b*x])/(4*h*(b*g - a*h)^4) + (d^4*q*r*Log[c + d*x])/(4*h*(d*
g - c*h)^4) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(4*h*(g + h*x)^4) - (b^4*p*r*Log[g + h*x])/(4*h*(b*g - a*h)
^4) - (d^4*q*r*Log[g + h*x])/(4*h*(d*g - c*h)^4)
Rule 2495
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^(m_.),
x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(h*(m + 1)), x] + (-Dist[(b*p*r)/(
h*(m + 1)), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(c + d*x
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]
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])
Rubi steps
\begin{align*} \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^5} \, dx &=-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h (g+h x)^4}+\frac{(b p r) \int \frac{1}{(a+b x) (g+h x)^4} \, dx}{4 h}+\frac{(d q r) \int \frac{1}{(c+d x) (g+h x)^4} \, dx}{4 h}\\ &=-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h (g+h x)^4}+\frac{(b p r) \int \left (\frac{b^4}{(b g-a h)^4 (a+b x)}-\frac{h}{(b g-a h) (g+h x)^4}-\frac{b h}{(b g-a h)^2 (g+h x)^3}-\frac{b^2 h}{(b g-a h)^3 (g+h x)^2}-\frac{b^3 h}{(b g-a h)^4 (g+h x)}\right ) \, dx}{4 h}+\frac{(d q r) \int \left (\frac{d^4}{(d g-c h)^4 (c+d x)}-\frac{h}{(d g-c h) (g+h x)^4}-\frac{d h}{(d g-c h)^2 (g+h x)^3}-\frac{d^2 h}{(d g-c h)^3 (g+h x)^2}-\frac{d^3 h}{(d g-c h)^4 (g+h x)}\right ) \, dx}{4 h}\\ &=\frac{b p r}{12 h (b g-a h) (g+h x)^3}+\frac{d q r}{12 h (d g-c h) (g+h x)^3}+\frac{b^2 p r}{8 h (b g-a h)^2 (g+h x)^2}+\frac{d^2 q r}{8 h (d g-c h)^2 (g+h x)^2}+\frac{b^3 p r}{4 h (b g-a h)^3 (g+h x)}+\frac{d^3 q r}{4 h (d g-c h)^3 (g+h x)}+\frac{b^4 p r \log (a+b x)}{4 h (b g-a h)^4}+\frac{d^4 q r \log (c+d x)}{4 h (d g-c h)^4}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h (g+h x)^4}-\frac{b^4 p r \log (g+h x)}{4 h (b g-a h)^4}-\frac{d^4 q r \log (g+h x)}{4 h (d g-c h)^4}\\ \end{align*}
Mathematica [A] time = 1.82202, size = 480, normalized size = 1.51 \[ \frac{\frac{r (g+h x) \left (2 (b g-a h)^3 (d g-c h)^3 (b d g (p+q)-h (a d q+b c p))-(g+h x) \left (6 (g+h x) \left (-(g+h x) \left (-\log (g+h x) \left (6 a^2 b^2 d^4 g^2 h^2 q-4 a^3 b d^4 g h^3 q+a^4 d^4 h^4 q-4 a b^3 d^4 g^3 h q+b^4 \left (6 c^2 d^2 g^2 h^2 p-4 c^3 d g h^3 p+c^4 h^4 p-4 c d^3 g^3 h p+d^4 g^4 (p+q)\right )\right )+b^4 p \log (a+b x) (d g-c h)^4+d^4 q (b g-a h)^4 \log (c+d x)\right )-(b g-a h) (c h-d g) \left (-3 a^2 b d^3 g h^2 q+a^3 d^3 h^3 q+3 a b^2 d^3 g^2 h q+b^3 \left (-\left (3 c^2 d g h^2 p-c^3 h^3 p-3 c d^2 g^2 h p+d^3 g^3 (p+q)\right )\right )\right )\right )+(b g-a h)^2 (d g-c h)^2 \left (-3 a^2 d^2 h^2 q+6 a b d^2 g h q-3 b^2 \left (c^2 h^2 p-2 c d g h p+d^2 g^2 (p+q)\right )\right )\right )\right )}{(b g-a h)^4 (d g-c h)^4}-6 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{24 h (g+h x)^4} \]
Antiderivative was successfully verified.
