\(\int \frac {(a-c x^4)^{3/2} (A+B x^2+C x^4)}{d+e x^2} \, dx\) [37]

Optimal result

Integrand size = 34, antiderivative size = 586 \[ \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx=-\frac {\left (9 a e^2 (C d-B e)-7 c d \left (C d^2-e (B d-A e)\right )\right ) x \sqrt {a-c x^4}}{21 e^4}+\frac {\left (11 a C e^2-9 c \left (C d^2-e (B d-A e)\right )\right ) x^3 \sqrt {a-c x^4}}{45 e^3}+\frac {c (C d-B e) x^5 \sqrt {a-c x^4}}{7 e^2}-\frac {c C x^7 \sqrt {a-c x^4}}{9 e}+\frac {a^{3/4} \left (4 a^2 C e^4+15 c^2 d^2 \left (C d^2-e (B d-A e)\right )-21 a c e^2 \left (C d^2-e (B d-A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 c^{3/4} e^5 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (7 \sqrt {a} e \left (4 a^2 C e^4+15 c^2 d^2 \left (C d^2-e (B d-A e)\right )-21 a c e^2 \left (C d^2-e (B d-A e)\right )\right )+5 \sqrt {c} \left (12 a^2 e^4 (C d-B e)+21 c^2 d^3 \left (C d^2-e (B d-A e)\right )-35 a c d e^2 \left (C d^2-e (B d-A e)\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{105 c^{3/4} e^6 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (c d^2-a e^2\right )^2 \left (C d^2-B d e+A e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d e^6 \sqrt {a-c x^4}} \] Output:

-1/21*(9*a*e^2*(-B*e+C*d)-7*c*d*(C*d^2-e*(-A*e+B*d)))*x*(-c*x^4+a)^(1/2)/e 
^4+1/45*(11*C*a*e^2-9*c*(C*d^2-e*(-A*e+B*d)))*x^3*(-c*x^4+a)^(1/2)/e^3+1/7 
*c*(-B*e+C*d)*x^5*(-c*x^4+a)^(1/2)/e^2-1/9*c*C*x^7*(-c*x^4+a)^(1/2)/e+1/15 
*a^(3/4)*(4*a^2*C*e^4+15*c^2*d^2*(C*d^2-e*(-A*e+B*d))-21*a*c*e^2*(C*d^2-e* 
(-A*e+B*d)))*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^(3/4)/e^5/ 
(-c*x^4+a)^(1/2)-1/105*a^(1/4)*(7*a^(1/2)*e*(4*a^2*C*e^4+15*c^2*d^2*(C*d^2 
-e*(-A*e+B*d))-21*a*c*e^2*(C*d^2-e*(-A*e+B*d)))+5*c^(1/2)*(12*a^2*e^4*(-B* 
e+C*d)+21*c^2*d^3*(C*d^2-e*(-A*e+B*d))-35*a*c*d*e^2*(C*d^2-e*(-A*e+B*d)))) 
*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(3/4)/e^6/(-c*x^4+a)^( 
1/2)+a^(1/4)*(-a*e^2+c*d^2)^2*(A*e^2-B*d*e+C*d^2)*(1-c*x^4/a)^(1/2)*Ellipt 
icPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/d,I)/c^(1/4)/d/e^6/(-c*x^4+a)^(1 
/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 14.29 (sec) , antiderivative size = 568, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right )}{d+e x^2} \, dx=\frac {-21 i \sqrt {a} d e \left (4 a^2 C e^4-21 a c e^2 \left (C d^2+e (-B d+A e)\right )+15 c^2 \left (C d^4+d^2 e (-B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+3 i d \left (28 a^{5/2} C e^5+60 a^2 \sqrt {c} e^4 (C d-B e)+105 \sqrt {a} c^2 d^2 e \left (C d^2+e (-B d+A e)\right )-175 a c^{3/2} d e^2 \left (C d^2+e (-B d+A e)\right )-147 a^{3/2} c e^3 \left (C d^2+e (-B d+A e)\right )+105 c^{5/2} \left (C d^5+d^3 e (-B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-\sqrt {c} \left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d e^2 x \left (a-c x^4\right ) \left (a e^2 \left (135 C d-135 B e-77 C e x^2\right )+c C \left (-105 d^3+63 d^2 e x^2-45 d e^2 x^4+35 e^3 x^6\right )+3 c e \left (7 A e \left (-5 d+3 e x^2\right )+B \left (35 d^2-21 d e x^2+15 e^2 x^4\right )\right )\right )+315 i \left (c d^2-a e^2\right )^2 \left (C d^2+e (-B d+A e)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{315 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \sqrt {c} d e^6 \sqrt {a-c x^4}} \] Input:

