3.393 \(\int \frac{1}{\left (d+e x^2\right )^2 \sqrt{a+b x^2-c x^4}} \, dx\)

Optimal. Leaf size=718 \[ -\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (e \left (b-\sqrt{4 a c+b^2}\right )+2 c d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{4 \sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac{e \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{4 \sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (e (2 b d-a e)+3 c d^2\right ) \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{2 \sqrt{2} \sqrt{c} d^2 \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{2 d \left (d+e x^2\right ) \left (-a e^2+b d e+c d^2\right )} \]

[Out]

-(e^2*x*Sqrt[a + b*x^2 - c*x^4])/(2*d*(c*d^2 + b*d*e - a*e^2)*(d + e*x^2)) + ((b
- Sqrt[b^2 + 4*a*c])*Sqrt[b + Sqrt[b^2 + 4*a*c]]*e*Sqrt[1 - (2*c*x^2)/(b - Sqrt
[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticE[ArcSin[(Sq
rt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt
[b^2 + 4*a*c])])/(4*Sqrt[2]*Sqrt[c]*d*(c*d^2 + e*(b*d - a*e))*Sqrt[a + b*x^2 - c
*x^4]) - (Sqrt[b + Sqrt[b^2 + 4*a*c]]*(2*c*d + (b - Sqrt[b^2 + 4*a*c])*e)*Sqrt[1
- (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])
]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b
^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(4*Sqrt[2]*Sqrt[c]*d*(c*d^2 + e*(b*d - a*
e))*Sqrt[a + b*x^2 - c*x^4]) + (Sqrt[b + Sqrt[b^2 + 4*a*c]]*(3*c*d^2 + e*(2*b*d
- a*e))*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt
[b^2 + 4*a*c])]*EllipticPi[-((b + Sqrt[b^2 + 4*a*c])*e)/(2*c*d), ArcSin[(Sqrt[2]
*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2
+ 4*a*c])])/(2*Sqrt[2]*Sqrt[c]*d^2*(c*d^2 + e*(b*d - a*e))*Sqrt[a + b*x^2 - c*x^
4])

