Integrand size = 34, antiderivative size = 597 \[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\frac {\left (C d^2-B d e+A e^2\right ) x \sqrt {a-c x^4}}{4 d e^2 \left (d+e x^2\right )^2}-\frac {\left (c d^2 \left (7 C d^2-e (3 B d+A e)\right )-a e^2 \left (5 C d^2-e (B d+3 A e)\right )\right ) x \sqrt {a-c x^4}}{8 d^2 e^2 \left (c d^2-a e^2\right ) \left (d+e x^2\right )}-\frac {a^{3/4} \sqrt [4]{c} \left (c d^2 \left (15 C d^2-e (3 B d+A e)\right )-a e^2 \left (13 C d^2-e (B d+3 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 e^3 \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (c d^2 \left (15 C d^2-e (3 B d+A e)\right )+2 \sqrt {a} \sqrt {c} d e \left (15 C d^2-e (3 B d+A e)\right )+a e^2 \left (13 C d^2-e (B d+3 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 d^2 e^4 \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (2 a c d^2 e^2 \left (11 C d^2-3 e (B d-A e)\right )-c^2 d^4 \left (15 C d^2-e (3 B d+A e)\right )-a^2 e^4 \left (3 C d^2+e (B d+3 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{c} d^3 e^4 \left (c d^2-a e^2\right ) \sqrt {a-c x^4}} \] Output:
1/4*(A*e^2-B*d*e+C*d^2)*x*(-c*x^4+a)^(1/2)/d/e^2/(e*x^2+d)^2-1/8*(c*d^2*(7 *C*d^2-e*(A*e+3*B*d))-a*e^2*(5*C*d^2-e*(3*A*e+B*d)))*x*(-c*x^4+a)^(1/2)/d^ 2/e^2/(-a*e^2+c*d^2)/(e*x^2+d)-1/8*a^(3/4)*c^(1/4)*(c*d^2*(15*C*d^2-e*(A*e +3*B*d))-a*e^2*(13*C*d^2-e*(3*A*e+B*d)))*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/ 4)*x/a^(1/4),I)/d^2/e^3/(-a*e^2+c*d^2)/(-c*x^4+a)^(1/2)+1/8*a^(1/4)*c^(1/4 )*(c*d^2*(15*C*d^2-e*(A*e+3*B*d))+2*a^(1/2)*c^(1/2)*d*e*(15*C*d^2-e*(A*e+3 *B*d))+a*e^2*(13*C*d^2-e*(3*A*e+B*d)))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4) *x/a^(1/4),I)/d^2/e^4/(c^(1/2)*d+a^(1/2)*e)/(-c*x^4+a)^(1/2)+1/8*a^(1/4)*( 2*a*c*d^2*e^2*(11*C*d^2-3*e*(-A*e+B*d))-c^2*d^4*(15*C*d^2-e*(A*e+3*B*d))-a ^2*e^4*(3*C*d^2+e*(3*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^3/e^4/(-a*e^2+c*d^2)/(-c*x^4+a)^(1/ 2)
Result contains complex when optimal does not.
Time = 14.04 (sec) , antiderivative size = 530, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\frac {\frac {d e^2 x \left (a-c x^4\right ) \left (2 d \left (c d^2-a e^2\right ) \left (C d^2+e (-B d+A e)\right )-\left (c \left (7 C d^4-d^2 e (3 B d+A e)\right )+a e^2 \left (-5 C d^2+e (B d+3 A e)\right )\right ) \left (d+e x^2\right )\right )}{\left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {i \sqrt {1-\frac {c x^4}{a}} \left (-\sqrt {a} \sqrt {c} d e \left (-15 c C d^4+13 a C d^2 e^2+c d^2 e (3 B d+A e)-a e^3 (B d+3 A e)\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-\sqrt {c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (2 \sqrt {a} \sqrt {c} d e \left (15 C d^2-e (3 B d+A e)\right )+c \left (15 C d^4-d^2 e (3 B d+A e)\right )-a e^2 \left (-13 C d^2+e (B d+3 A e)\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\left (-2 a c d^2 e^2 \left (11 C d^2+3 e (-B d+A e)\right )+c^2 \left (15 C d^6-d^4 e (3 B d+A e)\right )+a^2 e^4 \left (3 C d^2+e (B d+3 A e)\right )\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \left (-c d^2+a e^2\right )}}{8 d^3 e^4 \sqrt {a-c x^4}} \] Input:
Integrate[(Sqrt[a - c*x^4]*(A + B*x^2 + C*x^4))/(d + e*x^2)^3,x]
Output:
((d*e^2*x*(a - c*x^4)*(2*d*(c*d^2 - a*e^2)*(C*d^2 + e*(-(B*d) + A*e)) - (c *(7*C*d^4 - d^2*e*(3*B*d + A*e)) + a*e^2*(-5*C*d^2 + e*(B*d + 3*A*e)))*(d + e*x^2)))/((c*d^2 - a*e^2)*(d + e*x^2)^2) - (I*Sqrt[1 - (c*x^4)/a]*(-(Sqr t[a]*Sqrt[c]*d*e*(-15*c*C*d^4 + 13*a*C*d^2*e^2 + c*d^2*e*(3*B*d + A*e) - a *e^3*(B*d + 3*A*e))*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]) - Sqrt[c]*d*(Sqrt[c]*d - Sqrt[a]*e)*(2*Sqrt[a]*Sqrt[c]*d*e*(15*C*d^2 - e*( 3*B*d + A*e)) + c*(15*C*d^4 - d^2*e*(3*B*d + A*e)) - a*e^2*(-13*C*d^2 + e* (B*d + 3*A*e)))*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (-2 *a*c*d^2*e^2*(11*C*d^2 + 3*e*(-(B*d) + A*e)) + c^2*(15*C*d^6 - d^4*e*(3*B* d + A*e)) + a^2*e^4*(3*C*d^2 + e*(B*d + 3*A*e)))*EllipticPi[-((Sqrt[a]*e)/ (Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]))/(Sqrt[-(Sqrt[c] /Sqrt[a])]*(-(c*d^2) + a*e^2)))/(8*d^3*e^4*Sqrt[a - c*x^4])
