\(\int \frac {\sqrt {a-c x^4} (A+B x^2+C x^4)}{(d+e x^2)^2} \, dx\) [31]

Optimal result

Integrand size = 34, antiderivative size = 403 \[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^2} \, dx=\frac {C x \sqrt {a-c x^4}}{3 e^2}+\frac {\left (C d^2-B d e+A e^2\right ) x \sqrt {a-c x^4}}{2 d e^2 \left (d+e x^2\right )}+\frac {a^{3/4} \sqrt [4]{c} \left (5 C d^2-e (3 B d-A e)\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d e^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (4 a C d e^2-3 c d \left (5 C d^2-e (3 B d-A e)\right )-3 \sqrt {a} \sqrt {c} e \left (5 C d^2-e (3 B d-A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{6 \sqrt [4]{c} d e^4 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (c d^2 \left (5 C d^2-e (3 B d-A e)\right )-a e^2 \left (3 C d^2-e (B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{c} d^2 e^4 \sqrt {a-c x^4}} \] Output:

1/3*C*x*(-c*x^4+a)^(1/2)/e^2+1/2*(A*e^2-B*d*e+C*d^2)*x*(-c*x^4+a)^(1/2)/d/ 
e^2/(e*x^2+d)+1/2*a^(3/4)*c^(1/4)*(5*C*d^2-e*(-A*e+3*B*d))*(1-c*x^4/a)^(1/ 
2)*EllipticE(c^(1/4)*x/a^(1/4),I)/d/e^3/(-c*x^4+a)^(1/2)+1/6*a^(1/4)*(4*a* 
C*d*e^2-3*c*d*(5*C*d^2-e*(-A*e+3*B*d))-3*a^(1/2)*c^(1/2)*e*(5*C*d^2-e*(-A* 
e+3*B*d)))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(1/4)/d/e^4/ 
(-c*x^4+a)^(1/2)+1/2*a^(1/4)*(c*d^2*(5*C*d^2-e*(-A*e+3*B*d))-a*e^2*(3*C*d^ 
2-e*(A*e+B*d)))*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/ 
c^(1/2)/d,I)/c^(1/4)/d^2/e^4/(-c*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.26 (sec) , antiderivative size = 1282, normalized size of antiderivative = 3.18 \[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(5*a*Sqrt[-(Sqrt[c]/Sqrt[a])]*C*d^3*e^2*x - 3*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])] 
*d^2*e^3*x + 3*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*d*e^4*x + 2*a*Sqrt[-(Sqrt[c]/S 
qrt[a])]*C*d^2*e^3*x^3 - 5*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*C*d^3*e^2*x^5 + 3*B* 
Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^2*e^3*x^5 - 3*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d* 
e^4*x^5 - 2*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*C*d^2*e^3*x^7 - (3*I)*Sqrt[a]*Sqrt[ 
c]*d*e*(5*C*d^2 + e*(-3*B*d + A*e))*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*Ellipt 
icE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*d*(4*a*C*d*e^2 - 3*c*d* 
(5*C*d^2 - 3*B*d*e + A*e^2) - 3*Sqrt[a]*Sqrt[c]*e*(5*C*d^2 - 3*B*d*e + A*e 
^2))*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sq 
rt[a])]*x], -1] - (15*I)*c*C*d^5*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a] 
*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (9*I)*B*c*d 
^4*e*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[ 
Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*A*c*d^3*e^2*Sqrt[1 - (c*x^4)/a]*E 
llipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x 
], -1] + (9*I)*a*C*d^3*e^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(S 
qrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*a*B*d^2*e^3 
*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt 
[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*a*A*d*e^4*Sqrt[1 - (c*x^4)/a]*Ellipti 
cPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] 
 - (15*I)*c*C*d^4*e*x^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(S...
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 737, normalized size of antiderivative = 1.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2259, 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.

