Integrand size = 34, antiderivative size = 582 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=-\frac {\left (C d^2-B d e+A e^2\right ) x}{2 d \left (c d^2-a e^2\right ) \left (d+e x^2\right ) \sqrt {a-c x^4}}+\frac {x \left (d \left (A c \left (c d^2+2 a e^2\right )+a \left (a C e^2+c d (2 C d-3 B e)\right )\right )+c \left (B c d^3-2 A c d^2 e-3 a C d^2 e+2 a B d e^2-a A e^3\right ) x^2\right )}{2 a d \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {\sqrt [4]{c} \left (e \left (2 A c d^2+3 a C d^2+a A e^2\right )-B \left (c d^3+2 a d e^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} d \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {\left (A c^{3/2} d^2-a^{3/2} C d e+\sqrt {a} c d (B d-A e)+a \sqrt {c} \left (2 C d^2-e (2 B d-A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} \sqrt [4]{c} d \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (c d^2 \left (3 C d^2-e (5 B d-7 A e)\right )+a e^2 \left (3 C d^2-e (B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{c} d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \] Output:
-1/2*(A*e^2-B*d*e+C*d^2)*x/d/(-a*e^2+c*d^2)/(e*x^2+d)/(-c*x^4+a)^(1/2)+1/2 *x*(d*(A*c*(2*a*e^2+c*d^2)+a*(C*a*e^2+c*d*(-3*B*e+2*C*d)))+c*(-A*a*e^3-2*A *c*d^2*e+2*B*a*d*e^2+B*c*d^3-3*C*a*d^2*e)*x^2)/a/d/(-a*e^2+c*d^2)^2/(-c*x^ 4+a)^(1/2)+1/2*c^(1/4)*(e*(A*a*e^2+2*A*c*d^2+3*C*a*d^2)-B*(2*a*d*e^2+c*d^3 ))*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(1/4)/d/(-a*e^2+c*d^ 2)^2/(-c*x^4+a)^(1/2)+1/2*(A*c^(3/2)*d^2-a^(3/2)*C*d*e+a^(1/2)*c*d*(-A*e+B *d)+a*c^(1/2)*(2*C*d^2-e*(-A*e+2*B*d)))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4 )*x/a^(1/4),I)/a^(3/4)/c^(1/4)/d/(c^(1/2)*d+a^(1/2)*e)/(-a*e^2+c*d^2)/(-c* x^4+a)^(1/2)-1/2*a^(1/4)*(c*d^2*(3*C*d^2-e*(-7*A*e+5*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/(-a*e^2+c*d^2)^2/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.17 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d \left (a e^2 \left (C d^2+e (-B d+A e)\right ) x \left (a-c x^4\right )+d x \left (d+e x^2\right ) \left (a^2 C e^2+B c^2 d^2 x^2+A c \left (a e^2+c d \left (d-2 e x^2\right )\right )+a c \left (C d \left (d-2 e x^2\right )+B e \left (-2 d+e x^2\right )\right )\right )\right )-i \left (d+e x^2\right ) \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} \sqrt {c} d \left (-B c d^3+2 A c d^2 e+3 a C d^2 e-2 a B d e^2+a A e^3\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (A c^{3/2} d^2-a^{3/2} C d e+\sqrt {a} c d (B d-A e)+a \sqrt {c} \left (2 C d^2+e (-2 B d+A e)\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+a \left (c \left (-3 C d^4+d^2 e (5 B d-7 A e)\right )+a e^2 \left (-3 C d^2+e (B d+A e)\right )\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{2 a \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \left (c d^3-a d e^2\right )^2 \left (d+e x^2\right ) \sqrt {a-c x^4}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/((d + e*x^2)^2*(a - c*x^4)^(3/2)),x]
Output:
(Sqrt[-(Sqrt[c]/Sqrt[a])]*d*(a*e^2*(C*d^2 + e*(-(B*d) + A*e))*x*(a - c*x^4 ) + d*x*(d + e*x^2)*(a^2*C*e^2 + B*c^2*d^2*x^2 + A*c*(a*e^2 + c*d*(d - 2*e *x^2)) + a*c*(C*d*(d - 2*e*x^2) + B*e*(-2*d + e*x^2)))) - I*(d + e*x^2)*Sq rt[1 - (c*x^4)/a]*(Sqrt[a]*Sqrt[c]*d*(-(B*c*d^3) + 2*A*c*d^2*e + 3*a*C*d^2 *e - 2*a*B*d*e^2 + a*A*e^3)*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x ], -1] + d*(Sqrt[c]*d - Sqrt[a]*e)*(A*c^(3/2)*d^2 - a^(3/2)*C*d*e + Sqrt[a ]*c*d*(B*d - A*e) + a*Sqrt[c]*(2*C*d^2 + e*(-2*B*d + A*e)))*EllipticF[I*Ar cSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + a*(c*(-3*C*d^4 + d^2*e*(5*B*d - 7 *A*e)) + a*e^2*(-3*C*d^2 + e*(B*d + A*e)))*EllipticPi[-((Sqrt[a]*e)/(Sqrt[ c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]))/(2*a*Sqrt[-(Sqrt[c]/S qrt[a])]*(c*d^3 - a*d*e^2)^2*(d + e*x^2)*Sqrt[a - c*x^4])
