\(\int \frac {A+B x^2}{(d+e x^2)^2 (a+b x^2+c x^4)^{3/2}} \, dx\) [179]

Optimal result

Integrand size = 33, antiderivative size = 1247 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Output:

-1/2*e*(-A*e+B*d)*x/d/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)/(c*x^4+b*x^2+a)^(1/2)- 
1/2*x*(a*(-A*e+B*d)*(b*e+2*c*d)*(2*a*c*e-b^2*e+b*c*d)-2*(c*(-2*a*c+b^2)*d- 
b*(-3*a*c+b^2)*e)*(2*a*B*d*e+A*(c*d^2-e*(a*e+b*d)))+c*(a*(-A*e+B*d)*(-b*e+ 
2*c*d)*(b*e+2*c*d)-2*(2*a*c*e-b^2*e+b*c*d)*(2*a*B*d*e+A*(c*d^2-e*(a*e+b*d) 
)))*x^2)/a/(-4*a*c+b^2)/d/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)^(1/2)+1/2* 
c^(1/2)*(a*(-A*e+B*d)*(-b*e+2*c*d)*(b*e+2*c*d)-2*(2*a*c*e-b^2*e+b*c*d)*(2* 
a*B*d*e+A*(c*d^2-e*(a*e+b*d))))*x*(c*x^4+b*x^2+a)^(1/2)/a/(-4*a*c+b^2)/d/( 
a*e^2-b*d*e+c*d^2)^2/(a^(1/2)+c^(1/2)*x^2)+1/4*e^(3/2)*(A*e*(7*c*d^2-e*(-a 
*e+4*b*d))-B*d*(5*c*d^2-e*(a*e+2*b*d)))*arctan((a*e^2-b*d*e+c*d^2)^(1/2)*x 
/d^(1/2)/e^(1/2)/(c*x^4+b*x^2+a)^(1/2))/d^(3/2)/(a*e^2-b*d*e+c*d^2)^(5/2)- 
1/2*c^(1/4)*(a*(-A*e+B*d)*(-b*e+2*c*d)*(b*e+2*c*d)-2*(2*a*c*e-b^2*e+b*c*d) 
*(2*a*B*d*e+A*(c*d^2-e*(a*e+b*d))))*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a) 
/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))) 
,1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4)/(-4*a*c+b^2)/d/(a*e^2-b*d*e+c*d^ 
2)^2/(c*x^4+b*x^2+a)^(1/2)+1/2*c^(1/4)*(a*c^(1/2)*e*(-2*A*e+B*d)-A*c^(1/2) 
*d*(-b*e+c*d)+a^(1/2)*(-A*e+B*d)*(-b*e+c*d))*(a^(1/2)+c^(1/2)*x^2)*((c*x^4 
+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)* 
x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4)/(b-2*a^(1/2)*c^(1/2))/ 
d/(c^(1/2)*d-a^(1/2)*e)/(a*e^2-b*d*e+c*d^2)/(c*x^4+b*x^2+a)^(1/2)-1/8*e*(c 
^(1/2)*d+a^(1/2)*e)*(A*e*(7*c*d^2-e*(-a*e+4*b*d))-B*d*(5*c*d^2-e*(a*e+2...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 17.92 (sec) , antiderivative size = 8031, normalized size of antiderivative = 6.44 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x^2)/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^(3/2)),x]
 

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 3.48 (sec) , antiderivative size = 2112, normalized size of antiderivative = 1.69, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2258, 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.

\(\displaystyle \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2258

\(\displaystyle \int \left (\frac {A \left (-c e (a e+2 b d)+b^2 e^2+c^2 d^2\right )-c x^2 \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+a B e (2 c d-b e)}{\left (a+b x^2+c x^4\right )^{3/2} \left (a e^2-b d e+c d^2\right )^2}+\frac {e (A e-B d)}{\left (d+e x^2\right )^2 \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}+\frac {e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(B d-A e) x \sqrt {c x^4+b x^2+a} e^3}{2 d \left (c d^2-b e d+a e^2\right )^2 \left (e x^2+d\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e^2}{2 d \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}+\frac {\sqrt {c} (B d-A e) x \sqrt {c x^4+b x^2+a} e^2}{2 d \left (c d^2-b e d+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {(B d-A e) \left (3 c d^2-e (2 b d-a e)\right ) \arctan \left (\frac {\sqrt {c d^2-b e d+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+b x^2+a}}\right ) e^{3/2}}{4 d^{3/2} \left (c d^2-b e d+a e^2\right )^{5/2}}+\frac {\left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \arctan \left (\frac {\sqrt {c d^2-b e d+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+b x^2+a}}\right ) e^{3/2}}{2 \sqrt {d} \left (c d^2-b e d+a e^2\right )^{5/2}}+\frac {\sqrt [4]{c} \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {\sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^4+b x^2+a}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2-e (2 b d-a e)\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {a^{3/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {\sqrt [4]{c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {\sqrt [4]{c} \left (a^{3/2} B \sqrt {c} e^2+a (2 B c d-b B e-A c e) e+A (c d-b e)^2-\sqrt {a} \sqrt {c} \left (B c d^2-A e (2 c d-b e)\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}+\frac {\sqrt {c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right )\right ) x \sqrt {c x^4+b x^2+a}}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {x \left (-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right )\right ) x^2+a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}\)

Input:

Int[(A + B*x^2)/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^(3/2)),x]
 

Output:

(x*(a*b*c*(A*e*(2*c*d - b*e) - B*(c*d^2 - a*e^2)) + (b^2 - 2*a*c)*(a*B*e*( 
2*c*d - b*e) + A*(c^2*d^2 + b^2*e^2 - c*e*(2*b*d + a*e))) - c*(a*B*(2*c^2* 
d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^ 
2 - b*c*(c*d^2 - 3*a*e^2)))*x^2))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2) 
^2*Sqrt[a + b*x^2 + c*x^4]) + (Sqrt[c]*e^2*(B*d - A*e)*x*Sqrt[a + b*x^2 + 
c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) + (Sqrt[c] 
*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d*e - 4*a*c^2 
*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2)))*x*Sqrt[a + b*x^2 + c*x^4])/(a*(b^ 
2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3*(B*d 
- A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x^ 
2)) - (e^(3/2)*(B*d - A*e)*(3*c*d^2 - e*(2*b*d - a*e))*ArcTan[(Sqrt[c*d^2 
- b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(4*d^(3/2) 
*(c*d^2 - b*d*e + a*e^2)^(5/2)) + (e^(3/2)*(A*e*(2*c*d - b*e) - B*(c*d^2 - 
 a*e^2))*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + 
b*x^2 + c*x^4])])/(2*Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^(5/2)) - (a^(1/4)*c^( 
1/4)*e^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqr 
t[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sq 
rt[a]*Sqrt[c]))/4])/(2*d*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[a + b*x^2 + c*x^4] 
) - (c^(1/4)*(a*B*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)) + A*(2*b^2*c*d 
*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2)))*(Sqrt[a] + Sqrt[c]...
 

Defintions of rubi rules used

rule 2009

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

rule 2258

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ 
(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(d + e 
*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e}, x] 
&& PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(8275\) vs. \(2(1105)=2210\).

Time = 2.17 (sec) , antiderivative size = 8276, normalized size of antiderivative = 6.64

method	result	size
default	\(\text {Expression too large to display}\)	\(8276\)
elliptic	\(\text {Expression too large to display}\)	\(9725\)

Input:

int((B*x^2+A)/(e*x^2+d)^2/(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x**2+A)/(e*x**2+d)**2/(c*x**4+b*x**2+a)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima 
")
 

Output:

integrate((B*x^2 + A)/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)^2), x)
 

Giac [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)^2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x^2+d\right )}^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:

int((A + B*x^2)/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^(3/2)),x)
 

Output:

int((A + B*x^2)/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} e^{2} x^{12}+2 b c \,e^{2} x^{10}+2 c^{2} d e \,x^{10}+2 a c \,e^{2} x^{8}+b^{2} e^{2} x^{8}+4 b c d e \,x^{8}+c^{2} d^{2} x^{8}+2 a b \,e^{2} x^{6}+4 a c d e \,x^{6}+2 b^{2} d e \,x^{6}+2 b c \,d^{2} x^{6}+a^{2} e^{2} x^{4}+4 a b d e \,x^{4}+2 a c \,d^{2} x^{4}+b^{2} d^{2} x^{4}+2 a^{2} d e \,x^{2}+2 a b \,d^{2} x^{2}+a^{2} d^{2}}d x \right ) a +\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} e^{2} x^{12}+2 b c \,e^{2} x^{10}+2 c^{2} d e \,x^{10}+2 a c \,e^{2} x^{8}+b^{2} e^{2} x^{8}+4 b c d e \,x^{8}+c^{2} d^{2} x^{8}+2 a b \,e^{2} x^{6}+4 a c d e \,x^{6}+2 b^{2} d e \,x^{6}+2 b c \,d^{2} x^{6}+a^{2} e^{2} x^{4}+4 a b d e \,x^{4}+2 a c \,d^{2} x^{4}+b^{2} d^{2} x^{4}+2 a^{2} d e \,x^{2}+2 a b \,d^{2} x^{2}+a^{2} d^{2}}d x \right ) b \] Input:

int((B*x^2+A)/(e*x^2+d)^2/(c*x^4+b*x^2+a)^(3/2),x)
 

Output:

int(sqrt(a + b*x**2 + c*x**4)/(a**2*d**2 + 2*a**2*d*e*x**2 + a**2*e**2*x** 
4 + 2*a*b*d**2*x**2 + 4*a*b*d*e*x**4 + 2*a*b*e**2*x**6 + 2*a*c*d**2*x**4 + 
 4*a*c*d*e*x**6 + 2*a*c*e**2*x**8 + b**2*d**2*x**4 + 2*b**2*d*e*x**6 + b** 
2*e**2*x**8 + 2*b*c*d**2*x**6 + 4*b*c*d*e*x**8 + 2*b*c*e**2*x**10 + c**2*d 
**2*x**8 + 2*c**2*d*e*x**10 + c**2*e**2*x**12),x)*a + int((sqrt(a + b*x**2 
 + c*x**4)*x**2)/(a**2*d**2 + 2*a**2*d*e*x**2 + a**2*e**2*x**4 + 2*a*b*d** 
2*x**2 + 4*a*b*d*e*x**4 + 2*a*b*e**2*x**6 + 2*a*c*d**2*x**4 + 4*a*c*d*e*x* 
*6 + 2*a*c*e**2*x**8 + b**2*d**2*x**4 + 2*b**2*d*e*x**6 + b**2*e**2*x**8 + 
 2*b*c*d**2*x**6 + 4*b*c*d*e*x**8 + 2*b*c*e**2*x**10 + c**2*d**2*x**8 + 2* 
c**2*d*e*x**10 + c**2*e**2*x**12),x)*b