[In]
Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^5,x]
[Out]
(-6*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (r*(g + h*x)*(2*(b*g - a*h)^3*(d*g - c*h)^3*(b*d*g*(p + q) - h*(b*c
*p + a*d*q)) - (g + h*x)*((b*g - a*h)^2*(d*g - c*h)^2*(6*a*b*d^2*g*h*q - 3*a^2*d^2*h^2*q - 3*b^2*(-2*c*d*g*h*p
+ c^2*h^2*p + d^2*g^2*(p + q))) + 6*(g + h*x)*(-((b*g - a*h)*(-(d*g) + c*h)*(3*a*b^2*d^3*g^2*h*q - 3*a^2*b*d^
3*g*h^2*q + a^3*d^3*h^3*q - b^3*(-3*c*d^2*g^2*h*p + 3*c^2*d*g*h^2*p - c^3*h^3*p + d^3*g^3*(p + q)))) - (g + h*
x)*(b^4*(d*g - c*h)^4*p*Log[a + b*x] + d^4*(b*g - a*h)^4*q*Log[c + d*x] - (-4*a*b^3*d^4*g^3*h*q + 6*a^2*b^2*d^
4*g^2*h^2*q - 4*a^3*b*d^4*g*h^3*q + a^4*d^4*h^4*q + b^4*(-4*c*d^3*g^3*h*p + 6*c^2*d^2*g^2*h^2*p - 4*c^3*d*g*h^
3*p + c^4*h^4*p + d^4*g^4*(p + q)))*Log[g + h*x])))))/((b*g - a*h)^4*(d*g - c*h)^4))/(24*h*(g + h*x)^4)
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Maple [F] time = 0.509, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) }{ \left ( hx+g \right ) ^{5}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^5,x)
[Out]
int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^5,x)
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Maxima [B] time = 1.41247, size = 1048, normalized size = 3.3 \begin{align*} \frac{{\left ({\left (\frac{6 \, b^{3} \log \left (b x + a\right )}{b^{4} g^{4} - 4 \, a b^{3} g^{3} h + 6 \, a^{2} b^{2} g^{2} h^{2} - 4 \, a^{3} b g h^{3} + a^{4} h^{4}} - \frac{6 \, b^{3} \log \left (h x + g\right )}{b^{4} g^{4} - 4 \, a b^{3} g^{3} h + 6 \, a^{2} b^{2} g^{2} h^{2} - 4 \, a^{3} b g h^{3} + a^{4} h^{4}} + \frac{6 \, b^{2} h^{2} x^{2} + 11 \, b^{2} g^{2} - 7 \, a b g h + 2 \, a^{2} h^{2} + 3 \,{\left (5 \, b^{2} g h - a b h^{2}\right )} x}{b^{3} g^{6} - 3 \, a b^{2} g^{5} h + 3 \, a^{2} b g^{4} h^{2} - a^{3} g^{3} h^{3} +{\left (b^{3} g^{3} h^{3} - 3 \, a b^{2} g^{2} h^{4} + 3 \, a^{2} b g h^{5} - a^{3} h^{6}\right )} x^{3} + 3 \,{\left (b^{3} g^{4} h^{2} - 3 \, a b^{2} g^{3} h^{3} + 3 \, a^{2} b g^{2} h^{4} - a^{3} g h^{5}\right )} x^{2} + 3 \,{\left (b^{3} g^{5} h - 3 \, a b^{2} g^{4} h^{2} + 3 \, a^{2} b g^{3} h^{3} - a^{3} g^{2} h^{4}\right )} x}\right )} b f p +{\left (\frac{6 \, d^{3} \log \left (d x + c\right )}{d^{4} g^{4} - 4 \, c d^{3} g^{3} h + 6 \, c^{2} d^{2} g^{2} h^{2} - 4 \, c^{3} d g h^{3} + c^{4} h^{4}} - \frac{6 \, d^{3} \log \left (h x + g\right )}{d^{4} g^{4} - 4 \, c d^{3} g^{3} h + 6 \, c^{2} d^{2} g^{2} h^{2} - 4 \, c^{3} d g h^{3} + c^{4} h^{4}} + \frac{6 \, d^{2} h^{2} x^{2} + 11 \, d^{2} g^{2} - 7 \, c d g h + 2 \, c^{2} h^{2} + 3 \,{\left (5 \, d^{2} g h - c d h^{2}\right )} x}{d^{3} g^{6} - 3 \, c d^{2} g^{5} h + 3 \, c^{2} d g^{4} h^{2} - c^{3} g^{3} h^{3} +{\left (d^{3} g^{3} h^{3} - 3 \, c d^{2} g^{2} h^{4} + 3 \, c^{2} d g h^{5} - c^{3} h^{6}\right )} x^{3} + 3 \,{\left (d^{3} g^{4} h^{2} - 3 \, c d^{2} g^{3} h^{3} + 3 \, c^{2} d g^{2} h^{4} - c^{3} g h^{5}\right )} x^{2} + 3 \,{\left (d^{3} g^{5} h - 3 \, c d^{2} g^{4} h^{2} + 3 \, c^{2} d g^{3} h^{3} - c^{3} g^{2} h^{4}\right )} x}\right )} d f q\right )} r}{24 \, f h} - \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{4 \,{\left (h x + g\right )}^{4} h} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^5,x, algorithm="maxima")
[Out]
1/24*((6*b^3*log(b*x + a)/(b^4*g^4 - 4*a*b^3*g^3*h + 6*a^2*b^2*g^2*h^2 - 4*a^3*b*g*h^3 + a^4*h^4) - 6*b^3*log(
h*x + g)/(b^4*g^4 - 4*a*b^3*g^3*h + 6*a^2*b^2*g^2*h^2 - 4*a^3*b*g*h^3 + a^4*h^4) + (6*b^2*h^2*x^2 + 11*b^2*g^2
- 7*a*b*g*h + 2*a^2*h^2 + 3*(5*b^2*g*h - a*b*h^2)*x)/(b^3*g^6 - 3*a*b^2*g^5*h + 3*a^2*b*g^4*h^2 - a^3*g^3*h^3
+ (b^3*g^3*h^3 - 3*a*b^2*g^2*h^4 + 3*a^2*b*g*h^5 - a^3*h^6)*x^3 + 3*(b^3*g^4*h^2 - 3*a*b^2*g^3*h^3 + 3*a^2*b*