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Rubi [A]  time = 1.81231, antiderivative size = 718, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (e \left (b-\sqrt{4 a c+b^2}\right )+2 c d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{4 \sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac{e \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{4 \sqrt{2} \sqrt{c} d \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}+\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (e (2 b d-a e)+3 c d^2\right ) \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{2 \sqrt{2} \sqrt{c} d^2 \sqrt{a+b x^2-c x^4} \left (e (b d-a e)+c d^2\right )}-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{2 d \left (d+e x^2\right ) \left (e (b d-a e)+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(e^2*x*Sqrt[a + b*x^2 - c*x^4])/(2*d*(c*d^2 + e*(b*d - a*e))*(d + e*x^2)) + ((b
- Sqrt[b^2 + 4*a*c])*Sqrt[b + Sqrt[b^2 + 4*a*c]]*e*Sqrt[1 - (2*c*x^2)/(b - Sqrt
[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticE[ArcSin[(Sq
rt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt
[b^2 + 4*a*c])])/(4*Sqrt[2]*Sqrt[c]*d*(c*d^2 + e*(b*d - a*e))*Sqrt[a + b*x^2 - c
*x^4]) - (Sqrt[b + Sqrt[b^2 + 4*a*c]]*(2*c*d + (b - Sqrt[b^2 + 4*a*c])*e)*Sqrt[1
- (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])
]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b
^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(4*Sqrt[2]*Sqrt[c]*d*(c*d^2 + e*(b*d - a*
e))*Sqrt[a + b*x^2 - c*x^4]) + (Sqrt[b + Sqrt[b^2 + 4*a*c]]*(3*c*d^2 + e*(2*b*d
- a*e))*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt
[b^2 + 4*a*c])]*EllipticPi[-((b + Sqrt[b^2 + 4*a*c])*e)/(2*c*d), ArcSin[(Sqrt[2]
*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2
+ 4*a*c])])/(2*Sqrt[2]*Sqrt[c]*d^2*(c*d^2 + e*(b*d - a*e))*Sqrt[a + b*x^2 - c*x^
4])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x**2+d)**2/(-c*x**4+b*x**2+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 6.49224, size = 1341, normalized size = 1.87 \[ \frac{\left (e x^2+d\right ) \sqrt{-c x^4+b x^2+a} \left (-\frac{c e x^2}{2 d \left (c d^2+b e d-a e^2\right ) \sqrt{-c x^4+b x^2+a}}-\frac{c}{2 \left (c d^2+b e d-a e^2\right ) \sqrt{-c x^4+b x^2+a}}+\frac{3 c d^2+2 b e d-a e^2}{2 d \left (c d^2+b e d-a e^2\right ) \left (e x^2+d\right ) \sqrt{-c x^4+b x^2+a}}\right ) \left (\frac{i a \sqrt{\frac{2 c x^2}{\sqrt{b^2+4 a c}-b}+1} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|-\frac{b+\sqrt{b^2+4 a c}}{\sqrt{b^2+4 a c}-b}\right ) e^2}{\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} d \sqrt{-c x^4+b x^2+a}}+\frac{i \left (\sqrt{b^2+4 a c}-b\right ) \sqrt{\frac{2 c x^2}{\sqrt{b^2+4 a c}-b}+1} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} \left (E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|-\frac{b+\sqrt{b^2+4 a c}}{\sqrt{b^2+4 a c}-b}\right )-F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|-\frac{b+\sqrt{b^2+4 a c}}{\sqrt{b^2+4 a c}-b}\right )\right ) e}{2 \sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} \sqrt{-c x^4+b x^2+a}}-\frac{i \sqrt{2} b \sqrt{\frac{2 c x^2}{\sqrt{b^2+4 a c}-b}+1} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|-\frac{b+\sqrt{b^2+4 a c}}{\sqrt{b^2+4 a c}-b}\right ) e}{\sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} \sqrt{-c x^4+b x^2+a}}+\frac{i c d \sqrt{\frac{2 c x^2}{\sqrt{b^2+4 a c}-b}+1} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|-\frac{b+\sqrt{b^2+4 a c}}{\sqrt{b^2+4 a c}-b}\right )}{\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} \sqrt{-c x^4+b x^2+a}}-\frac{3 i c d \sqrt{\frac{2 c x^2}{\sqrt{b^2+4 a c}-b}+1} \sqrt{1-\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}} \Pi \left (-\frac{\left (b+\sqrt{b^2+4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|-\frac{b+\sqrt{b^2+4 a c}}{\sqrt{b^2+4 a c}-b}\right )}{\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} \sqrt{-c x^4+b x^2+a}}\right )}{-c e^2 x^4-2 c d e x^2+2 c d^2-a e^2+2 b d e}-\frac{e^2 x \sqrt{-c x^4+b x^2+a}}{2 d \left (c d^2+b e d-a e^2\right ) \left (e x^2+d\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(e^2*x*Sqrt[a + b*x^2 - c*x^4])/(2*d*(c*d^2 + b*d*e - a*e^2)*(d + e*x^2)) + ((d
+ e*x^2)*Sqrt[a + b*x^2 - c*x^4]*(-c/(2*(c*d^2 + b*d*e - a*e^2)*Sqrt[a + b*x^2
- c*x^4]) - (c*e*x^2)/(2*d*(c*d^2 + b*d*e - a*e^2)*Sqrt[a + b*x^2 - c*x^4]) + (3
*c*d^2 + 2*b*d*e - a*e^2)/(2*d*(c*d^2 + b*d*e - a*e^2)*(d + e*x^2)*Sqrt[a + b*x^
2 - c*x^4]))*(((I/2)*(-b + Sqrt[b^2 + 4*a*c])*e*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^
2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*(EllipticE[I*ArcSinh[Sq
rt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], -((b + Sqrt[b^2 + 4*a*c])/(-b + Sqr
t[b^2 + 4*a*c]))] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c])
)]*x], -((b + Sqrt[b^2 + 4*a*c])/(-b + Sqrt[b^2 + 4*a*c]))]))/(Sqrt[2]*Sqrt[-(c/
(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[a + b*x^2 - c*x^4]) + (I*c*d*Sqrt[1 + (2*c*x^2)/(
-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticF[I
*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], -((b + Sqrt[b^2 + 4*a*c]
)/(-b + Sqrt[b^2 + 4*a*c]))])/(Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[a
+ b*x^2 - c*x^4]) - ((3*I)*c*d*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqr
t[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticPi[-((b + Sqrt[b^2 + 4*a*c])*e)
/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], -((b + Sqrt[b
^2 + 4*a*c])/(-b + Sqrt[b^2 + 4*a*c]))])/(Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c
]))]*Sqrt[a + b*x^2 - c*x^4]) - (I*Sqrt[2]*b*e*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^2
+ 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticPi[-((b + Sqrt[b
^2 + 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x]
, -((b + Sqrt[b^2 + 4*a*c])/(-b + Sqrt[b^2 + 4*a*c]))])/(Sqrt[-(c/(b + Sqrt[b^2
+ 4*a*c]))]*Sqrt[a + b*x^2 - c*x^4]) + (I*a*e^2*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^
2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticPi[-((b + Sqrt[
b^2 + 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x
], -((b + Sqrt[b^2 + 4*a*c])/(-b + Sqrt[b^2 + 4*a*c]))])/(Sqrt[2]*Sqrt[-(c/(b +
Sqrt[b^2 + 4*a*c]))]*d*Sqrt[a + b*x^2 - c*x^4])))/(2*c*d^2 + 2*b*d*e - a*e^2 - 2
*c*d*e*x^2 - c*e^2*x^4)

