\(\int \frac {(a-c x^4)^{3/2} (A+B x^2+C x^4)}{(d+e x^2)^2} \, dx\) [38]

Optimal result

Integrand size = 34, antiderivative size = 614 \[ \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^2} \, dx=\frac {\left (9 a C e^2-7 c \left (3 C d^2-e (2 B d-A e)\right )\right ) x \sqrt {a-c x^4}}{21 e^4}+\frac {c (2 C d-B e) x^3 \sqrt {a-c x^4}}{5 e^3}-\frac {c C x^5 \sqrt {a-c x^4}}{7 e^2}-\frac {\left (c d^2-a e^2\right ) \left (C d^2-B d e+A e^2\right ) x \sqrt {a-c x^4}}{2 d e^4 \left (d+e x^2\right )}-\frac {a^{3/4} \sqrt [4]{c} \left (5 c d^2 \left (9 C d^2-e (7 B d-5 A e)\right )-a e^2 \left (33 C d^2-e (19 B d-5 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{10 d e^5 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (120 a^2 C d e^4-35 a c d e^2 \left (27 C d^2-e (17 B d-7 A e)\right )+105 c^2 d^3 \left (9 C d^2-e (7 B d-5 A e)\right )+105 \sqrt {a} c^{3/2} d^2 e \left (9 C d^2-e (7 B d-5 A e)\right )-21 a^{3/2} \sqrt {c} e^3 \left (33 C d^2-e (19 B d-5 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{210 \sqrt [4]{c} d e^6 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (c d^2-a e^2\right ) \left (c d^2 \left (9 C d^2-e (7 B d-5 A e)\right )-a e^2 \left (3 C d^2-e (B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{c} d^2 e^6 \sqrt {a-c x^4}} \] Output:

1/21*(9*C*a*e^2-7*c*(3*C*d^2-e*(-A*e+2*B*d)))*x*(-c*x^4+a)^(1/2)/e^4+1/5*c 
*(-B*e+2*C*d)*x^3*(-c*x^4+a)^(1/2)/e^3-1/7*c*C*x^5*(-c*x^4+a)^(1/2)/e^2-1/ 
2*(-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)-1/ 
10*a^(3/4)*c^(1/4)*(5*c*d^2*(9*C*d^2-e*(-5*A*e+7*B*d))-a*e^2*(33*C*d^2-e*( 
-5*A*e+19*B*d)))*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/d/e^5/(- 
c*x^4+a)^(1/2)+1/210*a^(1/4)*(120*a^2*C*d*e^4-35*a*c*d*e^2*(27*C*d^2-e*(-7 
*A*e+17*B*d))+105*c^2*d^3*(9*C*d^2-e*(-5*A*e+7*B*d))+105*a^(1/2)*c^(3/2)*d 
^2*e*(9*C*d^2-e*(-5*A*e+7*B*d))-21*a^(3/2)*c^(1/2)*e^3*(33*C*d^2-e*(-5*A*e 
+19*B*d)))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(1/4)/d/e^6/ 
(-c*x^4+a)^(1/2)-1/2*a^(1/4)*(-a*e^2+c*d^2)*(c*d^2*(9*C*d^2-e*(-5*A*e+7*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^6/(-c*x^4+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 14.07 (sec) , antiderivative size = 2295, normalized size of antiderivative = 3.74 \[ \int \frac {\left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right )}{\left (d+e x^2\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(-315*a*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*C*d^5*e^2*x + 245*a*B*Sqrt[-(Sqrt[c]/Sq 
rt[a])]*c*d^4*e^3*x - 175*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^3*e^4*x + 195*a 
^2*Sqrt[-(Sqrt[c]/Sqrt[a])]*C*d^3*e^4*x - 105*a^2*B*Sqrt[-(Sqrt[c]/Sqrt[a] 
)]*d^2*e^5*x + 105*a^2*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*d*e^6*x - 126*a*Sqrt[-(S 
qrt[c]/Sqrt[a])]*c*C*d^4*e^3*x^3 + 98*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^3*e 
^4*x^3 - 70*a*A*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^2*e^5*x^3 + 90*a^2*Sqrt[-(Sqr 
t[c]/Sqrt[a])]*C*d^2*e^5*x^3 + 315*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*C*d^5*e^2* 
x^5 - 245*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d^4*e^3*x^5 + 175*A*Sqrt[-(Sqrt[c 
]/Sqrt[a])]*c^2*d^3*e^4*x^5 - 141*a*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*C*d^3*e^4*x 
^5 + 63*a*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d^2*e^5*x^5 - 105*a*A*Sqrt[-(Sqrt[c 
]/Sqrt[a])]*c*d*e^6*x^5 + 126*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*C*d^4*e^3*x^7 - 
 98*B*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*d^3*e^4*x^7 + 70*A*Sqrt[-(Sqrt[c]/Sqrt[ 
a])]*c^2*d^2*e^5*x^7 - 120*a*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*C*d^2*e^5*x^7 - 54 
*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*C*d^3*e^4*x^9 + 42*B*Sqrt[-(Sqrt[c]/Sqrt[a]) 
]*c^2*d^2*e^5*x^9 + 30*Sqrt[-(Sqrt[c]/Sqrt[a])]*c^2*C*d^2*e^5*x^11 - (21*I 
)*Sqrt[a]*Sqrt[c]*d*e*(a*e^2*(33*C*d^2 - 19*B*d*e + 5*A*e^2) - 5*c*d^2*(9* 
C*d^2 - 7*B*d*e + 5*A*e^2))*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[I*Ar 
cSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*d*(120*a^2*C*d*e^4 - 21*a^(3/2) 
*Sqrt[c]*e^3*(33*C*d^2 - 19*B*d*e + 5*A*e^2) + 105*c^2*d^3*(9*C*d^2 - 7*B* 
d*e + 5*A*e^2) + 105*Sqrt[a]*c^(3/2)*d^2*e*(9*C*d^2 - 7*B*d*e + 5*A*e^2...
 

