\(\int \frac {\sqrt {a+b x^2} (A+B x^2+C x^4)}{(c+d x^2)^{5/2} (e+f x^2)} \, dx\) [12]

Optimal result

Integrand size = 44, antiderivative size = 637 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {\left (c^2 C-B c d+A d^2\right ) x \sqrt {a+b x^2}}{3 c d (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {\left (a d \left (2 A d^3 e+c^3 C f+c d^2 (B e-5 A f)-c^2 (4 C d e-2 B d f)\right )-b c \left (A d^3 e+2 c^3 C f+2 c d^2 (B e-2 A f)-c^2 (5 C d e-B d f)\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} d^{3/2} (b c-a d) (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (3 b^2 c^3 d \left (C e^2-f (B e-A f)\right )+3 a^2 c d^3 \left (C e^2-f (B e-A f)\right )-a b \left (A d^4 e^2+c^4 C f^2-c d^3 e (B e+2 A f)-c^3 d f (2 C e+B f)+c^2 d^2 \left (7 C e^2-4 B e f+7 A f^2\right )\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} d^{3/2} (b c-a d) (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} (b e-a f) \left (C e^2-B e f+A f^2\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.95 (sec) , antiderivative size = 3341, normalized size of antiderivative = 5.24 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Result too large to show} \] Input:

Integrate[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4))/((c + d*x^2)^(5/2)*(e + f* 
x^2)),x]
 

Output:

(4*a*b*Sqrt[b/a]*c^4*C*d^2*e^2*f*x - a*b*Sqrt[b/a]*B*c^3*d^3*e^2*f*x - 3*a 
^2*Sqrt[b/a]*c^3*C*d^3*e^2*f*x - 2*a*A*b*Sqrt[b/a]*c^2*d^4*e^2*f*x + 3*a^2 
*A*Sqrt[b/a]*c*d^5*e^2*f*x - a*b*Sqrt[b/a]*c^5*C*d*e*f^2*x - 2*a*b*Sqrt[b/ 
a]*B*c^4*d^2*e*f^2*x + 5*a*A*b*Sqrt[b/a]*c^3*d^3*e*f^2*x + 3*a^2*Sqrt[b/a] 
*B*c^3*d^3*e*f^2*x - 6*a^2*A*Sqrt[b/a]*c^2*d^4*e*f^2*x + 4*a*b*(b/a)^(3/2) 
*c^4*C*d^2*e^2*f*x^3 - a*b*(b/a)^(3/2)*B*c^3*d^3*e^2*f*x^3 + 2*a*b*Sqrt[b/ 
a]*c^3*C*d^3*e^2*f*x^3 - 2*a*A*b*(b/a)^(3/2)*c^2*d^4*e^2*f*x^3 - 2*a*b*Sqr 
t[b/a]*B*c^2*d^4*e^2*f*x^3 - 4*a^2*Sqrt[b/a]*c^2*C*d^4*e^2*f*x^3 + 2*a*A*b 
*Sqrt[b/a]*c*d^5*e^2*f*x^3 + a^2*Sqrt[b/a]*B*c*d^5*e^2*f*x^3 + 2*a^2*A*Sqr 
t[b/a]*d^6*e^2*f*x^3 - a*b*(b/a)^(3/2)*c^5*C*d*e*f^2*x^3 - 2*a*b*(b/a)^(3/ 
2)*B*c^4*d^2*e*f^2*x^3 - 2*a*b*Sqrt[b/a]*c^4*C*d^2*e*f^2*x^3 + 5*a*A*b*(b/ 
a)^(3/2)*c^3*d^3*e*f^2*x^3 + 2*a*b*Sqrt[b/a]*B*c^3*d^3*e*f^2*x^3 + a^2*Sqr 
t[b/a]*c^3*C*d^3*e*f^2*x^3 - 2*a*A*b*Sqrt[b/a]*c^2*d^4*e*f^2*x^3 + 2*a^2*S 
qrt[b/a]*B*c^2*d^4*e*f^2*x^3 - 5*a^2*A*Sqrt[b/a]*c*d^5*e*f^2*x^3 + 5*a*b*( 
b/a)^(3/2)*c^3*C*d^3*e^2*f*x^5 - 2*a*b*(b/a)^(3/2)*B*c^2*d^4*e^2*f*x^5 - 4 
*a*b*Sqrt[b/a]*c^2*C*d^4*e^2*f*x^5 - a*A*b*(b/a)^(3/2)*c*d^5*e^2*f*x^5 + a 
*b*Sqrt[b/a]*B*c*d^5*e^2*f*x^5 + 2*a*A*b*Sqrt[b/a]*d^6*e^2*f*x^5 - 2*a*b*( 
b/a)^(3/2)*c^4*C*d^2*e*f^2*x^5 - a*b*(b/a)^(3/2)*B*c^3*d^3*e*f^2*x^5 + a*b 
*Sqrt[b/a]*c^3*C*d^3*e*f^2*x^5 + 4*a*A*b*(b/a)^(3/2)*c^2*d^4*e*f^2*x^5 + 2 
*a*b*Sqrt[b/a]*B*c^2*d^4*e*f^2*x^5 - 5*a*A*b*Sqrt[b/a]*c*d^5*e*f^2*x^5 ...
 