Time = 1.42 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.34, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{3/2} \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {-A e^2+B d e-C d^2}{\sqrt {a-c x^4} \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}+\frac {e \left (B \left (a e^2+c d^2\right )-2 d e (a C+A c)\right )}{\sqrt {a-c x^4} \left (d+e x^2\right ) \left (c d^2-a e^2\right )^2}+\frac {-c x^2 \left (2 d e (a C+A c)-B \left (a e^2+c d^2\right )\right )+A c \left (a e^2+c d^2\right )+a \left (a C e^2+c d (C d-2 B e)\right )}{\left (a-c x^4\right )^{3/2} \left (c d^2-a e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/4} \sqrt [4]{c} e \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 \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {\sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} B \sqrt {c}+a C+A c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} \sqrt [4]{c} \sqrt {a-c x^4} \left (\sqrt {a} e+\sqrt {c} d\right )^2}+\frac {\sqrt [4]{a} \sqrt [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 ) \left (A e^2-B d e+C d^2\right )}{2 d \sqrt {a-c x^4} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (c d^2-a e^2\right )}+\frac {\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 (2 d e (a C+A c)-B \left (a e^2+c d^2\right )\right )}{2 \sqrt [4]{a} \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}-\frac {\sqrt [4]{a} e \sqrt {1-\frac {c x^4}{a}} \left (2 d e (a C+A c)-B \left (a e^2+c d^2\right )\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 \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}-\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 \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {e^2 x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )^2}+\frac {x \left (-c x^2 \left (2 d e (a C+A c)-B \left (a e^2+c d^2\right )\right )+A c \left (a e^2+c d^2\right )+a \left (a C e^2+c d (C d-2 B e)\right )\right )}{2 a \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}\) |
Input:
Int[(A + B*x^2 + C*x^4)/((d + e*x^2)^2*(a - c*x^4)^(3/2)),x]
Output:
(x*(A*c*(c*d^2 + a*e^2) + a*(a*C*e^2 + c*d*(C*d - 2*B*e)) - c*(2*(A*c + a* C)*d*e - B*(c*d^2 + a*e^2))*x^2))/(2*a*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) + (e^2*(C*d^2 - B*d*e + A*e^2)*x*Sqrt[a - c*x^4])/(2*d*(c*d^2 - a*e^2)^2*( d + e*x^2)) + (a^(3/4)*c^(1/4)*e*(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*(c*d^2 - a*e^2)^2*Sqrt [a - c*x^4]) + (c^(1/4)*(2*(A*c + a*C)*d*e - B*(c*d^2 + a*e^2))*Sqrt[1 - ( c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(1/4)*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) + ((Sqrt[a]*B*Sqrt[c] + A*c + a*C)*Sqrt[1 - (c*x ^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(3/4)*c^(1/4)*(Sqr t[c]*d + Sqrt[a]*e)^2*Sqrt[a - c*x^4]) + (a^(1/4)*c^(1/4)*(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2 *d*(Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 - a*e^2)*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]*EllipticPi[-((S qrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*c^(1/4)*d^2*( c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) - (a^(1/4)*e*(2*(A*c + a*C)*d*e - B*(c*d ^2 + a*e^2))*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), Ar cSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^ 4])
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1421 vs. \(2 (516 ) = 1032\).