g^2*h^4 - a^3*g*h^5)*x^2 + 3*(b^3*g^5*h - 3*a*b^2*g^4*h^2 + 3*a^2*b*g^3*h^3 - a^3*g^2*h^4)*x))*b*f*p + (6*d^3*
log(d*x + c)/(d^4*g^4 - 4*c*d^3*g^3*h + 6*c^2*d^2*g^2*h^2 - 4*c^3*d*g*h^3 + c^4*h^4) - 6*d^3*log(h*x + g)/(d^4
*g^4 - 4*c*d^3*g^3*h + 6*c^2*d^2*g^2*h^2 - 4*c^3*d*g*h^3 + c^4*h^4) + (6*d^2*h^2*x^2 + 11*d^2*g^2 - 7*c*d*g*h
+ 2*c^2*h^2 + 3*(5*d^2*g*h - c*d*h^2)*x)/(d^3*g^6 - 3*c*d^2*g^5*h + 3*c^2*d*g^4*h^2 - c^3*g^3*h^3 + (d^3*g^3*h
^3 - 3*c*d^2*g^2*h^4 + 3*c^2*d*g*h^5 - c^3*h^6)*x^3 + 3*(d^3*g^4*h^2 - 3*c*d^2*g^3*h^3 + 3*c^2*d*g^2*h^4 - c^3
*g*h^5)*x^2 + 3*(d^3*g^5*h - 3*c*d^2*g^4*h^2 + 3*c^2*d*g^3*h^3 - c^3*g^2*h^4)*x))*d*f*q)*r/(f*h) - 1/4*log(((b
*x + a)^p*(d*x + c)^q*f)^r*e)/((h*x + g)^4*h)
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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(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^5,x, algorithm="fricas")
[Out]
Timed out
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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(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/(h*x+g)**5,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 2.77682, size = 6884, normalized size = 21.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^5,x, algorithm="giac")
[Out]
-1/4*p*r*log(b*x + a)/(h^5*x^4 + 4*g*h^4*x^3 + 6*g^2*h^3*x^2 + 4*g^3*h^2*x + g^4*h) - 1/4*q*r*log(d*x + c)/(h^
5*x^4 + 4*g*h^4*x^3 + 6*g^2*h^3*x^2 + 4*g^3*h^2*x + g^4*h) - 1/4*(b^4*d^4*g^4*p*r - 4*b^4*c*d^3*g^3*h*p*r + 6*
b^4*c^2*d^2*g^2*h^2*p*r - 4*b^4*c^3*d*g*h^3*p*r + b^4*c^4*h^4*p*r + b^4*d^4*g^4*q*r - 4*a*b^3*d^4*g^3*h*q*r +
6*a^2*b^2*d^4*g^2*h^2*q*r - 4*a^3*b*d^4*g*h^3*q*r + a^4*d^4*h^4*q*r)*log(h*x + g)/(b^4*d^4*g^8*h - 4*b^4*c*d^3
*g^7*h^2 - 4*a*b^3*d^4*g^7*h^2 + 6*b^4*c^2*d^2*g^6*h^3 + 16*a*b^3*c*d^3*g^6*h^3 + 6*a^2*b^2*d^4*g^6*h^3 - 4*b^
