\(\int \frac {1}{(d+e x^2)^3 (a-c x^4)^{3/2}} \, dx\) [419]

Optimal result

Integrand size = 22, antiderivative size = 562 \[ \int \frac {1}{\left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2}} \, dx=\frac {c x \left (d-e x^2\right )}{2 a \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2 \sqrt {a-c x^4}}+\frac {e^2 \left (4 c d^2+a e^2\right ) x \sqrt {a-c x^4}}{4 a d \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}+\frac {3 e^2 \left (4 c^2 d^4+7 a c d^2 e^2-a^2 e^4\right ) x \sqrt {a-c x^4}}{8 a d^2 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )}+\frac {3 \sqrt [4]{c} e \left (4 c^2 d^4+7 a c d^2 e^2-a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{a} d^2 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{c} \left (4 c^2 d^4-8 \sqrt {a} c^{3/2} d^3 e+19 a c d^2 e^2-2 a^{3/2} \sqrt {c} d e^3-3 a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 a^{3/4} d^2 \left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (\sqrt {c} d+\sqrt {a} e\right )^3 \sqrt {a-c x^4}}-\frac {3 \sqrt [4]{a} e^2 \left (21 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}} \] Output:

1/2*c*x*(-e*x^2+d)/a/(-a*e^2+c*d^2)/(e*x^2+d)^2/(-c*x^4+a)^(1/2)+1/4*e^2*( 
a*e^2+4*c*d^2)*x*(-c*x^4+a)^(1/2)/a/d/(-a*e^2+c*d^2)^2/(e*x^2+d)^2+3/8*e^2 
*(-a^2*e^4+7*a*c*d^2*e^2+4*c^2*d^4)*x*(-c*x^4+a)^(1/2)/a/d^2/(-a*e^2+c*d^2 
)^3/(e*x^2+d)+3/8*c^(1/4)*e*(-a^2*e^4+7*a*c*d^2*e^2+4*c^2*d^4)*(1-c*x^4/a) 
^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(1/4)/d^2/(-a*e^2+c*d^2)^3/(-c*x^4 
+a)^(1/2)+1/8*c^(1/4)*(4*c^2*d^4-8*a^(1/2)*c^(3/2)*d^3*e+19*a*c*d^2*e^2-2* 
a^(3/2)*c^(1/2)*d*e^3-3*a^2*e^4)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^( 
1/4),I)/a^(3/4)/d^2/(c^(1/2)*d-a^(1/2)*e)^2/(c^(1/2)*d+a^(1/2)*e)^3/(-c*x^ 
4+a)^(1/2)-3/8*a^(1/4)*e^2*(a^2*e^4-2*a*c*d^2*e^2+21*c^2*d^4)*(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/(-a 
*e^2+c*d^2)^3/(-c*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.10 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d x \left (2 a d e^4 \left (c d^2-a e^2\right ) \left (a-c x^4\right )+a e^4 \left (17 c d^2-3 a e^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )+4 c^2 d^2 \left (d+e x^2\right )^2 \left (c d^2 \left (d-3 e x^2\right )+a e^2 \left (3 d-e x^2\right )\right )\right )+i \left (d+e x^2\right )^2 \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d e \left (-4 c^2 d^4-7 a c d^2 e^2+a^2 e^4\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\left (-4 c^3 d^6+12 \sqrt {a} c^{5/2} d^5 e-27 a c^2 d^4 e^2+21 a^{3/2} c^{3/2} d^3 e^3+a^2 c d^2 e^4-3 a^{5/2} \sqrt {c} d e^5\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 a e^2 \left (21 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{8 a \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \left (c d^3-a d e^2\right )^3 \left (d+e x^2\right )^2 \sqrt {a-c x^4}} \] Input:

Integrate[1/((d + e*x^2)^3*(a - c*x^4)^(3/2)),x]
 

Output:

(Sqrt[-(Sqrt[c]/Sqrt[a])]*d*x*(2*a*d*e^4*(c*d^2 - a*e^2)*(a - c*x^4) + a*e 
^4*(17*c*d^2 - 3*a*e^2)*(d + e*x^2)*(a - c*x^4) + 4*c^2*d^2*(d + e*x^2)^2* 
(c*d^2*(d - 3*e*x^2) + a*e^2*(3*d - e*x^2))) + I*(d + e*x^2)^2*Sqrt[1 - (c 
*x^4)/a]*(3*Sqrt[a]*Sqrt[c]*d*e*(-4*c^2*d^4 - 7*a*c*d^2*e^2 + a^2*e^4)*Ell 
ipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (-4*c^3*d^6 + 12*Sqrt[ 
a]*c^(5/2)*d^5*e - 27*a*c^2*d^4*e^2 + 21*a^(3/2)*c^(3/2)*d^3*e^3 + a^2*c*d 
^2*e^4 - 3*a^(5/2)*Sqrt[c]*d*e^5)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[ 
a])]*x], -1] + 3*a*e^2*(21*c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*EllipticPi[- 
((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]))/(8 
*a*Sqrt[-(Sqrt[c]/Sqrt[a])]*(c*d^3 - a*d*e^2)^3*(d + e*x^2)^2*Sqrt[a - c*x 
^4])
 

