3.25 \(\int (g+h x)^4 \log (e (f (a+b x)^p (c+d x)^q)^r) \, dx\)

Optimal. Leaf size=334 \[ -\frac{p r x (b g-a h)^4}{5 b^4}-\frac{p r (g+h x)^2 (b g-a h)^3}{10 b^3 h}-\frac{p r (g+h x)^3 (b g-a h)^2}{15 b^2 h}-\frac{p r (b g-a h)^5 \log (a+b x)}{5 b^5 h}+\frac{(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}-\frac{p r (g+h x)^4 (b g-a h)}{20 b h}-\frac{q r (g+h x)^3 (d g-c h)^2}{15 d^2 h}-\frac{q r (g+h x)^2 (d g-c h)^3}{10 d^3 h}-\frac{q r x (d g-c h)^4}{5 d^4}-\frac{q r (d g-c h)^5 \log (c+d x)}{5 d^5 h}-\frac{q r (g+h x)^4 (d g-c h)}{20 d h}-\frac{p r (g+h x)^5}{25 h}-\frac{q r (g+h x)^5}{25 h} \]

[Out]

-((b*g - a*h)^4*p*r*x)/(5*b^4) - ((d*g - c*h)^4*q*r*x)/(5*d^4) - ((b*g - a*h)^3*p*r*(g + h*x)^2)/(10*b^3*h) -
((d*g - c*h)^3*q*r*(g + h*x)^2)/(10*d^3*h) - ((b*g - a*h)^2*p*r*(g + h*x)^3)/(15*b^2*h) - ((d*g - c*h)^2*q*r*(
g + h*x)^3)/(15*d^2*h) - ((b*g - a*h)*p*r*(g + h*x)^4)/(20*b*h) - ((d*g - c*h)*q*r*(g + h*x)^4)/(20*d*h) - (p*
r*(g + h*x)^5)/(25*h) - (q*r*(g + h*x)^5)/(25*h) - ((b*g - a*h)^5*p*r*Log[a + b*x])/(5*b^5*h) - ((d*g - c*h)^5
*q*r*Log[c + d*x])/(5*d^5*h) + ((g + h*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*h)

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Rubi [A]  time = 0.185563, antiderivative size = 334, 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, 43} \[ -\frac{p r x (b g-a h)^4}{5 b^4}-\frac{p r (g+h x)^2 (b g-a h)^3}{10 b^3 h}-\frac{p r (g+h x)^3 (b g-a h)^2}{15 b^2 h}-\frac{p r (b g-a h)^5 \log (a+b x)}{5 b^5 h}+\frac{(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}-\frac{p r (g+h x)^4 (b g-a h)}{20 b h}-\frac{q r (g+h x)^3 (d g-c h)^2}{15 d^2 h}-\frac{q r (g+h x)^2 (d g-c h)^3}{10 d^3 h}-\frac{q r x (d g-c h)^4}{5 d^4}-\frac{q r (d g-c h)^5 \log (c+d x)}{5 d^5 h}-\frac{q r (g+h x)^4 (d g-c h)}{20 d h}-\frac{p r (g+h x)^5}{25 h}-\frac{q r (g+h x)^5}{25 h} \]

Antiderivative was successfully verified.

[In]

Int[(g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

-((b*g - a*h)^4*p*r*x)/(5*b^4) - ((d*g - c*h)^4*q*r*x)/(5*d^4) - ((b*g - a*h)^3*p*r*(g + h*x)^2)/(10*b^3*h) -
((d*g - c*h)^3*q*r*(g + h*x)^2)/(10*d^3*h) - ((b*g - a*h)^2*p*r*(g + h*x)^3)/(15*b^2*h) - ((d*g - c*h)^2*q*r*(
g + h*x)^3)/(15*d^2*h) - ((b*g - a*h)*p*r*(g + h*x)^4)/(20*b*h) - ((d*g - c*h)*q*r*(g + h*x)^4)/(20*d*h) - (p*
r*(g + h*x)^5)/(25*h) - (q*r*(g + h*x)^5)/(25*h) - ((b*g - a*h)^5*p*r*Log[a + b*x])/(5*b^5*h) - ((d*g - c*h)^5
*q*r*Log[c + d*x])/(5*d^5*h) + ((g + h*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*h)