4*c^3*d*g^5*h^4 - 24*a*b^3*c^2*d^2*g^5*h^4 - 24*a^2*b^2*c*d^3*g^5*h^4 - 4*a^3*b*d^4*g^5*h^4 + b^4*c^4*g^4*h^5
+ 16*a*b^3*c^3*d*g^4*h^5 + 36*a^2*b^2*c^2*d^2*g^4*h^5 + 16*a^3*b*c*d^3*g^4*h^5 + a^4*d^4*g^4*h^5 - 4*a*b^3*c^4
*g^3*h^6 - 24*a^2*b^2*c^3*d*g^3*h^6 - 24*a^3*b*c^2*d^2*g^3*h^6 - 4*a^4*c*d^3*g^3*h^6 + 6*a^2*b^2*c^4*g^2*h^7 +
16*a^3*b*c^3*d*g^2*h^7 + 6*a^4*c^2*d^2*g^2*h^7 - 4*a^3*b*c^4*g*h^8 - 4*a^4*c^3*d*g*h^8 + a^4*c^4*h^9) + 1/8*(
b^4*d^4*g^4*p*r - 4*b^4*c*d^3*g^3*h*p*r + 6*b^4*c^2*d^2*g^2*h^2*p*r - 4*b^4*c^3*d*g*h^3*p*r + b^4*c^4*h^4*p*r
+ b^4*d^4*g^4*q*r - 4*a*b^3*d^4*g^3*h*q*r + 6*a^2*b^2*d^4*g^2*h^2*q*r - 4*a^3*b*d^4*g*h^3*q*r + a^4*d^4*h^4*q*
r)*log(abs(b*d*x^2 + b*c*x + a*d*x + a*c))/(b^4*d^4*g^8*h - 4*b^4*c*d^3*g^7*h^2 - 4*a*b^3*d^4*g^7*h^2 + 6*b^4*
c^2*d^2*g^6*h^3 + 16*a*b^3*c*d^3*g^6*h^3 + 6*a^2*b^2*d^4*g^6*h^3 - 4*b^4*c^3*d*g^5*h^4 - 24*a*b^3*c^2*d^2*g^5*
h^4 - 24*a^2*b^2*c*d^3*g^5*h^4 - 4*a^3*b*d^4*g^5*h^4 + b^4*c^4*g^4*h^5 + 16*a*b^3*c^3*d*g^4*h^5 + 36*a^2*b^2*c
^2*d^2*g^4*h^5 + 16*a^3*b*c*d^3*g^4*h^5 + a^4*d^4*g^4*h^5 - 4*a*b^3*c^4*g^3*h^6 - 24*a^2*b^2*c^3*d*g^3*h^6 - 2
4*a^3*b*c^2*d^2*g^3*h^6 - 4*a^4*c*d^3*g^3*h^6 + 6*a^2*b^2*c^4*g^2*h^7 + 16*a^3*b*c^3*d*g^2*h^7 + 6*a^4*c^2*d^2
*g^2*h^7 - 4*a^3*b*c^4*g*h^8 - 4*a^4*c^3*d*g*h^8 + a^4*c^4*h^9) + 1/24*(6*b^3*d^3*g^3*h^3*p*r*x^3 - 18*b^3*c*d
^2*g^2*h^4*p*r*x^3 + 18*b^3*c^2*d*g*h^5*p*r*x^3 - 6*b^3*c^3*h^6*p*r*x^3 + 6*b^3*d^3*g^3*h^3*q*r*x^3 - 18*a*b^2
*d^3*g^2*h^4*q*r*x^3 + 18*a^2*b*d^3*g*h^5*q*r*x^3 - 6*a^3*d^3*h^6*q*r*x^3 + 21*b^3*d^3*g^4*h^2*p*r*x^2 - 63*b^
3*c*d^2*g^3*h^3*p*r*x^2 - 3*a*b^2*d^3*g^3*h^3*p*r*x^2 + 63*b^3*c^2*d*g^2*h^4*p*r*x^2 + 9*a*b^2*c*d^2*g^2*h^4*p
*r*x^2 - 21*b^3*c^3*g*h^5*p*r*x^2 - 9*a*b^2*c^2*d*g*h^5*p*r*x^2 + 3*a*b^2*c^3*h^6*p*r*x^2 + 21*b^3*d^3*g^4*h^2