Rubi [A] (verified)

Time = 3.29 (sec) , antiderivative size = 1066, normalized size of antiderivative = 1.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1557, 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.

\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^{3/2} \left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1557

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4} e^4}{8 d^2 \left (c d^2-a e^2\right )^3 \left (e x^2+d\right )}+\frac {c x \sqrt {a-c x^4} e^4}{\left (c d^2-a e^2\right )^3 \left (e x^2+d\right )}+\frac {x \sqrt {a-c x^4} e^4}{4 d \left (c d^2-a e^2\right )^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 ) \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}{8 d^2 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}+\frac {a^{3/4} c^{5/4} \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}{\left (c d^2-a e^2\right )^3 \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 ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) e^2}{8 d^2 \left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (\sqrt {c} d+\sqrt {a} e\right )^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{5/4} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) e^2}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} c^{3/4} \left (3 c d^2-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 ) e^2}{d \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} c^{3/4} \left (3 c d^2+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 ) e^2}{d \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}-\frac {3 \sqrt [4]{a} \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 ) e^2}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}+\frac {c^{5/4} \left (3 c d^2+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 ) e}{2 \sqrt [4]{a} \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}+\frac {c^{5/4} \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} \left (\sqrt {c} d+\sqrt {a} e\right )^3 \sqrt {a-c x^4}}+\frac {c^2 x \left (d \left (c d^2+3 a e^2\right )-e \left (3 c d^2+a e^2\right ) x^2\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}\)

Input:

Int[1/((d + e*x^2)^3*(a - c*x^4)^(3/2)),x]
 

Output:

(c^2*x*(d*(c*d^2 + 3*a*e^2) - e*(3*c*d^2 + a*e^2)*x^2))/(2*a*(c*d^2 - a*e^ 
2)^3*Sqrt[a - c*x^4]) + (e^4*x*Sqrt[a - c*x^4])/(4*d*(c*d^2 - a*e^2)^2*(d 
+ e*x^2)^2) + (c*e^4*x*Sqrt[a - c*x^4])/((c*d^2 - a*e^2)^3*(d + e*x^2)) + 
(3*e^4*(3*c*d^2 - a*e^2)*x*Sqrt[a - c*x^4])/(8*d^2*(c*d^2 - a*e^2)^3*(d + 
e*x^2)) + (a^(3/4)*c^(5/4)*e^3*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/ 
4)*x)/a^(1/4)], -1])/((c*d^2 - a*e^2)^3*Sqrt[a - c*x^4]) + (3*a^(3/4)*c^(1 
/4)*e^3*(3*c*d^2 - a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x) 
/a^(1/4)], -1])/(8*d^2*(c*d^2 - a*e^2)^3*Sqrt[a - c*x^4]) + (c^(5/4)*e*(3* 
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)^3*Sqrt[a - c*x^4]) + (c^(5/4)*Sqrt[1 - (c* 
x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(3/4)*(Sqrt[c]*d 
+ Sqrt[a]*e)^3*Sqrt[a - c*x^4]) + (a^(1/4)*c^(1/4)*e^2*(7*c*d^2 - 2*Sqrt[a 
]*Sqrt[c]*d*e - 3*a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/ 
a^(1/4)], -1])/(8*d^2*(Sqrt[c]*d - Sqrt[a]*e)^2*(Sqrt[c]*d + Sqrt[a]*e)^3* 
Sqrt[a - c*x^4]) + (a^(1/4)*c^(5/4)*e^2*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcS 
in[(c^(1/4)*x)/a^(1/4)], -1])/((Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 - a*e^2)^2*S 
qrt[a - c*x^4]) - (a^(1/4)*c^(3/4)*e^2*(3*c*d^2 - a*e^2)*Sqrt[1 - (c*x^4)/ 
a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1] 
)/(d*(c*d^2 - a*e^2)^3*Sqrt[a - c*x^4]) - (a^(1/4)*c^(3/4)*e^2*(3*c*d^2 + 
a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSi...
 