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

Output:

((-21*I)*Sqrt[a]*d*e*(4*a^2*C*e^4 - 21*a*c*e^2*(C*d^2 + e*(-(B*d) + A*e)) 
+ 15*c^2*(C*d^4 + d^2*e*(-(B*d) + A*e)))*Sqrt[1 - (c*x^4)/a]*EllipticE[I*A 
rcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (3*I)*d*(28*a^(5/2)*C*e^5 + 60*a 
^2*Sqrt[c]*e^4*(C*d - B*e) + 105*Sqrt[a]*c^2*d^2*e*(C*d^2 + e*(-(B*d) + A* 
e)) - 175*a*c^(3/2)*d*e^2*(C*d^2 + e*(-(B*d) + A*e)) - 147*a^(3/2)*c*e^3*( 
C*d^2 + e*(-(B*d) + A*e)) + 105*c^(5/2)*(C*d^5 + d^3*e*(-(B*d) + A*e)))*Sq 
rt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - S 
qrt[c]*(Sqrt[-(Sqrt[c]/Sqrt[a])]*d*e^2*x*(a - c*x^4)*(a*e^2*(135*C*d - 135 
*B*e - 77*C*e*x^2) + c*C*(-105*d^3 + 63*d^2*e*x^2 - 45*d*e^2*x^4 + 35*e^3* 
x^6) + 3*c*e*(7*A*e*(-5*d + 3*e*x^2) + B*(35*d^2 - 21*d*e*x^2 + 15*e^2*x^4 
))) + (315*I)*(c*d^2 - a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e))*Sqrt[1 - (c*x^4 
)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[ 
a])]*x], -1]))/(315*Sqrt[-(Sqrt[c]/Sqrt[a])]*Sqrt[c]*d*e^6*Sqrt[a - c*x^4] 
)
 

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 1115, normalized size of antiderivative = 1.90, 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.

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

\(\Big \downarrow \) 2259

\(\displaystyle \int \left (\frac {x^2 \left (a^2 C e^4-2 a c e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^4-d^2 e (B d-A e)\right )\right )}{e^5 \sqrt {a-c x^4}}-\frac {a^2 e^4 (C d-B e)-2 a c d e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^5-d^3 e (B d-A e)\right )}{e^6 \sqrt {a-c x^4}}+\frac {a^2 A e^6-a^2 B d e^5+a^2 C d^2 e^4-2 a A c d^2 e^4+2 a B c d^3 e^3-2 a c C d^4 e^2+A c^2 d^4 e^2-B c^2 d^5 e+c^2 C d^6}{e^6 \sqrt {a-c x^4} \left (d+e x^2\right )}-\frac {c x^4 \left (-2 a e^2 (C d-B e)-c d e (B d-A e)+c C d^3\right )}{e^4 \sqrt {a-c x^4}}+\frac {c x^6 \left (-2 a C e^2-c e (B d-A e)+c C d^2\right )}{e^3 \sqrt {a-c x^4}}-\frac {c^2 x^8 (C d-B e)}{e^2 \sqrt {a-c x^4}}+\frac {c^2 C x^{10}}{e \sqrt {a-c x^4}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c C \sqrt {a-c x^4} x^7}{9 e}+\frac {c (C d-B e) \sqrt {a-c x^4} x^5}{7 e^2}-\frac {\left (c C d^2-2 a C e^2-c e (B d-A e)\right ) \sqrt {a-c x^4} x^3}{5 e^3}-\frac {7 a C \sqrt {a-c x^4} x^3}{45 e}+\frac {5 a (C d-B e) \sqrt {a-c x^4} x}{21 e^2}+\frac {\left (c C d^3-c e (B d-A e) d-2 a e^2 (C d-B e)\right ) \sqrt {a-c x^4} x}{3 e^4}+\frac {3 a^{7/4} \left (c C d^2-2 a C e^2-c e (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 )}{5 c^{3/4} e^3 \sqrt {a-c x^4}}+\frac {a^{3/4} \left (a^2 C e^4-2 a c \left (C d^2-e (B d-A e)\right ) e^2+c^2 \left (C d^4-d^2 e (B d-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 )}{c^{3/4} e^5 \sqrt {a-c x^4}}+\frac {7 a^{11/4} 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 )}{15 c^{3/4} e \sqrt {a-c x^4}}-\frac {5 a^{9/4} (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 )}{21 \sqrt [4]{c} e^2 \sqrt {a-c x^4}}-\frac {3 a^{7/4} \left (c C d^2-2 a C e^2-c e (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 )}{5 c^{3/4} e^3 \sqrt {a-c x^4}}-\frac {a^{5/4} \left (c C d^3-c e (B d-A e) d-2 a e^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 )}{3 \sqrt [4]{c} e^4 \sqrt {a-c x^4}}-\frac {a^{3/4} \left (a^2 C e^4-2 a c \left (C d^2-e (B d-A e)\right ) e^2+c^2 \left (C d^4-d^2 e (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 )}{c^{3/4} e^5 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (a^2 (C d-B e) e^4-2 a c d \left (C d^2-e (B d-A e)\right ) e^2+c^2 \left (C d^5-d^3 e (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 )}{\sqrt [4]{c} e^6 \sqrt {a-c x^4}}-\frac {7 a^{11/4} C \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15 c^{3/4} e \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (c d^2-a e^2\right )^2 \left (C d^2-B e d+A e^2\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),x]
 