*q*r*x^2 - 3*b^3*c*d^2*g^3*h^3*q*r*x^2 - 63*a*b^2*d^3*g^3*h^3*q*r*x^2 + 9*a*b^2*c*d^2*g^2*h^4*q*r*x^2 + 63*a^2
*b*d^3*g^2*h^4*q*r*x^2 - 9*a^2*b*c*d^2*g*h^5*q*r*x^2 - 21*a^3*d^3*g*h^5*q*r*x^2 + 3*a^3*c*d^2*h^6*q*r*x^2 + 26
*b^3*d^3*g^5*h*p*r*x - 78*b^3*c*d^2*g^4*h^2*p*r*x - 10*a*b^2*d^3*g^4*h^2*p*r*x + 78*b^3*c^2*d*g^3*h^3*p*r*x +
30*a*b^2*c*d^2*g^3*h^3*p*r*x + 2*a^2*b*d^3*g^3*h^3*p*r*x - 26*b^3*c^3*g^2*h^4*p*r*x - 30*a*b^2*c^2*d*g^2*h^4*p
*r*x - 6*a^2*b*c*d^2*g^2*h^4*p*r*x + 10*a*b^2*c^3*g*h^5*p*r*x + 6*a^2*b*c^2*d*g*h^5*p*r*x - 2*a^2*b*c^3*h^6*p*
r*x + 26*b^3*d^3*g^5*h*q*r*x - 10*b^3*c*d^2*g^4*h^2*q*r*x - 78*a*b^2*d^3*g^4*h^2*q*r*x + 2*b^3*c^2*d*g^3*h^3*q
*r*x + 30*a*b^2*c*d^2*g^3*h^3*q*r*x + 78*a^2*b*d^3*g^3*h^3*q*r*x - 6*a*b^2*c^2*d*g^2*h^4*q*r*x - 30*a^2*b*c*d^
2*g^2*h^4*q*r*x - 26*a^3*d^3*g^2*h^4*q*r*x + 6*a^2*b*c^2*d*g*h^5*q*r*x + 10*a^3*c*d^2*g*h^5*q*r*x - 2*a^3*c^2*
d*h^6*q*r*x + 11*b^3*d^3*g^6*p*r - 33*b^3*c*d^2*g^5*h*p*r - 7*a*b^2*d^3*g^5*h*p*r + 33*b^3*c^2*d*g^4*h^2*p*r +
21*a*b^2*c*d^2*g^4*h^2*p*r + 2*a^2*b*d^3*g^4*h^2*p*r - 11*b^3*c^3*g^3*h^3*p*r - 21*a*b^2*c^2*d*g^3*h^3*p*r -
6*a^2*b*c*d^2*g^3*h^3*p*r + 7*a*b^2*c^3*g^2*h^4*p*r + 6*a^2*b*c^2*d*g^2*h^4*p*r - 2*a^2*b*c^3*g*h^5*p*r + 11*b
^3*d^3*g^6*q*r - 7*b^3*c*d^2*g^5*h*q*r - 33*a*b^2*d^3*g^5*h*q*r + 2*b^3*c^2*d*g^4*h^2*q*r + 21*a*b^2*c*d^2*g^4
*h^2*q*r + 33*a^2*b*d^3*g^4*h^2*q*r - 6*a*b^2*c^2*d*g^3*h^3*q*r - 21*a^2*b*c*d^2*g^3*h^3*q*r - 11*a^3*d^3*g^3*
h^3*q*r + 6*a^2*b*c^2*d*g^2*h^4*q*r + 7*a^3*c*d^2*g^2*h^4*q*r - 2*a^3*c^2*d*g*h^5*q*r - 6*b^3*d^3*g^6*r*log(f)
+ 18*b^3*c*d^2*g^5*h*r*log(f) + 18*a*b^2*d^3*g^5*h*r*log(f) - 18*b^3*c^2*d*g^4*h^2*r*log(f) - 54*a*b^2*c*d^2*
g^4*h^2*r*log(f) - 18*a^2*b*d^3*g^4*h^2*r*log(f) + 6*b^3*c^3*g^3*h^3*r*log(f) + 54*a*b^2*c^2*d*g^3*h^3*r*log(f
) + 54*a^2*b*c*d^2*g^3*h^3*r*log(f) + 6*a^3*d^3*g^3*h^3*r*log(f) - 18*a*b^2*c^3*g^2*h^4*r*log(f) - 54*a^2*b*c^