Rubi [A] (verified)

Time = 1.84 (sec) , antiderivative size = 1217, normalized size of antiderivative = 1.98, 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.

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

\(\Big \downarrow \) 2259

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

\(\Big \downarrow \) 2009

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

Output:

(-5*a*C*x*Sqrt[a - c*x^4])/(21*e^2) - ((3*c*C*d^2 - 2*a*C*e^2 - c*e*(2*B*d 
 - A*e))*x*Sqrt[a - c*x^4])/(3*e^4) + (c*(2*C*d - B*e)*x^3*Sqrt[a - c*x^4] 
)/(5*e^3) - (c*C*x^5*Sqrt[a - c*x^4])/(7*e^2) - ((c*d^2 - a*e^2)*(C*d^2 - 
B*d*e + A*e^2)*x*Sqrt[a - c*x^4])/(2*d*e^4*(d + e*x^2)) - (3*a^(7/4)*c^(1/ 
4)*(2*C*d - B*e)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)] 
, -1])/(5*e^3*Sqrt[a - c*x^4]) - (a^(3/4)*c^(1/4)*(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])/(2*d*e^5*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) - 2*a*e^2*(2*C*d - B*e))*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[ 
(c^(1/4)*x)/a^(1/4)], -1])/(e^5*Sqrt[a - c*x^4]) + (5*a^(9/4)*C*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^(1/4)*(2*C*d - B*e)*Sqrt[1 - (c*x^4)/a]*Ellipt 
icF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*e^3*Sqrt[a - c*x^4]) - (a^(1/4)*c 
^(1/4)*(c*d^2 - a*e^2)^2*(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*Ellip 
ticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*d*e^6*(Sqrt[c]*d + Sqrt[a]*e)*Sq 
rt[a - c*x^4]) + (a^(5/4)*(3*c*C*d^2 - 2*a*C*e^2 - c*e*(2*B*d - A*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)*c^(1/4)*(4*c*C*d^3 - c*d*e*(3*B*d - 2*A*e) - 
2*a*e^2*(2*C*d - B*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^ 
(1/4)], -1])/(e^5*Sqrt[a - c*x^4]) + (a^(1/4)*(a^2*C*e^4 + c^2*(5*C*d^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. 1145 vs. \(2 (544 ) = 1088\).

Time = 4.40 (sec) , antiderivative size = 1146, normalized size of antiderivative = 1.87

method	result	size
risch	\(\text {Expression too large to display}\)	\(1146\)
default	\(\text {Expression too large to display}\)	\(1718\)
elliptic	\(\text {Expression too large to display}\)	\(2521\)

Input:

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

Output:

-1/105*x*(15*C*c*e^2*x^4+21*B*c*e^2*x^2-42*C*c*d*e*x^2+35*A*c*e^2-70*B*c*d 
*e-45*C*a*e^2+105*C*c*d^2)*(-c*x^4+a)^(1/2)/e^4-1/105/e^4*(-105/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)/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))-105*(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*(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)*(...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(115*sqrt(a - c*x**4)*a**2*e**3*x - 56*sqrt(a - c*x**4)*a*b*d*e**2*x + 72* 
sqrt(a - c*x**4)*a*c*d**2*e*x + 30*sqrt(a - c*x**4)*a*c*d*e**2*x**3 + 147* 
sqrt(a - c*x**4)*b*c*d**2*e*x**3 - 63*sqrt(a - c*x**4)*b*c*d*e**2*x**5 - 1 
89*sqrt(a - c*x**4)*c**2*d**3*x**3 + 81*sqrt(a - c*x**4)*c**2*d**2*e*x**5 
- 45*sqrt(a - c*x**4)*c**2*d*e**2*x**7 + 200*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**3*d**2*e**3 + 200*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**3*d*e**4*x**2 + 5 
6*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*b*d**3*e**2 + 56*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*b*d**2*e**3*x**2 - 72*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 
*c*d**4*e - 72*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*c*d**3*e**2*x**2 + 115* 
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**2*c*d*e**4 + 115*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**2*c*e**5*x**2 - 497*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*...