Output:

(5*a*(C*d - B*e)*x*Sqrt[a - c*x^4])/(21*e^2) + ((c*C*d^3 - c*d*e*(B*d - A* 
e) - 2*a*e^2*(C*d - B*e))*x*Sqrt[a - c*x^4])/(3*e^4) - (7*a*C*x^3*Sqrt[a - 
 c*x^4])/(45*e) - ((c*C*d^2 - 2*a*C*e^2 - c*e*(B*d - A*e))*x^3*Sqrt[a - c* 
x^4])/(5*e^3) + (c*(C*d - B*e)*x^5*Sqrt[a - c*x^4])/(7*e^2) - (c*C*x^7*Sqr 
t[a - c*x^4])/(9*e) + (7*a^(11/4)*C*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[( 
c^(1/4)*x)/a^(1/4)], -1])/(15*c^(3/4)*e*Sqrt[a - c*x^4]) + (3*a^(7/4)*(c*C 
*d^2 - 2*a*C*e^2 - c*e*(B*d - A*e))*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[( 
c^(1/4)*x)/a^(1/4)], -1])/(5*c^(3/4)*e^3*Sqrt[a - c*x^4]) + (a^(3/4)*(a^2* 
C*e^4 - 2*a*c*e^2*(C*d^2 - e*(B*d - A*e)) + c^2*(C*d^4 - d^2*e*(B*d - A*e) 
))*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4 
)*e^5*Sqrt[a - c*x^4]) - (7*a^(11/4)*C*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSi 
n[(c^(1/4)*x)/a^(1/4)], -1])/(15*c^(3/4)*e*Sqrt[a - c*x^4]) - (5*a^(9/4)*( 
C*d - B*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1]) 
/(21*c^(1/4)*e^2*Sqrt[a - c*x^4]) - (3*a^(7/4)*(c*C*d^2 - 2*a*C*e^2 - c*e* 
(B*d - A*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1 
])/(5*c^(3/4)*e^3*Sqrt[a - c*x^4]) - (a^(5/4)*(c*C*d^3 - c*d*e*(B*d - A*e) 
 - 2*a*e^2*(C*d - B*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a 
^(1/4)], -1])/(3*c^(1/4)*e^4*Sqrt[a - c*x^4]) - (a^(3/4)*(a^2*C*e^4 - 2*a* 
c*e^2*(C*d^2 - e*(B*d - A*e)) + c^2*(C*d^4 - d^2*e*(B*d - A*e)))*Sqrt[1 - 
(c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*e^5*Sqr...
 

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 [A] (verified)