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 979, normalized size of antiderivative = 1.54, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {7276, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {\sqrt {a+b x^2} \left (A f^2-B e f+C e^2\right )}{f^2 \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )}-\frac {\sqrt {a+b x^2} (C e-B f)}{f^2 \left (c+d x^2\right )^{5/2}}+\frac {C x^2 \sqrt {a+b x^2}}{f \left (c+d x^2\right )^{5/2}}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(Sqrt[a + b*x^2]*(A + B*x^2 + C*x^4))/((c + d*x^2)^(5/2)*(e + f*x^2)), 
x]
 

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6012\) vs. \(2(613)=1226\).

Time = 8.85 (sec) , antiderivative size = 6013, normalized size of antiderivative = 9.44

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(6013\)
default	\(\text {Expression too large to display}\)	\(6326\)

Input:

int((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(5/2)/(f*x^2+e),x,method=_RE 
TURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(5/2)/(f*x^2+e),x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(1/2)*(C*x**4+B*x**2+A)/(d*x**2+c)**(5/2)/(f*x**2+e) 
,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(5/2)/(f*x^2+e),x, alg 
orithm="maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)*sqrt(b*x^2 + a)/((d*x^2 + c)^(5/2)*(f*x^2 + 
e)), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(5/2)/(f*x^2+e),x, alg 
orithm="giac")
 

Output:

integrate((C*x^4 + B*x^2 + A)*sqrt(b*x^2 + a)/((d*x^2 + c)^(5/2)*(f*x^2 + 
e)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (C\,x^4+B\,x^2+A\right )}{{\left (d\,x^2+c\right )}^{5/2}\,\left (f\,x^2+e\right )} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4))/((c + d*x^2)^(5/2)*(e + f*x^2) 
),x)
 

Output:

int(((a + b*x^2)^(1/2)*(A + B*x^2 + C*x^4))/((c + d*x^2)^(5/2)*(e + f*x^2) 
), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {too large to display} \] Input:

int((b*x^2+a)^(1/2)*(C*x^4+B*x^2+A)/(d*x^2+c)^(5/2)/(f*x^2+e),x)
 

Output:

( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*x - sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*b**2*x + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(2*a**2*c**3 
*d*e*f + 2*a**2*c**3*d*f**2*x**2 + 6*a**2*c**2*d**2*e*f*x**2 + 6*a**2*c**2 
*d**2*f**2*x**4 + 6*a**2*c*d**3*e*f*x**4 + 6*a**2*c*d**3*f**2*x**6 + 2*a** 
2*d**4*e*f*x**6 + 2*a**2*d**4*f**2*x**8 - 2*a*b*c**4*e*f - 2*a*b*c**4*f**2 
*x**2 + a*b*c**3*d*e**2 - 3*a*b*c**3*d*e*f*x**2 - 4*a*b*c**3*d*f**2*x**4 + 
 3*a*b*c**2*d**2*e**2*x**2 + 3*a*b*c**2*d**2*e*f*x**4 + 3*a*b*c*d**3*e**2* 
x**4 + 7*a*b*c*d**3*e*f*x**6 + 4*a*b*c*d**3*f**2*x**8 + a*b*d**4*e**2*x**6 
 + 3*a*b*d**4*e*f*x**8 + 2*a*b*d**4*f**2*x**10 - 2*b**2*c**4*e*f*x**2 - 2* 
b**2*c**4*f**2*x**4 + b**2*c**3*d*e**2*x**2 - 5*b**2*c**3*d*e*f*x**4 - 6*b 
**2*c**3*d*f**2*x**6 + 3*b**2*c**2*d**2*e**2*x**4 - 3*b**2*c**2*d**2*e*f*x 
**6 - 6*b**2*c**2*d**2*f**2*x**8 + 3*b**2*c*d**3*e**2*x**6 + b**2*c*d**3*e 
*f*x**8 - 2*b**2*c*d**3*f**2*x**10 + b**2*d**4*e**2*x**8 + b**2*d**4*e*f*x 
**10),x)*a**2*b*c**3*d**2*f**2 + 4*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)* 
x**6)/(2*a**2*c**3*d*e*f + 2*a**2*c**3*d*f**2*x**2 + 6*a**2*c**2*d**2*e*f* 
x**2 + 6*a**2*c**2*d**2*f**2*x**4 + 6*a**2*c*d**3*e*f*x**4 + 6*a**2*c*d**3 
*f**2*x**6 + 2*a**2*d**4*e*f*x**6 + 2*a**2*d**4*f**2*x**8 - 2*a*b*c**4*e*f 
 - 2*a*b*c**4*f**2*x**2 + a*b*c**3*d*e**2 - 3*a*b*c**3*d*e*f*x**2 - 4*a*b* 
c**3*d*f**2*x**4 + 3*a*b*c**2*d**2*e**2*x**2 + 3*a*b*c**2*d**2*e*f*x**4 + 
3*a*b*c*d**3*e**2*x**4 + 7*a*b*c*d**3*e*f*x**6 + 4*a*b*c*d**3*f**2*x**8...