\(\int \frac {(a-c x^4)^{3/2} (A+B x^2+C x^4)}{(d+e x^2)^3} \, dx\) [39]

Optimal result

Integrand size = 34, antiderivative size = 648 \[ \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\frac {c (3 C d-B e) x \sqrt {a-c x^4}}{3 e^4}-\frac {c C x^3 \sqrt {a-c x^4}}{5 e^3}-\frac {\left (c d^2-a e^2\right ) \left (C d^2-B d e+A e^2\right ) x \sqrt {a-c x^4}}{4 d e^4 \left (d+e x^2\right )^2}+\frac {\left (c d^2 \left (15 C d^2-e (11 B d-7 A e)\right )-a e^2 \left (5 C d^2-e (B d+3 A e)\right )\right ) x \sqrt {a-c x^4}}{8 d^2 e^4 \left (d+e x^2\right )}+\frac {a^{3/4} \sqrt [4]{c} \left (5 c d^2 \left (63 C d^2-5 e (7 B d-3 A e)\right )-a e^2 \left (81 C d^2-5 e (B d+3 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{40 d^2 e^5 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (15 c^{3/2} d^3 \left (63 C d^2-5 e (7 B d-3 A e)\right )+15 \sqrt {a} c d^2 e \left (63 C d^2-5 e (7 B d-3 A e)\right )-5 a \sqrt {c} d e^2 \left (99 C d^2-e (31 B d-3 A e)\right )-3 a^{3/2} e^3 \left (81 C d^2-5 e (B d+3 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{120 d^2 e^6 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (c^2 d^4 \left (63 C d^2-5 e (7 B d-3 A e)\right )-2 a c d^2 e^2 \left (27 C d^2-e (11 B d-3 A e)\right )+a^2 e^4 \left (3 C d^2+e (B d+3 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{c} d^3 e^6 \sqrt {a-c x^4}} \] Output:

1/3*c*(-B*e+3*C*d)*x*(-c*x^4+a)^(1/2)/e^4-1/5*c*C*x^3*(-c*x^4+a)^(1/2)/e^3 
-1/4*(-a*e^2+c*d^2)*(A*e^2-B*d*e+C*d^2)*x*(-c*x^4+a)^(1/2)/d/e^4/(e*x^2+d) 
^2+1/8*(c*d^2*(15*C*d^2-e*(-7*A*e+11*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^4/(e*x^2+d)+1/40*a^(3/4)*c^(1/4)*(5*c*d^2*(63*C*d 
^2-5*e*(-3*A*e+7*B*d))-a*e^2*(81*C*d^2-5*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^5/(-c*x^4+a)^(1/2)-1/120*a^(1/4)*c^( 
1/4)*(15*c^(3/2)*d^3*(63*C*d^2-5*e*(-3*A*e+7*B*d))+15*a^(1/2)*c*d^2*e*(63* 
C*d^2-5*e*(-3*A*e+7*B*d))-5*a*c^(1/2)*d*e^2*(99*C*d^2-e*(-3*A*e+31*B*d))-3 
*a^(3/2)*e^3*(81*C*d^2-5*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^6/(-c*x^4+a)^(1/2)+1/8*a^(1/4)*(c^2*d^4*(63*C*d^2-5* 
e*(-3*A*e+7*B*d))-2*a*c*d^2*e^2*(27*C*d^2-e*(-3*A*e+11*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^6/(-c*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 14.43 (sec) , antiderivative size = 579, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\frac {-\frac {d e^2 x \left (a-c x^4\right ) \left (-15 a e^2 \left (-C d^2 \left (3 d+5 e x^2\right )+e \left (B d \left (-d+e x^2\right )+A e \left (5 d+3 e x^2\right )\right )\right )+c d^2 \left (-3 C \left (105 d^3+147 d^2 e x^2+24 d e^2 x^4-8 e^3 x^6\right )+5 e \left (-3 A e \left (5 d+7 e x^2\right )+B \left (35 d^2+49 d e x^2+8 e^2 x^4\right )\right )\right )\right )}{\left (d+e x^2\right )^2}-\frac {i \sqrt {1-\frac {c x^4}{a}} \left (-3 \sqrt {a} \sqrt {c} d e \left (-5 c \left (63 C d^4+5 d^2 e (-7 B d+3 A e)\right )+a e^2 \left (81 C d^2-5 e (B d+3 A e)\right )\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-\sqrt {c} d \left (-5 a \sqrt {c} d e^2 \left (99 C d^2+e (-31 B d+3 A e)\right )+15 \sqrt {a} c d^2 e \left (63 C d^2+5 e (-7 B d+3 A e)\right )+15 c^{3/2} \left (63 C d^5+5 d^3 e (-7 B d+3 A e)\right )+3 a^{3/2} e^3 \left (-81 C d^2+5 e (B d+3 A e)\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+15 \left (-2 a c d^2 e^2 \left (27 C d^2+e (-11 B d+3 A e)\right )+c^2 \left (63 C d^6+5 d^4 e (-7 B d+3 A e)\right )+a^2 e^4 \left (3 C d^2+e (B d+3 A e)\right )\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}}{120 d^3 e^6 \sqrt {a-c x^4}} \] Input:

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

Output:

(-((d*e^2*x*(a - c*x^4)*(-15*a*e^2*(-(C*d^2*(3*d + 5*e*x^2)) + e*(B*d*(-d 
+ e*x^2) + A*e*(5*d + 3*e*x^2))) + c*d^2*(-3*C*(105*d^3 + 147*d^2*e*x^2 + 
24*d*e^2*x^4 - 8*e^3*x^6) + 5*e*(-3*A*e*(5*d + 7*e*x^2) + B*(35*d^2 + 49*d 
*e*x^2 + 8*e^2*x^4)))))/(d + e*x^2)^2) - (I*Sqrt[1 - (c*x^4)/a]*(-3*Sqrt[a 
]*Sqrt[c]*d*e*(-5*c*(63*C*d^4 + 5*d^2*e*(-7*B*d + 3*A*e)) + a*e^2*(81*C*d^ 
2 - 5*e*(B*d + 3*A*e)))*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], - 
1] - Sqrt[c]*d*(-5*a*Sqrt[c]*d*e^2*(99*C*d^2 + e*(-31*B*d + 3*A*e)) + 15*S 
qrt[a]*c*d^2*e*(63*C*d^2 + 5*e*(-7*B*d + 3*A*e)) + 15*c^(3/2)*(63*C*d^5 + 
5*d^3*e*(-7*B*d + 3*A*e)) + 3*a^(3/2)*e^3*(-81*C*d^2 + 5*e*(B*d + 3*A*e))) 
*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + 15*(-2*a*c*d^2*e^2 
*(27*C*d^2 + e*(-11*B*d + 3*A*e)) + c^2*(63*C*d^6 + 5*d^4*e*(-7*B*d + 3*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 
])])/(120*d^3*e^6*Sqrt[a - c*x^4])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1506\) vs. \(2(648)=1296\).

Time = 2.56 (sec) , antiderivative size = 1506, normalized size of antiderivative = 2.32, 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.

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

\(\Big \downarrow \) 2259

\(\displaystyle \int \left (\frac {a^2 C e^4-2 a c e^2 \left (6 C d^2-e (3 B d-A e)\right )+c^2 \left (15 C d^4-2 d^2 e (5 B d-3 A e)\right )}{e^6 \sqrt {a-c x^4} \left (d+e x^2\right )}+\frac {c \left (2 a e^2 (3 C d-B e)+3 c d e (2 B d-A e)-10 c C d^3\right )}{e^6 \sqrt {a-c x^4}}+\frac {\left (a e^2-c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{e^6 \sqrt {a-c x^4} \left (d+e x^2\right )^3}+\frac {c x^2 \left (-2 a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{e^5 \sqrt {a-c x^4}}+\frac {\left (c d^2-a e^2\right ) \left (a e^2 (2 C d-B e)+c d e (5 B d-4 A e)-6 c C d^3\right )}{e^6 \sqrt {a-c x^4} \left (d+e x^2\right )^2}+\frac {c^2 x^4 (B e-3 C d)}{e^4 \sqrt {a-c x^4}}+\frac {c^2 C x^6}{e^3 \sqrt {a-c x^4}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c C \sqrt {a-c x^4} x^3}{5 e^3}+\frac {c (3 C d-B e) \sqrt {a-c x^4} x}{3 e^4}-\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^4 \left (e x^2+d\right )}+\frac {\left (6 c C d^3-c e (5 B d-4 A e) d-a e^2 (2 C d-B e)\right ) \sqrt {a-c x^4} x}{2 d e^4 \left (e x^2+d\right )}-\frac {\left (c d^2-a e^2\right ) \left (C d^2-B e d+A e^2\right ) \sqrt {a-c x^4} x}{4 d e^4 \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^5 \sqrt {a-c x^4}}+\frac {a^{3/4} \sqrt [4]{c} \left (6 c C d^2-2 a C e^2-c 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 )}{e^5 \sqrt {a-c x^4}}+\frac {a^{3/4} \sqrt [4]{c} \left (6 c C d^3-c e (5 B d-4 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^5 \sqrt {a-c x^4}}+\frac {3 a^{7/4} \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 )}{5 e^3 \sqrt {a-c x^4}}-\frac {a^{5/4} 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 )}{3 e^4 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right ) \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^6 \sqrt {a-c x^4}}-\frac {a^{3/4} \sqrt [4]{c} \left (6 c C d^2-2 a C e^2-c e (3 B d-A 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 )}{e^5 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (6 c C d^3-c e (5 B d-4 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^6 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{3/4} \left (2 a e^2 (3 C d-B e)-c d \left (10 C d^2-6 B e d+3 A e^2\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 )}{e^6 \sqrt {a-c x^4}}-\frac {3 a^{7/4} \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 )}{5 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^6 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (3 c d^2-a e^2\right ) \left (6 c C d^3-c e (5 B d-4 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^6 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (a^2 C e^4-2 a c \left (6 C d^2-e (3 B d-A e)\right ) e^2+c^2 \left (15 C d^4-2 d^2 e (5 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 )}{\sqrt [4]{c} d e^6 \sqrt {a-c x^4}}\)