2*d*g^2*h^4*r*log(f) - 18*a^3*c*d^2*g^2*h^4*r*log(f) + 18*a^2*b*c^3*g*h^5*r*log(f) + 18*a^3*c^2*d*g*h^5*r*log(
f) - 6*a^3*c^3*h^6*r*log(f) - 6*b^3*d^3*g^6 + 18*b^3*c*d^2*g^5*h + 18*a*b^2*d^3*g^5*h - 18*b^3*c^2*d*g^4*h^2 -
54*a*b^2*c*d^2*g^4*h^2 - 18*a^2*b*d^3*g^4*h^2 + 6*b^3*c^3*g^3*h^3 + 54*a*b^2*c^2*d*g^3*h^3 + 54*a^2*b*c*d^2*g
^3*h^3 + 6*a^3*d^3*g^3*h^3 - 18*a*b^2*c^3*g^2*h^4 - 54*a^2*b*c^2*d*g^2*h^4 - 18*a^3*c*d^2*g^2*h^4 + 18*a^2*b*c
^3*g*h^5 + 18*a^3*c^2*d*g*h^5 - 6*a^3*c^3*h^6)/(b^3*d^3*g^6*h^5*x^4 - 3*b^3*c*d^2*g^5*h^6*x^4 - 3*a*b^2*d^3*g^
5*h^6*x^4 + 3*b^3*c^2*d*g^4*h^7*x^4 + 9*a*b^2*c*d^2*g^4*h^7*x^4 + 3*a^2*b*d^3*g^4*h^7*x^4 - b^3*c^3*g^3*h^8*x^
4 - 9*a*b^2*c^2*d*g^3*h^8*x^4 - 9*a^2*b*c*d^2*g^3*h^8*x^4 - a^3*d^3*g^3*h^8*x^4 + 3*a*b^2*c^3*g^2*h^9*x^4 + 9*
a^2*b*c^2*d*g^2*h^9*x^4 + 3*a^3*c*d^2*g^2*h^9*x^4 - 3*a^2*b*c^3*g*h^10*x^4 - 3*a^3*c^2*d*g*h^10*x^4 + a^3*c^3*
h^11*x^4 + 4*b^3*d^3*g^7*h^4*x^3 - 12*b^3*c*d^2*g^6*h^5*x^3 - 12*a*b^2*d^3*g^6*h^5*x^3 + 12*b^3*c^2*d*g^5*h^6*
x^3 + 36*a*b^2*c*d^2*g^5*h^6*x^3 + 12*a^2*b*d^3*g^5*h^6*x^3 - 4*b^3*c^3*g^4*h^7*x^3 - 36*a*b^2*c^2*d*g^4*h^7*x
^3 - 36*a^2*b*c*d^2*g^4*h^7*x^3 - 4*a^3*d^3*g^4*h^7*x^3 + 12*a*b^2*c^3*g^3*h^8*x^3 + 36*a^2*b*c^2*d*g^3*h^8*x^
3 + 12*a^3*c*d^2*g^3*h^8*x^3 - 12*a^2*b*c^3*g^2*h^9*x^3 - 12*a^3*c^2*d*g^2*h^9*x^3 + 4*a^3*c^3*g*h^10*x^3 + 6*
b^3*d^3*g^8*h^3*x^2 - 18*b^3*c*d^2*g^7*h^4*x^2 - 18*a*b^2*d^3*g^7*h^4*x^2 + 18*b^3*c^2*d*g^6*h^5*x^2 + 54*a*b^
2*c*d^2*g^6*h^5*x^2 + 18*a^2*b*d^3*g^6*h^5*x^2 - 6*b^3*c^3*g^5*h^6*x^2 - 54*a*b^2*c^2*d*g^5*h^6*x^2 - 54*a^2*b
*c*d^2*g^5*h^6*x^2 - 6*a^3*d^3*g^5*h^6*x^2 + 18*a*b^2*c^3*g^4*h^7*x^2 + 54*a^2*b*c^2*d*g^4*h^7*x^2 + 18*a^3*c*
d^2*g^4*h^7*x^2 - 18*a^2*b*c^3*g^3*h^8*x^2 - 18*a^3*c^2*d*g^3*h^8*x^2 + 6*a^3*c^3*g^2*h^9*x^2 + 4*b^3*d^3*g^9*
