\(\int \frac {x^2}{(a+b x^2)^2 (c+d x^2)} \, dx\) [685]

Optimal result

Integrand size = 22, antiderivative size = 104 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {x}{2 (b c-a d) \left (a+b x^2\right )}+\frac {(b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} (b c-a d)^2}-\frac {\sqrt {c} \sqrt {d} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{(b c-a d)^2} \] Output:

-1/2*x/(-a*d+b*c)/(b*x^2+a)+1/2*(a*d+b*c)*arctan(b^(1/2)*x/a^(1/2))/a^(1/2 
)/b^(1/2)/(-a*d+b*c)^2-c^(1/2)*d^(1/2)*arctan(d^(1/2)*x/c^(1/2))/(-a*d+b*c 
)^2
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {x}{2 (-b c+a d) \left (a+b x^2\right )}+\frac {(b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} (-b c+a d)^2}-\frac {\sqrt {c} \sqrt {d} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{(b c-a d)^2} \] Input:

Integrate[x^2/((a + b*x^2)^2*(c + d*x^2)),x]
 

Output:

x/(2*(-(b*c) + a*d)*(a + b*x^2)) + ((b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a] 
])/(2*Sqrt[a]*Sqrt[b]*(-(b*c) + a*d)^2) - (Sqrt[c]*Sqrt[d]*ArcTan[(Sqrt[d] 
*x)/Sqrt[c]])/(b*c - a*d)^2
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {373, 397, 218}

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.

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

\(\Big \downarrow \) 373

\(\displaystyle \frac {\int \frac {c-d x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}dx}{2 (b c-a d)}-\frac {x}{2 \left (a+b x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {(a d+b c) \int \frac {1}{b x^2+a}dx}{b c-a d}-\frac {2 c d \int \frac {1}{d x^2+c}dx}{b c-a d}}{2 (b c-a d)}-\frac {x}{2 \left (a+b x^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (a d+b c)}{\sqrt {a} \sqrt {b} (b c-a d)}-\frac {2 \sqrt {c} \sqrt {d} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{b c-a d}}{2 (b c-a d)}-\frac {x}{2 \left (a+b x^2\right ) (b c-a d)}\)

Input:

Int[x^2/((a + b*x^2)^2*(c + d*x^2)),x]
 

Output:

-1/2*x/((b*c - a*d)*(a + b*x^2)) + (((b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a 
]])/(Sqrt[a]*Sqrt[b]*(b*c - a*d)) - (2*Sqrt[c]*Sqrt[d]*ArcTan[(Sqrt[d]*x)/ 
Sqrt[c]])/(b*c - a*d))/(2*(b*c - a*d))
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 373

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 
1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1))   Int[(e 
*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 
 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 
 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, 
m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (a d +b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{\left (a d -b c \right )^{2}}-\frac {c d \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{\left (a d -b c \right )^{2} \sqrt {c d}}\)	\(85\)
risch	\(\frac {x}{2 \left (a d -b c \right ) \left (b \,x^{2}+a \right )}-\frac {\ln \left (a \,b^{2} x -\left (-a b \right )^{\frac {3}{2}}\right ) a d}{4 \sqrt {-a b}\, \left (a d -b c \right )^{2}}-\frac {\ln \left (a \,b^{2} x -\left (-a b \right )^{\frac {3}{2}}\right ) b c}{4 \sqrt {-a b}\, \left (a d -b c \right )^{2}}+\frac {\ln \left (-a \,b^{2} x -\left (-a b \right )^{\frac {3}{2}}\right ) a d}{4 \sqrt {-a b}\, \left (a d -b c \right )^{2}}+\frac {\ln \left (-a \,b^{2} x -\left (-a b \right )^{\frac {3}{2}}\right ) b c}{4 \sqrt {-a b}\, \left (a d -b c \right )^{2}}+\frac {\sqrt {-c d}\, \ln \left (\left (-4 \left (-c d \right )^{\frac {3}{2}} a b d -4 \left (-c d \right )^{\frac {3}{2}} b^{2} c -a^{2} \sqrt {-c d}\, d^{3}-2 \sqrt {-c d}\, a b c \,d^{2}-5 b^{2} c^{2} \sqrt {-c d}\, d \right ) x -a^{2} c \,d^{3}+2 a b \,c^{2} d^{2}-b^{2} c^{3} d \right )}{2 \left (a d -b c \right )^{2}}-\frac {\sqrt {-c d}\, \ln \left (\left (4 \left (-c d \right )^{\frac {3}{2}} a b d +4 \left (-c d \right )^{\frac {3}{2}} b^{2} c +a^{2} \sqrt {-c d}\, d^{3}+2 \sqrt {-c d}\, a b c \,d^{2}+5 b^{2} c^{2} \sqrt {-c d}\, d \right ) x -a^{2} c \,d^{3}+2 a b \,c^{2} d^{2}-b^{2} c^{3} d \right )}{2 \left (a d -b c \right )^{2}}\)	\(403\)