Time = 1.09 (sec) , antiderivative size = 1422, normalized size of antiderivative = 2.44
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1422\) |
| elliptic | \(\text {Expression too large to display}\) | \(2169\) |
Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
C/e^2*(1/2/a*x/(-(x^4-a/c)*c)^(1/2)+1/2/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)*Ellip ticF(x*(c^(1/2)/a^(1/2))^(1/2),I))+1/e^2*(B*e-2*C*d)*(2*c*(1/4/a*e/(a*e^2- c*d^2)*x^3-1/4/a*d/(a*e^2-c*d^2)*x)/(-(x^4-a/c)*c)^(1/2)-1/2*c/a*d/(a*e^2- c*d^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)/a^(1/2)*e/(a*e^2-c*d^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)/a^(1/2)*e/(a*e^2-c*d^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/(a*e^2-c*d^2)*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)))+1/e^2*(A* e^2-B*d*e+C*d^2)*(1/2*e^4/(a*e^2-c*d^2)^2/d*x*(-c*x^4+a)^(1/2)/(e*x^2+d)+2 *c*(-1/2/a*c*d*e/(a*e^2-c*d^2)^2*x^3+1/4/a*(a*e^2+c*d^2)/(a*e^2-c*d^2)^2*x )/(-(x^4-a/c)*c)^(1/2)+1/(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)*e^2*c/(a*e^2-c*d^2)^2+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)*E...
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(3/2),x, algorithm="frica s")
Output:
Timed out
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((C*x**4+B*x**2+A)/(e*x**2+d)**2/(-c*x**4+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(3/2),x, algorithm="maxim a")
Output:
integrate((C*x^4 + B*x^2 + A)/((-c*x^4 + a)^(3/2)*(e*x^2 + d)^2), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(3/2),x, algorithm="giac" )
Output:
integrate((C*x^4 + B*x^2 + A)/((-c*x^4 + a)^(3/2)*(e*x^2 + d)^2), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (a-c\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((A + B*x^2 + C*x^4)/((a - c*x^4)^(3/2)*(d + e*x^2)^2),x)
Output:
int((A + B*x^2 + C*x^4)/((a - c*x^4)^(3/2)*(d + e*x^2)^2), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} e^{2} x^{12}+2 c^{2} d e \,x^{10}-2 a c \,e^{2} x^{8}+c^{2} d^{2} x^{8}-4 a c d e \,x^{6}+a^{2} e^{2} x^{4}-2 a c \,d^{2} x^{4}+2 a^{2} d e \,x^{2}+a^{2} d^{2}}d x \right ) a +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{c^{2} e^{2} x^{12}+2 c^{2} d e \,x^{10}-2 a c \,e^{2} x^{8}+c^{2} d^{2} x^{8}-4 a c d e \,x^{6}+a^{2} e^{2} x^{4}-2 a c \,d^{2} x^{4}+2 a^{2} d e \,x^{2}+a^{2} d^{2}}d x \right ) c +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} e^{2} x^{12}+2 c^{2} d e \,x^{10}-2 a c \,e^{2} x^{8}+c^{2} d^{2} x^{8}-4 a c d e \,x^{6}+a^{2} e^{2} x^{4}-2 a c \,d^{2} x^{4}+2 a^{2} d e \,x^{2}+a^{2} d^{2}}d x \right ) b \] Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(3/2),x)
Output:
int(sqrt(a - c*x**4)/(a**2*d**2 + 2*a**2*d*e*x**2 + a**2*e**2*x**4 - 2*a*c *d**2*x**4 - 4*a*c*d*e*x**6 - 2*a*c*e**2*x**8 + c**2*d**2*x**8 + 2*c**2*d* e*x**10 + c**2*e**2*x**12),x)*a + int((sqrt(a - c*x**4)*x**4)/(a**2*d**2 + 2*a**2*d*e*x**2 + a**2*e**2*x**4 - 2*a*c*d**2*x**4 - 4*a*c*d*e*x**6 - 2*a *c*e**2*x**8 + c**2*d**2*x**8 + 2*c**2*d*e*x**10 + c**2*e**2*x**12),x)*c + int((sqrt(a - c*x**4)*x**2)/(a**2*d**2 + 2*a**2*d*e*x**2 + a**2*e**2*x**4 - 2*a*c*d**2*x**4 - 4*a*c*d*e*x**6 - 2*a*c*e**2*x**8 + c**2*d**2*x**8 + 2 *c**2*d*e*x**10 + c**2*e**2*x**12),x)*b