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Maple [B]  time = 0.039, size = 1293, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x^2+d)^2/(-c*x^4+b*x^2+a)^(1/2),x)

[Out]

1/2*e^2/(a*e^2-b*d*e-c*d^2)/d*x*(-c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)+1/8*c/(a*e^2-b*
d*e-c*d^2)*2^(1/2)/(-b/a+1/a*(4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a-2/a*x^2*(4*a*
c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a+2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)/(-c*x^4+b*x^2+
a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b
+(4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/4*e*c/(a*e^2-b*d*e-c*d^2)/d*a*2^(1/2)/(-b/a+1/
a*(4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a-2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*
x^2/a+2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)/(-c*x^4+b*x^2+a)^(1/2)/(b+(4*a*c+b^2)^(1/
2))*EllipticF(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a
*c+b^2)^(1/2))/a/c)^(1/2))+1/4*e*c/(a*e^2-b*d*e-c*d^2)/d*a*2^(1/2)/(-b/a+1/a*(4*
a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a-2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)*(4+2*b*x^2/a
+2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)/(-c*x^4+b*x^2+a)^(1/2)/(b+(4*a*c+b^2)^(1/2))*E
llipticE(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4-2*b*(b+(4*a*c+b^
2)^(1/2))/a/c)^(1/2))+1/2/(a*e^2-b*d*e-c*d^2)/d^2*e^2*2^(1/2)/(-b/a+1/a*(4*a*c+b
^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a-1/2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/
a+1/2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)/(-c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(
1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(4*a*c+b^2)^(1/2))*a*e/d,(-1/2*(b+(
4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(4*a*c+b^2)^(1/2))/a)^(1/2))*a-1/(a*e^2-
b*d*e-c*d^2)/d*e*2^(1/2)/(-b/a+1/a*(4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a-1/2/a
*x^2*(4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a+1/2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)/
(-c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2)
,-2/(-b+(4*a*c+b^2)^(1/2))*a*e/d,(-1/2*(b+(4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((
-b+(4*a*c+b^2)^(1/2))/a)^(1/2))*b-3/2/(a*e^2-b*d*e-c*d^2)*2^(1/2)/(-b/a+1/a*(4*a
*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a-1/2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*
x^2/a+1/2/a*x^2*(4*a*c+b^2)^(1/2))^(1/2)/(-c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x
*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^(1/2),-2/(-b+(4*a*c+b^2)^(1/2))*a*e/d,(-1/2*
(b+(4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(4*a*c+b^2)^(1/2))/a)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*(e*x^2 + d)^2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*(e*x^2 + d)^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x^{2}\right )^{2} \sqrt{a + b x^{2} - c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x**2+d)**2/(-c*x**4+b*x**2+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-c*x^4 + b*x^2 + a)*(e*x^2 + d)^2),x, algorithm="giac")

[Out]

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