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

Optimal. Leaf size=703 \[ \frac{e^2 x \sqrt{a+b x^2+c x^4}}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e x \sqrt{a+b x^2+c x^4}}{2 d \left (\sqrt{a}+\sqrt{c} x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 d \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (3 c d^2-e (2 b d-a e)\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{8 \sqrt [4]{c} d^2 \sqrt{a+b x^2+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2-b d e+c d^2\right )}+\frac{\left (3 c d^2-e (2 b d-a e)\right ) \tan ^{-1}\left (\frac{x \sqrt{\frac{a e}{d}-b+\frac{c d}{e}}}{\sqrt{a+b x^2+c x^4}}\right )}{4 d^3 e \left (\frac{a e}{d}-b+\frac{c d}{e}\right )^{3/2}}-\frac{\sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}-\frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{2 \sqrt [4]{a} d \sqrt{a+b x^2+c x^4} \left (\sqrt{a} e-\sqrt{c} d\right )} \]

[Out]

-(Sqrt[c]*e*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)*(Sqrt[a] + S
qrt[c]*x^2)) + (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)) + ((3*c*d^2 - e*(2*b*d - a*e))*ArcTan[(Sqrt[-b + (c*d)/e + (a*e)/d]*x)/
Sqrt[a + b*x^2 + c*x^4]])/(4*d^3*e*(-b + (c*d)/e + (a*e)/d)^(3/2)) + (a^(1/4)*c^
(1/4)*e*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)
^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*d*
(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*(Sqrt[a] + Sqrt[c]*x
^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1
/4)*x)/a^(1/4)], 1/2 - b/(4*Sqrt[a]*Sqrt[c])])/(2*a^(1/4)*d*(-(Sqrt[c]*d) + Sqrt
[a]*e)*Sqrt[a + b*x^2 + c*x^4]) - (a^(1/4)*((Sqrt[c]*d)/Sqrt[a] + e)*(3*c*d^2 -
e*(2*b*d - a*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqr
t[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*Ar
cTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(8*c^(1/4)*d^2*(Sqrt[c]
*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4])

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Rubi [A]  time = 1.45479, antiderivative size = 889, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{x \sqrt{c x^4+b x^2+a} e^2}{2 d \left (c d^2-b e d+a e^2\right ) \left (e x^2+d\right )}+\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right ) e}{2 d \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}}-\frac{\sqrt{c} x \sqrt{c x^4+b x^2+a} e}{2 d \left (c d^2-b e d+a e^2\right ) \left (\sqrt{c} x^2+\sqrt{a}\right )}+\frac{\sqrt [4]{c} \left (3 c d^2-e (2 b d-a e)\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} d \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}}-\frac{\sqrt [4]{a} \sqrt [4]{c} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 d \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}}-\frac{\left (\sqrt{c} d+\sqrt{a} e\right ) \left (3 c d^2-e (2 b d-a e)\right ) \left (\sqrt{c} x^2+\sqrt{a}\right ) \sqrt{\frac{c x^4+b x^2+a}{\left (\sqrt{c} x^2+\sqrt{a}\right )^2}} \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2-b e d+a e^2\right ) \sqrt{c x^4+b x^2+a}}+\frac{\left (3 c d^2-e (2 b d-a e)\right ) \tan ^{-1}\left (\frac{\sqrt{-b+\frac{a e}{d}+\frac{c d}{e}} x}{\sqrt{c x^4+b x^2+a}}\right )}{4 d^3 \left (-b+\frac{a e}{d}+\frac{c d}{e}\right )^{3/2} e} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(Sqrt[c]*e*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)*(Sqrt[a] + S
qrt[c]*x^2)) + (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)) + ((3*c*d^2 - e*(2*b*d - a*e))*ArcTan[(Sqrt[-b + (c*d)/e + (a*e)/d]*x)/
Sqrt[a + b*x^2 + c*x^4]])/(4*d^3*e*(-b + (c*d)/e + (a*e)/d)^(3/2)) + (a^(1/4)*c^
(1/4)*e*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)
^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*d*
(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) - (a^(1/4)*c^(1/4)*((Sqrt[c]*d)
/Sqrt[a] + e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c
]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])
/(4*d*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*(3*c*d^2 - e*(
2*b*d - a*e))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c
]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])
/(4*a^(1/4)*d*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c
*x^4]) - ((Sqrt[c]*d + Sqrt[a]*e)*(3*c*d^2 - e*(2*b*d - a*e))*(Sqrt[a] + Sqrt[c]
*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-(Sqrt[c]*d
- Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(
Sqrt[a]*Sqrt[c]))/4])/(8*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 - b*
d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 143.104, size = 784, normalized size = 1.12 \[ \text{result too large to display} \]