Input:

int(x^2/(b*x^2+a)^2/(d*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

1/(a*d-b*c)^2*((1/2*a*d-1/2*b*c)*x/(b*x^2+a)+1/2*(a*d+b*c)/(a*b)^(1/2)*arc 
tan(b*x/(a*b)^(1/2)))-c*d/(a*d-b*c)^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 704, normalized size of antiderivative = 6.77 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\left [-\frac {{\left (a b c + a^{2} d + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, {\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d\right )} x}{4 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}, \frac {{\left (a b c + a^{2} d + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) - {\left (a b^{2} c - a^{2} b d\right )} x}{2 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}, -\frac {4 \, {\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (a b c + a^{2} d + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d\right )} x}{4 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}, \frac {{\left (a b c + a^{2} d + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 2 \, {\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - {\left (a b^{2} c - a^{2} b d\right )} x}{2 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2}\right )}}\right ] \] Input:

integrate(x^2/(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")
 

Output:

[-1/4*((a*b*c + a^2*d + (b^2*c + a*b*d)*x^2)*sqrt(-a*b)*log((b*x^2 - 2*sqr 
t(-a*b)*x - a)/(b*x^2 + a)) - 2*(a*b^2*x^2 + a^2*b)*sqrt(-c*d)*log((d*x^2 
- 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) + 2*(a*b^2*c - a^2*b*d)*x)/(a^2*b^3*c^2 
 - 2*a^3*b^2*c*d + a^4*b*d^2 + (a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*x 
^2), 1/2*((a*b*c + a^2*d + (b^2*c + a*b*d)*x^2)*sqrt(a*b)*arctan(sqrt(a*b) 
*x/a) + (a*b^2*x^2 + a^2*b)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d 
*x^2 + c)) - (a*b^2*c - a^2*b*d)*x)/(a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d 
^2 + (a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*x^2), -1/4*(4*(a*b^2*x^2 + 
a^2*b)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) + (a*b*c + a^2*d + (b^2*c + a*b*d)* 
x^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 2*(a*b^2*c 
 - a^2*b*d)*x)/(a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2 + (a*b^4*c^2 - 2*a 
^2*b^3*c*d + a^3*b^2*d^2)*x^2), 1/2*((a*b*c + a^2*d + (b^2*c + a*b*d)*x^2) 
*sqrt(a*b)*arctan(sqrt(a*b)*x/a) - 2*(a*b^2*x^2 + a^2*b)*sqrt(c*d)*arctan( 
sqrt(c*d)*x/c) - (a*b^2*c - a^2*b*d)*x)/(a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4 
*b*d^2 + (a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*x^2)]
 

Sympy [F(-1)]

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

integrate(x**2/(b*x**2+a)**2/(d*x**2+c),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.14 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {c d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {{\left (b c + a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b}} - \frac {x}{2 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )}} \] Input:

integrate(x^2/(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")
 

Output:

-c*d*arctan(d*x/sqrt(c*d))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d)) + 1 
/2*(b*c + a*d)*arctan(b*x/sqrt(a*b))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt 
(a*b)) - 1/2*x/(a*b*c - a^2*d + (b^2*c - a*b*d)*x^2)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.06 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {c d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d}} + \frac {{\left (b c + a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b}} - \frac {x}{2 \, {\left (b x^{2} + a\right )} {\left (b c - a d\right )}} \] Input:

integrate(x^2/(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")
 

Output:

-c*d*arctan(d*x/sqrt(c*d))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d)) + 1 
/2*(b*c + a*d)*arctan(b*x/sqrt(a*b))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt 
(a*b)) - 1/2*x/((b*x^2 + a)*(b*c - a*d))
 

Mupad [B] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 3153, normalized size of antiderivative = 30.32 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

int(x^2/((a + b*x^2)^2*(c + d*x^2)),x)
 

Output:

x/(2*(a + b*x^2)*(a*d - b*c)) + (atan((((-c*d)^(1/2)*(((-c*d)^(1/2)*((2*b^ 
6*c^5*d^2 - 8*a*b^5*c^4*d^3 + 2*a^4*b^2*c*d^6 + 12*a^2*b^4*c^3*d^4 - 8*a^3 
*b^3*c^2*d^5)/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (x 
*(-c*d)^(1/2)*(16*a^5*b^2*d^7 + 16*b^7*c^5*d^2 - 48*a*b^6*c^4*d^3 - 48*a^4 
*b^3*c*d^6 + 32*a^2*b^5*c^3*d^4 + 32*a^3*b^4*c^2*d^5))/(8*(a^2*d^2 + b^2*c 
^2 - 2*a*b*c*d)^2)))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (x*(a^2*b*d^5 + 
 5*b^3*c^2*d^3 + 2*a*b^2*c*d^4))/(4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))*1i)/ 
(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) - ((-c*d)^(1/2)*(((-c*d)^(1/2)*((2*b^6*c^5 
*d^2 - 8*a*b^5*c^4*d^3 + 2*a^4*b^2*c*d^6 + 12*a^2*b^4*c^3*d^4 - 8*a^3*b^3* 
c^2*d^5)/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x*(-c* 
d)^(1/2)*(16*a^5*b^2*d^7 + 16*b^7*c^5*d^2 - 48*a*b^6*c^4*d^3 - 48*a^4*b^3* 
c*d^6 + 32*a^2*b^5*c^3*d^4 + 32*a^3*b^4*c^2*d^5))/(8*(a^2*d^2 + b^2*c^2 - 
2*a*b*c*d)^2)))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(a^2*b*d^5 + 5*b^ 
3*c^2*d^3 + 2*a*b^2*c*d^4))/(4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))*1i)/(a^2* 
d^2 + b^2*c^2 - 2*a*b*c*d))/(((b^2*c^2*d^3)/2 + (a*b*c*d^4)/2)/(a^3*d^3 - 
b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) + ((-c*d)^(1/2)*(((-c*d)^(1/2)*(( 
2*b^6*c^5*d^2 - 8*a*b^5*c^4*d^3 + 2*a^4*b^2*c*d^6 + 12*a^2*b^4*c^3*d^4 - 8 
*a^3*b^3*c^2*d^5)/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) 
- (x*(-c*d)^(1/2)*(16*a^5*b^2*d^7 + 16*b^7*c^5*d^2 - 48*a*b^6*c^4*d^3 - 48 
*a^4*b^3*c*d^6 + 32*a^2*b^5*c^3*d^4 + 32*a^3*b^4*c^2*d^5))/(8*(a^2*d^2 ...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.09 \[ \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} d +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b c +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b d \,x^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} c \,x^{2}-2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} b -2 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a \,b^{2} x^{2}+a^{2} b d x -a \,b^{2} c x}{2 a b \left (a^{2} b \,d^{2} x^{2}-2 a \,b^{2} c d \,x^{2}+b^{3} c^{2} x^{2}+a^{3} d^{2}-2 a^{2} b c d +a \,b^{2} c^{2}\right )} \] Input:

int(x^2/(b*x^2+a)^2/(d*x^2+c),x)
 

Output:

(sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*d + sqrt(b)*sqrt(a)*at 
an((b*x)/(sqrt(b)*sqrt(a)))*a*b*c + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sq 
rt(a)))*a*b*d*x**2 + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**2*c* 
x**2 - 2*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b - 2*sqrt(d)* 
sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b**2*x**2 + a**2*b*d*x - a*b**2*c* 
x)/(2*a*b*(a**3*d**2 - 2*a**2*b*c*d + a**2*b*d**2*x**2 + a*b**2*c**2 - 2*a 
*b**2*c*d*x**2 + b**3*c**2*x**2))