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

\(\Big \downarrow \) 2259

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(C*x*Sqrt[a - c*x^4])/(3*e^2) + ((C*d^2 - B*d*e + A*e^2)*x*Sqrt[a - c*x^4] 
)/(2*d*e^2*(d + e*x^2)) + (a^(3/4)*c^(1/4)*(2*C*d - B*e)*Sqrt[1 - (c*x^4)/ 
a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(e^3*Sqrt[a - c*x^4]) + (a^ 
(3/4)*c^(1/4)*(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin 
[(c^(1/4)*x)/a^(1/4)], -1])/(2*d*e^3*Sqrt[a - c*x^4]) - (a^(5/4)*C*Sqrt[1 
- (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(3*c^(1/4)*e^2*Sq 
rt[a - c*x^4]) - (a^(3/4)*c^(1/4)*(2*C*d - B*e)*Sqrt[1 - (c*x^4)/a]*Ellipt 
icF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(e^3*Sqrt[a - c*x^4]) + (a^(1/4)*c^( 
1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*E 
llipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*d*e^4*Sqrt[a - c*x^4]) + (a^ 
(1/4)*(a*C*e^2 - c*(3*C*d^2 - e*(2*B*d - A*e)))*Sqrt[1 - (c*x^4)/a]*Ellipt 
icF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*e^4*Sqrt[a - c*x^4]) - (a^( 
1/4)*(3*c*d^2 - a*e^2)*(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*Ellipti 
cPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*c^(1/ 
4)*d^2*e^4*Sqrt[a - c*x^4]) + (a^(1/4)*(4*c*C*d^3 - c*d*e*(3*B*d - 2*A*e) 
- a*e^2*(2*C*d - B*e))*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[ 
c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d*e^4*Sqrt[a - c*x^4])
 

Defintions of rubi rules used

rule 2009

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

rule 2259

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (345 ) = 690\).

Time = 5.13 (sec) , antiderivative size = 970, normalized size of antiderivative = 2.41

method	result	size
default	\(\text {Expression too large to display}\)	\(970\)
risch	\(\text {Expression too large to display}\)	\(1114\)
elliptic	\(\text {Expression too large to display}\)	\(1396\)

Input:

int((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
 

Output:

C/e^2*(1/3*x*(-c*x^4+a)^(1/2)+2/3*a/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2 
/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x 
*(c^(1/2)/a^(1/2))^(1/2),I))+1/e^2*(B*e-2*C*d)*(c*d/e^2/(c^(1/2)/a^(1/2))^ 
(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+ 
a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)+1/e*c^(1/2)*a^(1/2)/(c^(1/ 
2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1 
/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-1/e*c^(1/2)*a^ 
(1/2)/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2 
/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I)+1/ 
d/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^( 
1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(c^(1/2)/a^(1/2))^(1/2),-a^(1/2) 
*e/c^(1/2)/d,(-c^(1/2)/a^(1/2))^(1/2)/(c^(1/2)/a^(1/2))^(1/2))*a-1/e^2*d/( 
c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2 
))^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(c^(1/2)/a^(1/2))^(1/2),-a^(1/2)*e/ 
c^(1/2)/d,(-c^(1/2)/a^(1/2))^(1/2)/(c^(1/2)/a^(1/2))^(1/2))*c)+1/e^2*(A*e^ 
2-B*d*e+C*d^2)*(1/2/d*x*(-c*x^4+a)^(1/2)/(e*x^2+d)-1/2*c/e^2/(c^(1/2)/a^(1 
/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c 
*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-1/2*c^(1/2)/d/e*a^(1/ 
2)/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^ 
(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)+1/...
 

Fricas [F(-1)]

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

integrate((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d)^2,x, algorithm="frica 
s")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(sqrt(a - c*x**4)*(A + B*x**2 + C*x**4)/(d + e*x**2)**2, x)
 

Maxima [F]

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

integrate((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d)^2,x, algorithm="maxim 
a")
 

Output:

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

Giac [F]

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

integrate((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d)^2,x, algorithm="giac" 
)
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(a - c*x**4)*a*e*x + 3*sqrt(a - c*x**4)*c*d*x**3 + 8*int(sqrt(a - c*x 
**4)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c 
*e**2*x**8),x)*a**2*d**2*e + 8*int(sqrt(a - c*x**4)/(a*d**2 + 2*a*d*e*x**2 
 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a**2*d*e**2* 
x**2 + int((sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - 
c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*c*d*e**2 + int((sqrt(a - c* 
x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x 
**6 - c*e**2*x**8),x)*a*c*e**3*x**2 - 9*int((sqrt(a - c*x**4)*x**6)/(a*d** 
2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8) 
,x)*b*c*d**2*e - 9*int((sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a* 
e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*b*c*d*e**2*x**2 + 
 15*int((sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d 
**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*c**2*d**3 + 15*int((sqrt(a - c*x 
**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x* 
*6 - c*e**2*x**8),x)*c**2*d**2*e*x**2 + int((sqrt(a - c*x**4)*x**2)/(a*d** 
2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8) 
,x)*a**2*d*e**2 + int((sqrt(a - c*x**4)*x**2)/(a*d**2 + 2*a*d*e*x**2 + a*e 
**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a**2*e**3*x**2 + 9 
*int((sqrt(a - c*x**4)*x**2)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2 
*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*b*d**2*e + 9*int((sqrt(a - c*x...