\(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{(g+h x)^3} \, dx\) [314]

Optimal result

Integrand size = 33, antiderivative size = 629 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx=\frac {3 B (b c-a d) h n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 (b g-a h)^2 (d g-c h) (g+h x)}+\frac {b^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 h (b g-a h)^2}-\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 h (g+h x)^2}+\frac {3 B^2 (b c-a d)^2 h n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {3 B (b c-a d) (2 b d g-b c h-a d h) n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \log \left (1-\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{2 (b g-a h)^2 (d g-c h)^2}+\frac {3 B^3 (b c-a d)^2 h n^3 \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {3 B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}-\frac {3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^3 \operatorname {PolyLog}\left (3,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2} \] Output:

3/2*B*(-a*d+b*c)*h*n*(b*x+a)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*h+b*g 
)^2/(-c*h+d*g)/(h*x+g)+1/2*b^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/h/(-a*h 
+b*g)^2-1/2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/h/(h*x+g)^2+3*B^2*(-a*d+b* 
c)^2*h*n^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))*ln(1-(-c*h+d*g)*(b*x+a)/(-a*h 
+b*g)/(d*x+c))/(-a*h+b*g)^2/(-c*h+d*g)^2+3/2*B*(-a*d+b*c)*(-a*d*h-b*c*h+2* 
b*d*g)*n*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2*ln(1-(-c*h+d*g)*(b*x+a)/(-a*h 
+b*g)/(d*x+c))/(-a*h+b*g)^2/(-c*h+d*g)^2+3*B^3*(-a*d+b*c)^2*h*n^3*polylog( 
2,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/(-a*h+b*g)^2/(-c*h+d*g)^2+3*B^2*( 
-a*d+b*c)*(-a*d*h-b*c*h+2*b*d*g)*n^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))*pol 
ylog(2,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/(-a*h+b*g)^2/(-c*h+d*g)^2-3* 
B^3*(-a*d+b*c)*(-a*d*h-b*c*h+2*b*d*g)*n^3*polylog(3,(-c*h+d*g)*(b*x+a)/(-a 
*h+b*g)/(d*x+c))/(-a*h+b*g)^2/(-c*h+d*g)^2
 

Mathematica [F]

\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx=\int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.53 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2973, 2953, 2798, 2804, 2009}

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.

\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(g+h x)^3} \, dx\)

\(\Big \downarrow \) 2973

\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(g+h x)^3}dx\)

\(\Big \downarrow \) 2953

\(\displaystyle (b c-a d) \int \frac {\left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{\left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}\)

\(\Big \downarrow \) 2798

\(\displaystyle (b c-a d) \left (\frac {3 B n \int \frac {(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 h (b c-a d)}-\frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{2 h (b c-a d) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^2}\right )\)

\(\Big \downarrow \) 2804

\(\displaystyle (b c-a d) \left (\frac {3 B n \int \left (\frac {b^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b g-a h)^2 (a+b x)}+\frac {(b c-a d) h (-2 b d g+b c h+a d h) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b g-a h)^2 (d g-c h) \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )}+\frac {(b c-a d)^2 h^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b g-a h) (d g-c h) \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^2}\right )d\frac {a+b x}{c+d x}}{2 h (b c-a d)}-\frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{2 h (b c-a d) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle (b c-a d) \left (\frac {3 B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B n (b g-a h)^2}+\frac {h^2 (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(c+d x) (b g-a h)^2 (d g-c h) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )}+\frac {2 B h^2 n (b c-a d)^2 \log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {2 B h n (b c-a d) (-a d h-b c h+2 b d g) \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {h (b c-a d) (-a d h-b c h+2 b d g) \log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(b g-a h)^2 (d g-c h)^2}+\frac {2 B^2 h^2 n^2 (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}-\frac {2 B^2 h n^2 (b c-a d) (-a d h-b c h+2 b d g) \operatorname {PolyLog}\left (3,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}\right )}{2 h (b c-a d)}-\frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{2 h (b c-a d) \left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^2}\right )\)

Input:

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

Output:

(b*c - a*d)*(-1/2*((b - (d*(a + b*x))/(c + d*x))^2*(A + B*Log[e*((a + b*x) 
/(c + d*x))^n])^3)/((b*c - a*d)*h*(b*g - a*h - ((d*g - c*h)*(a + b*x))/(c 
+ d*x))^2) + (3*B*n*(((b*c - a*d)^2*h^2*(a + b*x)*(A + B*Log[e*((a + b*x)/ 
(c + d*x))^n])^2)/((b*g - a*h)^2*(d*g - c*h)*(c + d*x)*(b*g - a*h - ((d*g 
- c*h)*(a + b*x))/(c + d*x))) + (b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n] 
)^3)/(3*B*(b*g - a*h)^2*n) + (2*B*(b*c - a*d)^2*h^2*n*(A + B*Log[e*((a + b 
*x)/(c + d*x))^n])*Log[1 - ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x)) 
])/((b*g - a*h)^2*(d*g - c*h)^2) + ((b*c - a*d)*h*(2*b*d*g - b*c*h - a*d*h 
)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - ((d*g - c*h)*(a + b*x)) 
/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2) + (2*B^2*(b*c - a 
*d)^2*h^2*n^2*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))]) 
/((b*g - a*h)^2*(d*g - c*h)^2) + (2*B*(b*c - a*d)*h*(2*b*d*g - b*c*h - a*d 
*h)*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, ((d*g - c*h)*(a + 
b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2) - (2*B^2*(b* 
c - a*d)*h*(2*b*d*g - b*c*h - a*d*h)*n^2*PolyLog[3, ((d*g - c*h)*(a + b*x) 
)/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2)))/(2*(b*c - a*d) 
*h))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2798

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_)*(( 
f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*(d + e*x)^(q + 
 1)*((a + b*Log[c*x^n])^p/((q + 1)*(e*f - d*g))), x] - Simp[b*n*(p/((q + 1) 
*(e*f - d*g)))   Int[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[e*f 
 - d*g, 0] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2804

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

rule 2953

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)   Sub 
st[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2 
)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n} 
, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
 

rule 2973

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] 
 :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr 
eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !Intege 
rQ[n]
 

Maple [F]

\[\int \frac {{\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{3}}{\left (h x +g \right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (h x + g\right )}^{3}} \,d x } \] Input:

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(h*x+g)^3,x, algorithm="fri 
cas")
 

Output:

integral((B^3*log((b*x + a)^n*e/(d*x + c)^n)^3 + 3*A*B^2*log((b*x + a)^n*e 
/(d*x + c)^n)^2 + 3*A^2*B*log((b*x + a)^n*e/(d*x + c)^n) + A^3)/(h^3*x^3 + 
 3*g*h^2*x^2 + 3*g^2*h*x + g^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (h x + g\right )}^{3}} \,d x } \] Input:

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(h*x+g)^3,x, algorithm="max 
ima")
 

Output:

1/2*B^3*log((d*x + c)^n)^3/(h^3*x^2 + 2*g*h^2*x + g^2*h) + 3/2*(b^2*e*n*lo 
g(b*x + a)/(b^2*g^2*h - 2*a*b*g*h^2 + a^2*h^3) - d^2*e*n*log(d*x + c)/(d^2 
*g^2*h - 2*c*d*g*h^2 + c^2*h^3) - (2*a*b*d^2*e*g*n - a^2*d^2*e*h*n - (2*c* 
d*e*g*n - c^2*e*h*n)*b^2)*log(h*x + g)/((d^2*g^2*h^2 - 2*c*d*g*h^3 + c^2*h 
^4)*a^2 - 2*(d^2*g^3*h - 2*c*d*g^2*h^2 + c^2*g*h^3)*a*b + (d^2*g^4 - 2*c*d 
*g^3*h + c^2*g^2*h^2)*b^2) + (b*c*e*n - a*d*e*n)/((d*g^2*h - c*g*h^2)*a - 
(d*g^3 - c*g^2*h)*b + ((d*g*h^2 - c*h^3)*a - (d*g^2*h - c*g*h^2)*b)*x))*A^ 
2*B/e - 3/2*A^2*B*log((b*x + a)^n*e/(d*x + c)^n)/(h^3*x^2 + 2*g*h^2*x + g^ 
2*h) - 1/2*A^3/(h^3*x^2 + 2*g*h^2*x + g^2*h) + integrate(1/2*(2*B^3*c*h*lo 
g(e)^3 + 6*A*B^2*c*h*log(e)^2 + 2*(B^3*d*h*x + B^3*c*h)*log((b*x + a)^n)^3 
 + 6*(B^3*c*h*log(e) + A*B^2*c*h + (B^3*d*h*log(e) + A*B^2*d*h)*x)*log((b* 
x + a)^n)^2 + 3*(2*A*B^2*c*h - (d*g*n - 2*c*h*log(e))*B^3 - ((h*n - 2*h*lo 
g(e))*B^3*d - 2*A*B^2*d*h)*x + 2*(B^3*d*h*x + B^3*c*h)*log((b*x + a)^n))*l 
og((d*x + c)^n)^2 + 2*(B^3*d*h*log(e)^3 + 3*A*B^2*d*h*log(e)^2)*x + 6*(B^3 
*c*h*log(e)^2 + 2*A*B^2*c*h*log(e) + (B^3*d*h*log(e)^2 + 2*A*B^2*d*h*log(e 
))*x)*log((b*x + a)^n) - 6*(B^3*c*h*log(e)^2 + 2*A*B^2*c*h*log(e) + (B^3*d 
*h*x + B^3*c*h)*log((b*x + a)^n)^2 + (B^3*d*h*log(e)^2 + 2*A*B^2*d*h*log(e 
))*x + 2*(B^3*c*h*log(e) + A*B^2*c*h + (B^3*d*h*log(e) + A*B^2*d*h)*x)*log 
((b*x + a)^n))*log((d*x + c)^n))/(d*h^4*x^4 + c*g^3*h + (3*d*g*h^3 + c*h^4 
)*x^3 + 3*(d*g^2*h^2 + c*g*h^3)*x^2 + (d*g^3*h + 3*c*g^2*h^2)*x), x)
 