Time = 3.36 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.09

method	result	size
risch	\(\frac {x \left (-35 C \,x^{6} c \,e^{3}-45 B c \,e^{3} x^{4}+45 C c d \,e^{2} x^{4}-63 A c \,e^{3} x^{2}+63 B c d \,e^{2} x^{2}+77 C a \,e^{3} x^{2}-63 C c \,d^{2} e \,x^{2}+105 A c d \,e^{2}+135 B a \,e^{3}-105 B c \,d^{2} e -135 a C d \,e^{2}+105 C c \,d^{3}\right ) \sqrt {-c \,x^{4}+a}}{315 e^{4}}-\frac {-\frac {105 \left (A \,a^{2} e^{6}-2 A a c \,d^{2} e^{4}+A \,c^{2} d^{4} e^{2}-B \,a^{2} d \,e^{5}+2 B a c \,d^{3} e^{3}-B \,c^{2} d^{5} e +C \,a^{2} d^{2} e^{4}-2 C a c \,d^{4} e^{2}+C \,c^{2} d^{6}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{e^{2} d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {5 \left (35 A a c d \,e^{4}-21 A \,c^{2} d^{3} e^{2}+12 B \,a^{2} e^{5}-35 B a c \,d^{2} e^{3}+21 B \,c^{2} d^{4} e -12 a^{2} C d \,e^{4}+35 C a c \,d^{3} e^{2}-21 C \,c^{2} d^{5}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {7 \left (21 A a c \,e^{4}-15 A \,c^{2} d^{2} e^{2}-21 a B c d \,e^{3}+15 B \,d^{3} c^{2} e -4 a^{2} C \,e^{4}+21 C a c \,d^{2} e^{2}-15 C \,c^{2} d^{4}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}}{105 e^{4}}\)	\(636\)
default	\(\text {Expression too large to display}\)	\(1132\)
elliptic	\(\text {Expression too large to display}\)	\(2741\)

Input:

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

Output:

1/315*x*(-35*C*c*e^3*x^6-45*B*c*e^3*x^4+45*C*c*d*e^2*x^4-63*A*c*e^3*x^2+63 
*B*c*d*e^2*x^2+77*C*a*e^3*x^2-63*C*c*d^2*e*x^2+105*A*c*d*e^2+135*B*a*e^3-1 
05*B*c*d^2*e-135*C*a*d*e^2+105*C*c*d^3)*(-c*x^4+a)^(1/2)/e^4-1/105/e^4*(-1 
05*(A*a^2*e^6-2*A*a*c*d^2*e^4+A*c^2*d^4*e^2-B*a^2*d*e^5+2*B*a*c*d^3*e^3-B* 
c^2*d^5*e+C*a^2*d^2*e^4-2*C*a*c*d^4*e^2+C*c^2*d^6)/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))-5*(35*A*a*c*d*e^4-21*A*c^2*d^3 
*e^2+12*B*a^2*e^5-35*B*a*c*d^2*e^3+21*B*c^2*d^4*e-12*C*a^2*d*e^4+35*C*a*c* 
d^3*e^2-21*C*c^2*d^5)/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)-7/e*(21*A*a*c*e^4-15*A*c^2*d^2*e^2-21*B*a*c*d*e^3+15*B*c 
^2*d^3*e-4*C*a^2*e^4+21*C*a*c*d^2*e^2-15*C*c^2*d^4)*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)/c^(1/2)*(EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*( 
c^(1/2)/a^(1/2))^(1/2),I)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(135*sqrt(a - c*x**4)*a*b*e**3*x - 30*sqrt(a - c*x**4)*a*c*d*e**2*x + 14*s 
qrt(a - c*x**4)*a*c*e**3*x**3 - 105*sqrt(a - c*x**4)*b*c*d**2*e*x + 63*sqr 
t(a - c*x**4)*b*c*d*e**2*x**3 - 45*sqrt(a - c*x**4)*b*c*e**3*x**5 + 105*sq 
rt(a - c*x**4)*c**2*d**3*x - 63*sqrt(a - c*x**4)*c**2*d**2*e*x**3 + 45*sqr 
t(a - c*x**4)*c**2*d*e**2*x**5 - 35*sqrt(a - c*x**4)*c**2*e**3*x**7 + 315* 
int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**3*e**4 - 
 135*int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*b 
*d*e**3 + 30*int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x 
)*a**2*c*d**2*e**2 + 105*int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c*d*x**4 - 
 c*e*x**6),x)*a*b*c*d**3*e - 105*int(sqrt(a - c*x**4)/(a*d + a*e*x**2 - c* 
d*x**4 - c*e*x**6),x)*a*c**2*d**4 - 357*int((sqrt(a - c*x**4)*x**4)/(a*d + 
 a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*c*e**4 + 441*int((sqrt(a - c*x**4 
)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c*d*e**3 - 126*int(( 
sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c**2*d* 
*2*e**2 - 315*int((sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e 
*x**6),x)*b*c**2*d**3*e + 315*int((sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 
- c*d*x**4 - c*e*x**6),x)*c**3*d**4 + 180*int((sqrt(a - c*x**4)*x**2)/(a*d 
 + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*b*e**4 - 12*int((sqrt(a - c*x** 
4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*c*d*e**3 - 84*int( 
(sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c...