Leaf count is larger than twice the leaf count of optimal. \(1223\) vs. \(2(597)=1194\).
Time = 2.27 (sec) , antiderivative size = 1223, normalized size of antiderivative = 2.05, 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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {-a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3}{e^4 \sqrt {a-c x^4} \left (d+e x^2\right )^2}+\frac {a C e^2-c \left (6 C d^2-e (3 B d-A e)\right )}{e^4 \sqrt {a-c x^4} \left (d+e x^2\right )}+\frac {\left (a e^2-c d^2\right ) \left (A e^2-B d e+C d^2\right )}{e^4 \sqrt {a-c x^4} \left (d+e x^2\right )^3}-\frac {c (B e-3 C d)}{e^4 \sqrt {a-c x^4}}-\frac {c C x^2}{e^3 \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \left (3 c d^2-a e^2\right ) \left (C d^2-B e d+A e^2\right ) \sqrt {a-c x^4} x}{8 d^2 e^2 \left (c d^2-a e^2\right ) \left (e x^2+d\right )}-\frac {\left (4 c C d^3-c e (3 B d-2 A e) d-a e^2 (2 C d-B e)\right ) \sqrt {a-c x^4} x}{2 d e^2 \left (c d^2-a e^2\right ) \left (e x^2+d\right )}+\frac {\left (C d^2-B e d+A e^2\right ) \sqrt {a-c x^4} x}{4 d e^2 \left (e x^2+d\right )^2}+\frac {3 a^{3/4} \sqrt [4]{c} \left (3 c d^2-a e^2\right ) \left (C d^2-B e d+A e^2\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 )}{8 d^2 e^3 \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}-\frac {a^{3/4} \sqrt [4]{c} \left (4 c C d^3-c e (3 B d-2 A e) d-a e^2 (2 C d-B 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 \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}-\frac {a^{3/4} \sqrt [4]{c} 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 )}{e^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{3/4} (3 C d-B e) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{e^4 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (7 c d^2-2 \sqrt {a} \sqrt {c} e d-3 a e^2\right ) \left (C d^2-B e d+A e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 d^2 e^4 \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (4 c C d^3-c e (3 B d-2 A e) d-a e^2 (2 C d-B e)\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 d e^4 \left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {a-c x^4}}+\frac {a^{3/4} \sqrt [4]{c} C \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{e^3 \sqrt {a-c x^4}}-\frac {3 \sqrt [4]{a} \left (C d^2-B e d+A e^2\right ) \left (5 c^2 d^4-2 a c e^2 d^2+a^2 e^4\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 )}{8 \sqrt [4]{c} d^3 e^4 \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (3 c d^2-a e^2\right ) \left (4 c C d^3-c e (3 B d-2 A e) d-a e^2 (2 C d-B e)\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 \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (a C e^2-c \left (6 C d^2-e (3 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 )}{\sqrt [4]{c} d e^4 \sqrt {a-c x^4}}\) |
Input:
Int[(Sqrt[a - c*x^4]*(A + B*x^2 + C*x^4))/(d + e*x^2)^3,x]
Output:
((C*d^2 - B*d*e + A*e^2)*x*Sqrt[a - c*x^4])/(4*d*e^2*(d + e*x^2)^2) + (3*( 3*c*d^2 - a*e^2)*(C*d^2 - B*d*e + A*e^2)*x*Sqrt[a - c*x^4])/(8*d^2*e^2*(c* d^2 - a*e^2)*(d + e*x^2)) - ((4*c*C*d^3 - c*d*e*(3*B*d - 2*A*e) - a*e^2*(2 *C*d - B*e))*x*Sqrt[a - c*x^4])/(2*d*e^2*(c*d^2 - a*e^2)*(d + e*x^2)) - (a ^(3/4)*c^(1/4)*C*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)] , -1])/(e^3*Sqrt[a - c*x^4]) + (3*a^(3/4)*c^(1/4)*(3*c*d^2 - a*e^2)*(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])/(8*d^2*e^3*(c*d^2 - a*e^2)*Sqrt[a - c*x^4]) - (a^(3/4)*c^(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] *EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*d*e^3*(c*d^2 - a*e^2)*Sqrt [a - c*x^4]) + (a^(3/4)*c^(1/4)*C*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^ (1/4)*x)/a^(1/4)], -1])/(e^3*Sqrt[a - c*x^4]) + (a^(1/4)*c^(3/4)*(3*C*d - B*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(e^4* Sqrt[a - c*x^4]) + (a^(1/4)*c^(1/4)*(7*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a *e^2)*(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])/(8*d^2*e^4*(Sqrt[c]*d + Sqrt[a]*e)*Sqrt[a - c*x^4]) - (a^(1/4)*c^(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]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*d*e^4* (Sqrt[c]*d + Sqrt[a]*e)*Sqrt[a - c*x^4]) - (3*a^(1/4)*(C*d^2 - B*d*e + A*e ^2)*(5*c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*Sqrt[1 - (c*x^4)/a]*EllipticP...
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. 1831 vs. \(2 (535 ) = 1070\).
Time = 2.09 (sec) , antiderivative size = 1832, normalized size of antiderivative = 3.07
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1832\) |
| elliptic | \(\text {Expression too large to display}\) | \(3049\) |
Input:
int((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
C/e^2*(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)*Elliptic E(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)+(B*e-2*C*d)/e^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)+1/2*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)*Ellip ticE(x*(c^(1/2)/a^(1/2))^(1/2),I)+1/2/d^2/(c^(1/2)/a^(1/2))^(1/2)*(1-c^...
\[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d)^3,x, algorithm="frica s")
Output:
integral((C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d ^2*e*x^2 + d^3), x)
\[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {\sqrt {a - c x^{4}} \left (A + B x^{2} + C x^{4}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \] Input:
integrate((-c*x**4+a)**(1/2)*(C*x**4+B*x**2+A)/(e*x**2+d)**3,x)
Output:
Integral(sqrt(a - c*x**4)*(A + B*x**2 + C*x**4)/(d + e*x**2)**3, x)
\[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d)^3,x, algorithm="maxim a")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)/(e*x^2 + d)^3, x)
\[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d)^3,x, algorithm="giac" )
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)/(e*x^2 + d)^3, x)
Timed out. \[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {\sqrt {a-c\,x^4}\,\left (C\,x^4+B\,x^2+A\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:
int(((a - c*x^4)^(1/2)*(A + B*x^2 + C*x^4))/(d + e*x^2)^3,x)
Output:
int(((a - c*x^4)^(1/2)*(A + B*x^2 + C*x^4))/(d + e*x^2)^3, x)
\[ \int \frac {\sqrt {a-c x^4} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\text {too large to display} \] Input:
int((-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A)/(e*x^2+d)^3,x)
Output:
( - sqrt(a - c*x**4)*a*e*x + 3*sqrt(a - c*x**4)*c*d*x**3 + 4*int(sqrt(a - c*x**4)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3 *x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10),x)*a**2*d**3*e + 8*int(sqrt(a - c*x**4)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e* *3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10), x)*a**2*d**2*e**2*x**2 + 4*int(sqrt(a - c*x**4)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e* *2*x**8 - c*e**3*x**10),x)*a**2*d*e**3*x**4 + int((sqrt(a - c*x**4)*x**6)/ (a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10),x)*a*c*d**2*e**2 + 2*int ((sqrt(a - c*x**4)*x**6)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e **3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10) ,x)*a*c*d*e**3*x**2 + int((sqrt(a - c*x**4)*x**6)/(a*d**3 + 3*a*d**2*e*x** 2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d* e**2*x**8 - c*e**3*x**10),x)*a*c*e**4*x**4 - 3*int((sqrt(a - c*x**4)*x**6) /(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10),x)*b*c*d**3*e - 6*int(( sqrt(a - c*x**4)*x**6)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e** 3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10),x )*b*c*d**2*e**2*x**2 - 3*int((sqrt(a - c*x**4)*x**6)/(a*d**3 + 3*a*d**2...