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]

a**(1/4)*c**(1/4)*e*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqr
t(a) + sqrt(c)*x**2)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*
sqrt(c)))/(2*d*sqrt(a + b*x**2 + c*x**4)*(a*e**2 - b*d*e + c*d**2)) - sqrt(c)*e*
x*sqrt(a + b*x**2 + c*x**4)/(2*d*(sqrt(a) + sqrt(c)*x**2)*(a*e**2 - b*d*e + c*d*
*2)) + e**2*x*sqrt(a + b*x**2 + c*x**4)/(2*d*(d + e*x**2)*(a*e**2 - b*d*e + c*d*
*2)) + (a*e**2 - 2*b*d*e + 3*c*d**2)*atan(x*sqrt(a*e/d - b + c*d/e)/sqrt(a + b*x
**2 + c*x**4))/(4*d**3*e*(a*e/d - b + c*d/e)**(3/2)) - c**(1/4)*sqrt((a + b*x**2
+ c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*(sqrt(a)*e + sq
rt(c)*d)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(4
*a**(1/4)*d*sqrt(a + b*x**2 + c*x**4)*(a*e**2 - b*d*e + c*d**2)) - c**(1/4)*sqrt
((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*(a*
e**2 - 2*b*d*e + 3*c*d**2)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sq
rt(a)*sqrt(c)))/(4*a**(1/4)*d*(sqrt(a)*e - sqrt(c)*d)*sqrt(a + b*x**2 + c*x**4)*
(a*e**2 - b*d*e + c*d**2)) + sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)
**2)*(sqrt(a) + sqrt(c)*x**2)*(sqrt(a)*e + sqrt(c)*d)*(a*e**2 - 2*b*d*e + 3*c*d*
*2)*elliptic_pi(-sqrt(a)*(e - sqrt(c)*d/sqrt(a))**2/(4*sqrt(c)*d*e), 2*atan(c**(
1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(8*a**(1/4)*c**(1/4)*d**2*(sqrt(a
)*e - sqrt(c)*d)*sqrt(a + b*x**2 + c*x**4)*(a*e**2 - b*d*e + c*d**2))

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Mathematica [C]  time = 3.09192, size = 1069, normalized size = 1.52 \[ \frac{2 i \sqrt{2} c \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (e x^2+d\right ) F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right ) d^2-6 i \sqrt{2} c \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (e x^2+d\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right ) d^2+4 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} e^2 x \left (c x^4+b x^2+a\right ) d+i \sqrt{2} \left (b-\sqrt{b^2-4 a c}\right ) e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (e x^2+d\right ) \left (E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )-F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right ) d+4 i \sqrt{2} b e \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (e x^2+d\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right ) d-2 i \sqrt{2} a e^2 \sqrt{\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (e x^2+d\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{8 \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} d \left (c d^3+e (a e-b d) d\right ) \left (e x^2+d\right ) \sqrt{c x^4+b x^2+a}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e^2*x*(a + b*x^2 + c*x^4) + I*Sqrt[2]*(b -
Sqrt[b^2 - 4*a*c])*d*e*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*
a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*(EllipticE[I*ArcS
inh[Sqrt[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])]) + (2*I)*Sqrt[2]*c*d^2*Sqrt[
(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b
- Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*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])] - (6*I)*Sqrt
[2]*c*d^2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1
+ (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*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 + Sqr
t[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (4*I)*Sqrt[2]*b*d*e*Sqrt[(b + Sqrt[b^
2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2
- 4*a*c])]*(d + e*x^2)*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])] - (2*I)*Sqrt[2]*a*e^2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b +
Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*Elli
pticPi[((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])])/(8*Sqrt[c/(b
+ Sqrt[b^2 - 4*a*c])]*d*(c*d^3 + d*e*(-(b*d) + a*e))*(d + e*x^2)*Sqrt[a + b*x^2
+ c*x^4])

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Maple [A]  time = 0.039, size = 1279, 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*x*(c*x^4+b*x^2+a)^(1/2)/d/(a*e^2-b*d*e+c*d^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))*EllipticE(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)*Ellip
ticPi(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)