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (g+h x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}-\frac{(b p r) \int \frac{(g+h x)^5}{a+b x} \, dx}{5 h}-\frac{(d q r) \int \frac{(g+h x)^5}{c+d x} \, dx}{5 h}\\ &=\frac{(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}-\frac{(b p r) \int \left (\frac{h (b g-a h)^4}{b^5}+\frac{(b g-a h)^5}{b^5 (a+b x)}+\frac{h (b g-a h)^3 (g+h x)}{b^4}+\frac{h (b g-a h)^2 (g+h x)^2}{b^3}+\frac{h (b g-a h) (g+h x)^3}{b^2}+\frac{h (g+h x)^4}{b}\right ) \, dx}{5 h}-\frac{(d q r) \int \left (\frac{h (d g-c h)^4}{d^5}+\frac{(d g-c h)^5}{d^5 (c+d x)}+\frac{h (d g-c h)^3 (g+h x)}{d^4}+\frac{h (d g-c h)^2 (g+h x)^2}{d^3}+\frac{h (d g-c h) (g+h x)^3}{d^2}+\frac{h (g+h x)^4}{d}\right ) \, dx}{5 h}\\ &=-\frac{(b g-a h)^4 p r x}{5 b^4}-\frac{(d g-c h)^4 q r x}{5 d^4}-\frac{(b g-a h)^3 p r (g+h x)^2}{10 b^3 h}-\frac{(d g-c h)^3 q r (g+h x)^2}{10 d^3 h}-\frac{(b g-a h)^2 p r (g+h x)^3}{15 b^2 h}-\frac{(d g-c h)^2 q r (g+h x)^3}{15 d^2 h}-\frac{(b g-a h) p r (g+h x)^4}{20 b h}-\frac{(d g-c h) q r (g+h x)^4}{20 d h}-\frac{p r (g+h x)^5}{25 h}-\frac{q r (g+h x)^5}{25 h}-\frac{(b g-a h)^5 p r \log (a+b x)}{5 b^5 h}-\frac{(d g-c h)^5 q r \log (c+d x)}{5 d^5 h}+\frac{(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 h}\\ \end{align*}

Mathematica [A]  time = 0.351812, size = 275, normalized size = 0.82 \[ \frac{-\frac{p r \left (30 b^2 (g+h x)^2 (b g-a h)^3+20 b^3 (g+h x)^3 (b g-a h)^2+15 b^4 (g+h x)^4 (b g-a h)+60 b h x (b g-a h)^4+60 (b g-a h)^5 \log (a+b x)+12 b^5 (g+h x)^5\right )}{60 b^5}+(g+h x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac{q r \left (30 d^2 (g+h x)^2 (d g-c h)^3+20 d^3 (g+h x)^3 (d g-c h)^2+15 d^4 (g+h x)^4 (d g-c h)+60 d h x (d g-c h)^4+60 (d g-c h)^5 \log (c+d x)+12 d^5 (g+h x)^5\right )}{60 d^5}}{5 h} \]

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

(-(p*r*(60*b*h*(b*g - a*h)^4*x + 30*b^2*(b*g - a*h)^3*(g + h*x)^2 + 20*b^3*(b*g - a*h)^2*(g + h*x)^3 + 15*b^4*
(b*g - a*h)*(g + h*x)^4 + 12*b^5*(g + h*x)^5 + 60*(b*g - a*h)^5*Log[a + b*x]))/(60*b^5) - (q*r*(60*d*h*(d*g -
c*h)^4*x + 30*d^2*(d*g - c*h)^3*(g + h*x)^2 + 20*d^3*(d*g - c*h)^2*(g + h*x)^3 + 15*d^4*(d*g - c*h)*(g + h*x)^
4 + 12*d^5*(g + h*x)^5 + 60*(d*g - c*h)^5*Log[c + d*x]))/(60*d^5) + (g + h*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)
^q)^r])/(5*h)

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Maple [F]  time = 0.393, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{4}\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)

[Out]

int((h*x+g)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)