Giac [F]

\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (h x + g\right )}^{3}} \,d x } \] Input:

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

Output:

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^3/(h*x + g)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx=\int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3}{{\left (g+h\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^3} \, dx=\text {too large to display} \] Input:

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

Output:

(24*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(a**5*c**2*d**3*g**3*h**5 + 3*a 
**5*c**2*d**3*g**2*h**6*x + 3*a**5*c**2*d**3*g*h**7*x**2 + a**5*c**2*d**3* 
h**8*x**3 + a**5*c*d**4*g**3*h**5*x + 3*a**5*c*d**4*g**2*h**6*x**2 + 3*a** 
5*c*d**4*g*h**7*x**3 + a**5*c*d**4*h**8*x**4 + 2*a**4*b*c**3*d**2*g**3*h** 
5 + 6*a**4*b*c**3*d**2*g**2*h**6*x + 6*a**4*b*c**3*d**2*g*h**7*x**2 + 2*a* 
*4*b*c**3*d**2*h**8*x**3 - 6*a**4*b*c**2*d**3*g**4*h**4 - 15*a**4*b*c**2*d 
**3*g**3*h**5*x - 9*a**4*b*c**2*d**3*g**2*h**6*x**2 + 3*a**4*b*c**2*d**3*g 
*h**7*x**3 + 3*a**4*b*c**2*d**3*h**8*x**4 - 6*a**4*b*c*d**4*g**4*h**4*x - 
17*a**4*b*c*d**4*g**3*h**5*x**2 - 15*a**4*b*c*d**4*g**2*h**6*x**3 - 3*a**4 
*b*c*d**4*g*h**7*x**4 + a**4*b*c*d**4*h**8*x**5 + 2*a**3*b**2*c**4*d*g**3* 
h**5 + 6*a**3*b**2*c**4*d*g**2*h**6*x + 6*a**3*b**2*c**4*d*g*h**7*x**2 + 2 
*a**3*b**2*c**4*d*h**8*x**3 - 9*a**3*b**2*c**3*d**2*g**4*h**4 - 23*a**3*b* 
*2*c**3*d**2*g**3*h**5*x - 15*a**3*b**2*c**3*d**2*g**2*h**6*x**2 + 3*a**3* 
b**2*c**3*d**2*g*h**7*x**3 + 4*a**3*b**2*c**3*d**2*h**8*x**4 + 12*a**3*b** 
2*c**2*d**3*g**5*h**3 + 21*a**3*b**2*c**2*d**3*g**4*h**4*x - 7*a**3*b**2*c 
**2*d**3*g**3*h**5*x**2 - 27*a**3*b**2*c**2*d**3*g**2*h**6*x**3 - 9*a**3*b 
**2*c**2*d**3*g*h**7*x**4 + 2*a**3*b**2*c**2*d**3*h**8*x**5 + a**3*b**2*c* 
d**4*g**6*h**2 + 15*a**3*b**2*c*d**4*g**5*h**3*x + 33*a**3*b**2*c*d**4*g** 
4*h**4*x**2 + 19*a**3*b**2*c*d**4*g**3*h**5*x**3 - 6*a**3*b**2*c*d**4*g**2 
*h**6*x**4 - 6*a**3*b**2*c*d**4*g*h**7*x**5 + a**3*b**2*d**5*g**6*h**2*...