h^2*x - 12*b^3*c*d^2*g^8*h^3*x - 12*a*b^2*d^3*g^8*h^3*x + 12*b^3*c^2*d*g^7*h^4*x + 36*a*b^2*c*d^2*g^7*h^4*x +
12*a^2*b*d^3*g^7*h^4*x - 4*b^3*c^3*g^6*h^5*x - 36*a*b^2*c^2*d*g^6*h^5*x - 36*a^2*b*c*d^2*g^6*h^5*x - 4*a^3*d^3
*g^6*h^5*x + 12*a*b^2*c^3*g^5*h^6*x + 36*a^2*b*c^2*d*g^5*h^6*x + 12*a^3*c*d^2*g^5*h^6*x - 12*a^2*b*c^3*g^4*h^7
*x - 12*a^3*c^2*d*g^4*h^7*x + 4*a^3*c^3*g^3*h^8*x + b^3*d^3*g^10*h - 3*b^3*c*d^2*g^9*h^2 - 3*a*b^2*d^3*g^9*h^2
+ 3*b^3*c^2*d*g^8*h^3 + 9*a*b^2*c*d^2*g^8*h^3 + 3*a^2*b*d^3*g^8*h^3 - b^3*c^3*g^7*h^4 - 9*a*b^2*c^2*d*g^7*h^4
- 9*a^2*b*c*d^2*g^7*h^4 - a^3*d^3*g^7*h^4 + 3*a*b^2*c^3*g^6*h^5 + 9*a^2*b*c^2*d*g^6*h^5 + 3*a^3*c*d^2*g^6*h^5
- 3*a^2*b*c^3*g^5*h^6 - 3*a^3*c^2*d*g^5*h^6 + a^3*c^3*g^4*h^7) + 1/8*(b^5*c*d^4*g^4*p*r - a*b^4*d^5*g^4*p*r -
4*b^5*c^2*d^3*g^3*h*p*r + 4*a*b^4*c*d^4*g^3*h*p*r + 6*b^5*c^3*d^2*g^2*h^2*p*r - 6*a*b^4*c^2*d^3*g^2*h^2*p*r -
4*b^5*c^4*d*g*h^3*p*r + 4*a*b^4*c^3*d^2*g*h^3*p*r + b^5*c^5*h^4*p*r - a*b^4*c^4*d*h^4*p*r - b^5*c*d^4*g^4*q*r
+ a*b^4*d^5*g^4*q*r + 4*a*b^4*c*d^4*g^3*h*q*r - 4*a^2*b^3*d^5*g^3*h*q*r - 6*a^2*b^3*c*d^4*g^2*h^2*q*r + 6*a^3
*b^2*d^5*g^2*h^2*q*r + 4*a^3*b^2*c*d^4*g*h^3*q*r - 4*a^4*b*d^5*g*h^3*q*r - a^4*b*c*d^4*h^4*q*r + a^5*d^5*h^4*q
*r)*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*g^8*h
- 4*b^4*c*d^3*g^7*h^2 - 4*a*b^3*d^4*g^7*h^2 + 6*b^4*c^2*d^2*g^6*h^3 + 16*a*b^3*c*d^3*g^6*h^3 + 6*a^2*b^2*d^4*g
^6*h^3 - 4*b^4*c^3*d*g^5*h^4 - 24*a*b^3*c^2*d^2*g^5*h^4 - 24*a^2*b^2*c*d^3*g^5*h^4 - 4*a^3*b*d^4*g^5*h^4 + b^4
*c^4*g^4*h^5 + 16*a*b^3*c^3*d*g^4*h^5 + 36*a^2*b^2*c^2*d^2*g^4*h^5 + 16*a^3*b*c*d^3*g^4*h^5 + a^4*d^4*g^4*h^5
- 4*a*b^3*c^4*g^3*h^6 - 24*a^2*b^2*c^3*d*g^3*h^6 - 24*a^3*b*c^2*d^2*g^3*h^6 - 4*a^4*c*d^3*g^3*h^6 + 6*a^2*b^2*
c^4*g^2*h^7 + 16*a^3*b*c^3*d*g^2*h^7 + 6*a^4*c^2*d^2*g^2*h^7 - 4*a^3*b*c^4*g*h^8 - 4*a^4*c^3*d*g*h^8 + a^4*c^4
*h^9)*abs(-b*c + a*d))