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Maxima [B]  time = 1.20345, size = 842, normalized size = 2.52 \begin{align*} \frac{1}{5} \,{\left (h^{4} x^{5} + 5 \, g h^{3} x^{4} + 10 \, g^{2} h^{2} x^{3} + 10 \, g^{3} h x^{2} + 5 \, g^{4} x\right )} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac{r{\left (\frac{60 \,{\left (5 \, a b^{4} f g^{4} p - 10 \, a^{2} b^{3} f g^{3} h p + 10 \, a^{3} b^{2} f g^{2} h^{2} p - 5 \, a^{4} b f g h^{3} p + a^{5} f h^{4} p\right )} \log \left (b x + a\right )}{b^{5}} + \frac{60 \,{\left (5 \, c d^{4} f g^{4} q - 10 \, c^{2} d^{3} f g^{3} h q + 10 \, c^{3} d^{2} f g^{2} h^{2} q - 5 \, c^{4} d f g h^{3} q + c^{5} f h^{4} q\right )} \log \left (d x + c\right )}{d^{5}} - \frac{12 \, b^{4} d^{4} f h^{4}{\left (p + q\right )} x^{5} - 15 \,{\left (a b^{3} d^{4} f h^{4} p -{\left (5 \, d^{4} f g h^{3}{\left (p + q\right )} - c d^{3} f h^{4} q\right )} b^{4}\right )} x^{4} - 20 \,{\left (5 \, a b^{3} d^{4} f g h^{3} p - a^{2} b^{2} d^{4} f h^{4} p -{\left (10 \, d^{4} f g^{2} h^{2}{\left (p + q\right )} - 5 \, c d^{3} f g h^{3} q + c^{2} d^{2} f h^{4} q\right )} b^{4}\right )} x^{3} - 30 \,{\left (10 \, a b^{3} d^{4} f g^{2} h^{2} p - 5 \, a^{2} b^{2} d^{4} f g h^{3} p + a^{3} b d^{4} f h^{4} p -{\left (10 \, d^{4} f g^{3} h{\left (p + q\right )} - 10 \, c d^{3} f g^{2} h^{2} q + 5 \, c^{2} d^{2} f g h^{3} q - c^{3} d f h^{4} q\right )} b^{4}\right )} x^{2} - 60 \,{\left (10 \, a b^{3} d^{4} f g^{3} h p - 10 \, a^{2} b^{2} d^{4} f g^{2} h^{2} p + 5 \, a^{3} b d^{4} f g h^{3} p - a^{4} d^{4} f h^{4} p -{\left (5 \, d^{4} f g^{4}{\left (p + q\right )} - 10 \, c d^{3} f g^{3} h q + 10 \, c^{2} d^{2} f g^{2} h^{2} q - 5 \, c^{3} d f g h^{3} q + c^{4} f h^{4} q\right )} b^{4}\right )} x}{b^{4} d^{4}}\right )}}{300 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="maxima")

[Out]

1/5*(h^4*x^5 + 5*g*h^3*x^4 + 10*g^2*h^2*x^3 + 10*g^3*h*x^2 + 5*g^4*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e) + 1
/300*r*(60*(5*a*b^4*f*g^4*p - 10*a^2*b^3*f*g^3*h*p + 10*a^3*b^2*f*g^2*h^2*p - 5*a^4*b*f*g*h^3*p + a^5*f*h^4*p)
*log(b*x + a)/b^5 + 60*(5*c*d^4*f*g^4*q - 10*c^2*d^3*f*g^3*h*q + 10*c^3*d^2*f*g^2*h^2*q - 5*c^4*d*f*g*h^3*q +
c^5*f*h^4*q)*log(d*x + c)/d^5 - (12*b^4*d^4*f*h^4*(p + q)*x^5 - 15*(a*b^3*d^4*f*h^4*p - (5*d^4*f*g*h^3*(p + q)
- c*d^3*f*h^4*q)*b^4)*x^4 - 20*(5*a*b^3*d^4*f*g*h^3*p - a^2*b^2*d^4*f*h^4*p - (10*d^4*f*g^2*h^2*(p + q) - 5*c
*d^3*f*g*h^3*q + c^2*d^2*f*h^4*q)*b^4)*x^3 - 30*(10*a*b^3*d^4*f*g^2*h^2*p - 5*a^2*b^2*d^4*f*g*h^3*p + a^3*b*d^
4*f*h^4*p - (10*d^4*f*g^3*h*(p + q) - 10*c*d^3*f*g^2*h^2*q + 5*c^2*d^2*f*g*h^3*q - c^3*d*f*h^4*q)*b^4)*x^2 - 6
0*(10*a*b^3*d^4*f*g^3*h*p - 10*a^2*b^2*d^4*f*g^2*h^2*p + 5*a^3*b*d^4*f*g*h^3*p - a^4*d^4*f*h^4*p - (5*d^4*f*g^
4*(p + q) - 10*c*d^3*f*g^3*h*q + 10*c^2*d^2*f*g^2*h^2*q - 5*c^3*d*f*g*h^3*q + c^4*f*h^4*q)*b^4)*x)/(b^4*d^4))/
f

