\(\int (a g+b g x) (A+B \log (\frac {e (c+d x)^2}{(a+b x)^2}))^2 \, dx\) [213]

Optimal result

Integrand size = 32, antiderivative size = 211 \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\frac {4 B^2 (b c-a d)^2 g \log (a+b x)}{b d^2}+\frac {2 B (b c-a d) g (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{d^2}+\frac {g (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{2 b}+\frac {2 B (b c-a d)^2 g \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{b d^2}-\frac {4 B^2 (b c-a d)^2 g \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.92 \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\frac {g \left ((a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2+\frac {4 B (b c-a d) \left (A b d x+B (b c-a d) \log ^2(c+d x)+B d (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )-(b c-a d) \log (c+d x) \left (A-2 B+2 B \log \left (\frac {d (a+b x)}{-b c+a d}\right )+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )+(-2 b B c+2 a B d) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{d^2}\right )}{2 b} \] Input:

Integrate[(a*g + b*g*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2952, 2756, 2789, 2751, 16, 2779, 2838}

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.

\(\displaystyle \int (a g+b g x) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2 \, dx\)

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

\(\displaystyle g (b c-a d)^2 \left (\frac {\left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{2 b \left (d-\frac {b (c+d x)}{a+b x}\right )^2}-\frac {2 B \left (\frac {b \left (\frac {(c+d x) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{d (a+b x) \left (d-\frac {b (c+d x)}{a+b x}\right )}-\frac {2 B \int \frac {1}{d-\frac {b (c+d x)}{a+b x}}d\frac {c+d x}{a+b x}}{d}\right )}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )}d\frac {c+d x}{a+b x}}{d}\right )}{b}\right )\)

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

\(\displaystyle g (b c-a d)^2 \left (\frac {\left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{2 b \left (d-\frac {b (c+d x)}{a+b x}\right )^2}-\frac {2 B \left (\frac {\frac {2 B \int \frac {(a+b x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{c+d x}d\frac {c+d x}{a+b x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{d}}{d}+\frac {b \left (\frac {(c+d x) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{d (a+b x) \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {2 B \log \left (d-\frac {b (c+d x)}{a+b x}\right )}{b d}\right )}{d}\right )}{b}\right )\)

\(\Big \downarrow \) 2838

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

Input:

Int[(a*g + b*g*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2,x]
 

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2756

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

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2952

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( 
m + 1)*(g/d)^m   Subst[Int[(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] && EqQ[ 
n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f 
 - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [F]

\[\int \left (b g x +a g \right ) {\left (A +B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )\right )}^{2}d x\]

Input:

int((b*g*x+a*g)*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2,x)
 

Output:

int((b*g*x+a*g)*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2,x)
 

Fricas [F]

\[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )} {\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2} \,d x } \] Input:

integrate((b*g*x+a*g)*(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="fri 
cas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\text {Timed out} \] Input:

integrate((b*g*x+a*g)*(A+B*ln(e*(d*x+c)**2/(b*x+a)**2))**2,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (208) = 416\).

Time = 0.17 (sec) , antiderivative size = 730, normalized size of antiderivative = 3.46 \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((b*g*x+a*g)*(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="max 
ima")
 

Output:

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

Giac [F]

\[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )} {\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2} \,d x } \] Input:

integrate((b*g*x+a*g)*(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="gia 
c")
 

Output:

integrate((b*g*x + a*g)*(B*log((d*x + c)^2*e/(b*x + a)^2) + A)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\int \left (a\,g+b\,g\,x\right )\,{\left (A+B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\right )}^2 \,d x \] Input:

int((a*g + b*g*x)*(A + B*log((e*(c + d*x)^2)/(a + b*x)^2))^2,x)
 

Output:

int((a*g + b*g*x)*(A + B*log((e*(c + d*x)^2)/(a + b*x)^2))^2, x)
 

Reduce [F]

\[ \int (a g+b g x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx =\text {Too large to display} \] Input:

int((b*g*x+a*g)*(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x)
 

Output:

(g*( - 4*int((log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b** 
2*x**2))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b**2*d**3 + 8*int((lo 
g((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*x)/(a*c 
 + a*d*x + b*c*x + b*d*x**2),x)*a*b**3*c*d**2 - 4*int((log((c**2*e + 2*c*d 
*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*x)/(a*c + a*d*x + b*c*x 
+ b*d*x**2),x)*b**4*c**2*d - 4*log(c + d*x)*a**3*d**2 + 8*log(c + d*x)*a** 
2*b*c*d + 8*log(c + d*x)*a**2*b*d**2 - 4*log(c + d*x)*a*b**2*c**2 - 16*log 
(c + d*x)*a*b**2*c*d + 8*log(c + d*x)*b**3*c**2 + log((c**2*e + 2*c*d*e*x 
+ d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*a*b**2*c*d + 2*log((c**2*e 
 + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*a*b**2*d**2*x 
 + log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2 
*b**3*d**2*x**2 + 2*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x 
 + b**2*x**2))*a**3*d**2 + 4*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 
+ 2*a*b*x + b**2*x**2))*a**2*b*d**2*x - 4*log((c**2*e + 2*c*d*e*x + d**2*e 
*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**2*b*d**2 + 4*log((c**2*e + 2*c*d*e 
*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a*b**2*c*d + 2*log((c**2*e 
 + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a*b**2*d**2*x**2 
 - 4*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))* 
a*b**2*d**2*x + 4*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + 
 b**2*x**2))*b**3*c*d*x + 2*a**3*d**2*x + a**2*b*d**2*x**2 - 4*a**2*b*d...