Input:

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

Output:

(c*(3*C*d - B*e)*x*Sqrt[a - c*x^4])/(3*e^4) - (c*C*x^3*Sqrt[a - c*x^4])/(5 
*e^3) - ((c*d^2 - a*e^2)*(C*d^2 - B*d*e + A*e^2)*x*Sqrt[a - c*x^4])/(4*d*e 
^4*(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^4*(d + e*x^2)) + ((6*c*C*d^3 - c*d*e*(5*B*d - 4*A*e) - 
a*e^2*(2*C*d - B*e))*x*Sqrt[a - c*x^4])/(2*d*e^4*(d + e*x^2)) + (3*a^(7/4) 
*c^(1/4)*C*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1]) 
/(5*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^5*Sqrt[a - c*x^4]) + (a^(3/4)*c^(1/4)*(6*c*C*d^2 - 2*a*C*e^2 
- c*e*(3*B*d - A*e))*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1 
/4)], -1])/(e^5*Sqrt[a - c*x^4]) + (a^(3/4)*c^(1/4)*(6*c*C*d^3 - c*d*e*(5* 
B*d - 4*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^5*Sqrt[a - c*x^4]) - (3*a^(7/4)*c^(1/4)*C 
*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*e^3*Sq 
rt[a - c*x^4]) - (a^(5/4)*c^(3/4)*(3*C*d - B*e)*Sqrt[1 - (c*x^4)/a]*Ellipt 
icF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(3*e^4*Sqrt[a - c*x^4]) - (a^(1/4)*c 
^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(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^6*Sqrt[a - c*x^4]) - (a^(3/4)*c^(1/4)*(6*c*C*d^2 - 
 2*a*C*e^2 - c*e*(3*B*d - A*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c...
 

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. 2099 vs. \(2 (578 ) = 1156\).

Time = 6.95 (sec) , antiderivative size = 2100, normalized size of antiderivative = 3.24

method	result	size
risch	\(\text {Expression too large to display}\)	\(2100\)
default	\(\text {Expression too large to display}\)	\(2446\)
elliptic	\(\text {Expression too large to display}\)	\(2466\)

Input:

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

Output:

-1/15*c*x*(3*C*e*x^2+5*B*e-15*C*d)*(-c*x^4+a)^(1/2)/e^4+1/15/e^4*(-15/e^2* 
(2*A*a*c*e^4-6*A*c^2*d^2*e^2-6*B*a*c*d*e^3+10*B*c^2*d^3*e-C*a^2*e^4+12*C*a 
*c*d^2*e^2-15*C*c^2*d^4)/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))+15/e^2*(4*A*a*c*d*e^4-4*A*c^2*d^3*e^2+B*a^2*e^5-6*B*a*c*d^2* 
e^3+5*B*c^2*d^4*e-2*C*a^2*d*e^4+8*C*a*c*d^3*e^2-6*C*c^2*d^5)*(1/2*e^2/(a*e 
^2-c*d^2)/d*x*(-c*x^4+a)^(1/2)/(e*x^2+d)+1/2*c/(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*e*c^(1/2)/(a*e^2- 
c*d^2)/d*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*e*c^(1/2)/(a*e^2-c*d^2)/d*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)+1/2/(a*e^2-c*d^2)/d^2*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)*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-3/2/(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)*EllipticPi(x*(c^(1/2)/a^(1/2))^(1/2),-a^(1/2)*...
 

Fricas [F(-1)]

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

integrate((-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A)/(e*x^2+d)^3,x, algorithm="frica 
s")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral((a - c*x**4)**(3/2)*(A + B*x**2 + C*x**4)/(d + e*x**2)**3, x)
 

Maxima [F]

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

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^3} \, dx=\text {too large to display} \] Input:

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

Output:

(3*sqrt(a - c*x**4)*a**2*e**3*x - 20*sqrt(a - c*x**4)*a*b*d*e**2*x + 36*sq 
rt(a - c*x**4)*a*c*d**2*e*x + 6*sqrt(a - c*x**4)*a*c*d*e**2*x**3 + 35*sqrt 
(a - c*x**4)*b*c*d**2*e*x**3 - 5*sqrt(a - c*x**4)*b*c*d*e**2*x**5 - 63*sqr 
t(a - c*x**4)*c**2*d**3*x**3 + 9*sqrt(a - c*x**4)*c**2*d**2*e*x**5 - 3*sqr 
t(a - c*x**4)*c**2*d*e**2*x**7 + 12*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**3*d**3*e**3 + 24*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**3*d**2*e**4*x** 
2 + 12*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**3*d*e**5*x**4 + 20*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*b*d**4*e**2 + 40*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*b*d**3*e**3*x**2 + 
 20*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*b*d**2*e**4*x**4 - 36*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...