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Fricas [B]  time = 1.54248, size = 1932, normalized size = 5.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="fricas")

[Out]

-1/300*(12*(b^5*d^5*h^4*p + b^5*d^5*h^4*q)*r*x^5 + 15*((5*b^5*d^5*g*h^3 - a*b^4*d^5*h^4)*p + (5*b^5*d^5*g*h^3
- b^5*c*d^4*h^4)*q)*r*x^4 + 20*((10*b^5*d^5*g^2*h^2 - 5*a*b^4*d^5*g*h^3 + a^2*b^3*d^5*h^4)*p + (10*b^5*d^5*g^2
*h^2 - 5*b^5*c*d^4*g*h^3 + b^5*c^2*d^3*h^4)*q)*r*x^3 + 30*((10*b^5*d^5*g^3*h - 10*a*b^4*d^5*g^2*h^2 + 5*a^2*b^
3*d^5*g*h^3 - a^3*b^2*d^5*h^4)*p + (10*b^5*d^5*g^3*h - 10*b^5*c*d^4*g^2*h^2 + 5*b^5*c^2*d^3*g*h^3 - b^5*c^3*d^
2*h^4)*q)*r*x^2 + 60*((5*b^5*d^5*g^4 - 10*a*b^4*d^5*g^3*h + 10*a^2*b^3*d^5*g^2*h^2 - 5*a^3*b^2*d^5*g*h^3 + a^4
*b*d^5*h^4)*p + (5*b^5*d^5*g^4 - 10*b^5*c*d^4*g^3*h + 10*b^5*c^2*d^3*g^2*h^2 - 5*b^5*c^3*d^2*g*h^3 + b^5*c^4*d
*h^4)*q)*r*x - 60*(b^5*d^5*h^4*p*r*x^5 + 5*b^5*d^5*g*h^3*p*r*x^4 + 10*b^5*d^5*g^2*h^2*p*r*x^3 + 10*b^5*d^5*g^3
*h*p*r*x^2 + 5*b^5*d^5*g^4*p*r*x + (5*a*b^4*d^5*g^4 - 10*a^2*b^3*d^5*g^3*h + 10*a^3*b^2*d^5*g^2*h^2 - 5*a^4*b*
d^5*g*h^3 + a^5*d^5*h^4)*p*r)*log(b*x + a) - 60*(b^5*d^5*h^4*q*r*x^5 + 5*b^5*d^5*g*h^3*q*r*x^4 + 10*b^5*d^5*g^
2*h^2*q*r*x^3 + 10*b^5*d^5*g^3*h*q*r*x^2 + 5*b^5*d^5*g^4*q*r*x + (5*b^5*c*d^4*g^4 - 10*b^5*c^2*d^3*g^3*h + 10*
b^5*c^3*d^2*g^2*h^2 - 5*b^5*c^4*d*g*h^3 + b^5*c^5*h^4)*q*r)*log(d*x + c) - 60*(b^5*d^5*h^4*x^5 + 5*b^5*d^5*g*h
^3*x^4 + 10*b^5*d^5*g^2*h^2*x^3 + 10*b^5*d^5*g^3*h*x^2 + 5*b^5*d^5*g^4*x)*log(e) - 60*(b^5*d^5*h^4*r*x^5 + 5*b
^5*d^5*g*h^3*r*x^4 + 10*b^5*d^5*g^2*h^2*r*x^3 + 10*b^5*d^5*g^3*h*r*x^2 + 5*b^5*d^5*g^4*r*x)*log(f))/(b^5*d^5)

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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((h*x+g)**4*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r),x)

[Out]

Timed out

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Giac [B]  time = 1.52229, size = 1715, normalized size = 5.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="giac")

[Out]