Defintions of rubi rules used

rule 1557

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Mod 
ule[{aa, cc}, Int[ExpandIntegrand[1/Sqrt[aa + cc*x^4], (d + e*x^2)^q*(aa + 
cc*x^4)^(p + 1/2), x] /. {aa -> a, cc -> c}, x]] /; FreeQ[{a, c, d, e}, x] 
&& NeQ[c*d^2 + a*e^2, 0] && ILtQ[q, 0] && IntegerQ[p + 1/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1323 vs. \(2 (488 ) = 976\).

Time = 1.20 (sec) , antiderivative size = 1324, normalized size of antiderivative = 2.36

method	result	size
default	\(\text {Expression too large to display}\)	\(1324\)
elliptic	\(\text {Expression too large to display}\)	\(1324\)

Input:

int(1/(e*x^2+d)^3/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2*c*(1/4*c*e*(a*e^2+3*c*d^2)/a/(a*e^2-c*d^2)^3*x^3-1/4*c*d*(3*a*e^2+c*d^2) 
/a/(a*e^2-c*d^2)^3*x)/(-(x^4-1/c*a)*c)^(1/2)+1/4*e^4/d/(a*e^2-c*d^2)^2*x*( 
-c*x^4+a)^(1/2)/(e*x^2+d)^2+1/8*e^4*(3*a*e^2-17*c*d^2)/(a*e^2-c*d^2)^3/d^2 
*x*(-c*x^4+a)^(1/2)/(e*x^2+d)-27/8/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)* 
c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*Ellipt 
icF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)*c^2*d/(a*e^2-c*d^2)^3*e^2-1/2/(1/a^(1/2 
)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2) 
^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)*c^3*d^3/a 
/(a*e^2-c*d^2)^3+1/8/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^( 
1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/ 
2)*c^(1/2))^(1/2),I)*e^4*c/d/(a*e^2-c*d^2)^3*a+21/8*a^(1/2)/(1/a^(1/2)*c^( 
1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2 
)/(-c*x^4+a)^(1/2)*c^(3/2)*e^3/(a*e^2-c*d^2)^3*EllipticF(x*(1/a^(1/2)*c^(1 
/2))^(1/2),I)-21/8*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)* 
x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*c^(3/2)*e^3/(a 
*e^2-c*d^2)^3*EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+3/2/a^(1/2)/(1/a^(1 
/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^ 
2)^(1/2)/(-c*x^4+a)^(1/2)*c^(5/2)*e/(a*e^2-c*d^2)^3*d^2*EllipticF(x*(1/a^( 
1/2)*c^(1/2))^(1/2),I)-3/2/a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)* 
c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*c^(...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x^2+d)^3/(-c*x^4+a)^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (a - c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{3}}\, dx \] Input:

integrate(1/(e*x**2+d)**3/(-c*x**4+a)**(3/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(1/((a - c*x**4)**(3/2)*(d + e*x**2)**3), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(1/(e*x^2+d)^3/(-c*x^4+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((-c*x^4 + a)^(3/2)*(e*x^2 + d)^3), x)
 

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(1/(e*x^2+d)^3/(-c*x^4+a)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((-c*x^4 + a)^(3/2)*(e*x^2 + d)^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int(1/((a - c*x^4)^(3/2)*(d + e*x^2)^3),x)
 

Output:

int(1/((a - c*x^4)^(3/2)*(d + e*x^2)^3), x)
 

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} e^{3} x^{14}+3 c^{2} d \,e^{2} x^{12}-2 a c \,e^{3} x^{10}+3 c^{2} d^{2} e \,x^{10}-6 a c d \,e^{2} x^{8}+c^{2} d^{3} x^{8}+a^{2} e^{3} x^{6}-6 a c \,d^{2} e \,x^{6}+3 a^{2} d \,e^{2} x^{4}-2 a c \,d^{3} x^{4}+3 a^{2} d^{2} e \,x^{2}+a^{2} d^{3}}d x \] Input:

int(1/(e*x^2+d)^3/(-c*x^4+a)^(3/2),x)
 

Output:

int(sqrt(a - c*x**4)/(a**2*d**3 + 3*a**2*d**2*e*x**2 + 3*a**2*d*e**2*x**4 
+ a**2*e**3*x**6 - 2*a*c*d**3*x**4 - 6*a*c*d**2*e*x**6 - 6*a*c*d*e**2*x**8 
 - 2*a*c*e**3*x**10 + c**2*d**3*x**8 + 3*c**2*d**2*e*x**10 + 3*c**2*d*e**2 
*x**12 + c**2*e**3*x**14),x)