-1/25*(h^4*p*r + h^4*q*r - 5*h^4*r*log(f) - 5*h^4)*x^5 - 1/20*(5*b*d*g*h^3*p*r - a*d*h^4*p*r + 5*b*d*g*h^3*q*r
- b*c*h^4*q*r - 20*b*d*g*h^3*r*log(f) - 20*b*d*g*h^3)*x^4/(b*d) + 1/5*(h^4*p*r*x^5 + 5*g*h^3*p*r*x^4 + 10*g^2
*h^2*p*r*x^3 + 10*g^3*h*p*r*x^2 + 5*g^4*p*r*x)*log(b*x + a) + 1/5*(h^4*q*r*x^5 + 5*g*h^3*q*r*x^4 + 10*g^2*h^2*
q*r*x^3 + 10*g^3*h*q*r*x^2 + 5*g^4*q*r*x)*log(d*x + c) - 1/15*(10*b^2*d^2*g^2*h^2*p*r - 5*a*b*d^2*g*h^3*p*r +
a^2*d^2*h^4*p*r + 10*b^2*d^2*g^2*h^2*q*r - 5*b^2*c*d*g*h^3*q*r + b^2*c^2*h^4*q*r - 30*b^2*d^2*g^2*h^2*r*log(f)
- 30*b^2*d^2*g^2*h^2)*x^3/(b^2*d^2) - 1/10*(10*b^3*d^3*g^3*h*p*r - 10*a*b^2*d^3*g^2*h^2*p*r + 5*a^2*b*d^3*g*h
^3*p*r - a^3*d^3*h^4*p*r + 10*b^3*d^3*g^3*h*q*r - 10*b^3*c*d^2*g^2*h^2*q*r + 5*b^3*c^2*d*g*h^3*q*r - b^3*c^3*h
^4*q*r - 20*b^3*d^3*g^3*h*r*log(f) - 20*b^3*d^3*g^3*h)*x^2/(b^3*d^3) - 1/5*(5*b^4*d^4*g^4*p*r - 10*a*b^3*d^4*g
^3*h*p*r + 10*a^2*b^2*d^4*g^2*h^2*p*r - 5*a^3*b*d^4*g*h^3*p*r + a^4*d^4*h^4*p*r + 5*b^4*d^4*g^4*q*r - 10*b^4*c
*d^3*g^3*h*q*r + 10*b^4*c^2*d^2*g^2*h^2*q*r - 5*b^4*c^3*d*g*h^3*q*r + b^4*c^4*h^4*q*r - 5*b^4*d^4*g^4*r*log(f)
- 5*b^4*d^4*g^4)*x/(b^4*d^4) + 1/10*(5*a*b^4*d^5*g^4*p*r - 10*a^2*b^3*d^5*g^3*h*p*r + 10*a^3*b^2*d^5*g^2*h^2*
p*r - 5*a^4*b*d^5*g*h^3*p*r + a^5*d^5*h^4*p*r + 5*b^5*c*d^4*g^4*q*r - 10*b^5*c^2*d^3*g^3*h*q*r + 10*b^5*c^3*d^
2*g^2*h^2*q*r - 5*b^5*c^4*d*g*h^3*q*r + b^5*c^5*h^4*q*r)*log(abs(b*d*x^2 + b*c*x + a*d*x + a*c))/(b^5*d^5) + 1
/10*(5*a*b^5*c*d^5*g^4*p*r - 5*a^2*b^4*d^6*g^4*p*r - 10*a^2*b^4*c*d^5*g^3*h*p*r + 10*a^3*b^3*d^6*g^3*h*p*r + 1
0*a^3*b^3*c*d^5*g^2*h^2*p*r - 10*a^4*b^2*d^6*g^2*h^2*p*r - 5*a^4*b^2*c*d^5*g*h^3*p*r + 5*a^5*b*d^6*g*h^3*p*r +
a^5*b*c*d^5*h^4*p*r - a^6*d^6*h^4*p*r - 5*b^6*c^2*d^4*g^4*q*r + 5*a*b^5*c*d^5*g^4*q*r + 10*b^6*c^3*d^3*g^3*h*
q*r - 10*a*b^5*c^2*d^4*g^3*h*q*r - 10*b^6*c^4*d^2*g^2*h^2*q*r + 10*a*b^5*c^3*d^3*g^2*h^2*q*r + 5*b^6*c^5*d*g*h
^3*q*r - 5*a*b^5*c^4*d^2*g*h^3*q*r - b^6*c^6*h^4*q*r + a*b^5*c^5*d*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^5*d^5*abs(-b*c + a*d))