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

Optimal. Leaf size=299 \[ -\frac{a^{3/4} \sqrt [4]{c} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d \sqrt{a-c x^4} \left (c d^2-a e^2\right )}+\frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \left (3 c d^2-a e^2\right ) \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{c} d^2 \sqrt{a-c x^4} \left (c d^2-a e^2\right )}-\frac{e^2 x \sqrt{a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d \sqrt{a-c x^4} \left (\sqrt{a} e+\sqrt{c} d\right )} \]

[Out]

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

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Rubi [A]  time = 0.517639, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{a^{3/4} \sqrt [4]{c} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d \sqrt{a-c x^4} \left (c d^2-a e^2\right )}+\frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \left (3 c d^2-a e^2\right ) \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{c} d^2 \sqrt{a-c x^4} \left (c d^2-a e^2\right )}-\frac{e^2 x \sqrt{a-c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}-\frac{\sqrt [4]{a} \sqrt [4]{c} \sqrt{1-\frac{c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 d \sqrt{a-c x^4} \left (\sqrt{a} e+\sqrt{c} d\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 69.3863, size = 265, normalized size = 0.89 \[ \frac{a^{\frac{3}{4}} \sqrt [4]{c} e \sqrt{1 - \frac{c x^{4}}{a}} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 d \sqrt{a - c x^{4}} \left (a e^{2} - c d^{2}\right )} - \frac{\sqrt [4]{a} \sqrt [4]{c} \sqrt{1 - \frac{c x^{4}}{a}} \left (\sqrt{a} e - \sqrt{c} d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 d \sqrt{a - c x^{4}} \left (a e^{2} - c d^{2}\right )} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{c x^{4}}{a}} \left (a e^{2} - 3 c d^{2}\right ) \Pi \left (- \frac{\sqrt{a} e}{\sqrt{c} d}; \operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{2 \sqrt [4]{c} d^{2} \sqrt{a - c x^{4}} \left (a e^{2} - c d^{2}\right )} + \frac{e^{2} x \sqrt{a - c x^{4}}}{2 d \left (d + e x^{2}\right ) \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/4)*c**(1/4)*e*sqrt(1 - c*x**4/a)*elliptic_e(asin(c**(1/4)*x/a**(1/4)), -1)
/(2*d*sqrt(a - c*x**4)*(a*e**2 - c*d**2)) - a**(1/4)*c**(1/4)*sqrt(1 - c*x**4/a)
*(sqrt(a)*e - sqrt(c)*d)*elliptic_f(asin(c**(1/4)*x/a**(1/4)), -1)/(2*d*sqrt(a -
c*x**4)*(a*e**2 - c*d**2)) + a**(1/4)*sqrt(1 - c*x**4/a)*(a*e**2 - 3*c*d**2)*el
liptic_pi(-sqrt(a)*e/(sqrt(c)*d), asin(c**(1/4)*x/a**(1/4)), -1)/(2*c**(1/4)*d**
2*sqrt(a - c*x**4)*(a*e**2 - c*d**2)) + e**2*x*sqrt(a - c*x**4)/(2*d*(d + e*x**2
)*(a*e**2 - c*d**2))

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Mathematica [C]  time = 1.56114, size = 508, normalized size = 1.7 \[ \frac{-3 i c d^3 \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-3 i c d^2 e x^2 \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+i a e^3 x^2 \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+c d e^2 x^5 \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}+i a d e^2 \sqrt{1-\frac{c x^4}{a}} \Pi \left (-\frac{\sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-a d e^2 x \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}}-i \sqrt{c} d \sqrt{1-\frac{c x^4}{a}} \left (d+e x^2\right ) \left (\sqrt{a} e-\sqrt{c} d\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+i \sqrt{a} \sqrt{c} d e \sqrt{1-\frac{c x^4}{a}} \left (d+e x^2\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{2 d^2 \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \sqrt{a-c x^4} \left (d+e x^2\right ) \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-(a*Sqrt[-(Sqrt[c]/Sqrt[a])]*d*e^2*x) + Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d*e^2*x^5 +
I*Sqrt[a]*Sqrt[c]*d*e*(d + e*x^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-
(Sqrt[c]/Sqrt[a])]*x], -1] - I*Sqrt[c]*d*(-(Sqrt[c]*d) + Sqrt[a]*e)*(d + e*x^2)*
Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)
*c*d^3*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt
[-(Sqrt[c]/Sqrt[a])]*x], -1] + I*a*d*e^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[
a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - (3*I)*c*d^2*e*x
^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(S
qrt[c]/Sqrt[a])]*x], -1] + I*a*e^3*x^2*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]
*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1])/(2*Sqrt[-(Sqrt[c]/
Sqrt[a])]*d^2*(c*d^2 - a*e^2)*(d + e*x^2)*Sqrt[a - c*x^4])

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Maple [B]  time = 0.033, size = 523, 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+a)^(1/2),x)

[Out]

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)/(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)-1/2*e*c^(1/2)/(a*e^
2-c*d^2)/d*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)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1
/2),I)+1/2*e*c^(1/2)/(a*e^2-c*d^2)/d*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)*Elliptic
E(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+1/2/(a*e^2-c*d^2)/d^2*e^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)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1/2),-e*a^(1/2)/d/c^(1/2),(-1/a^(1/2)*c^
(1/2))^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2))*a-3/2/(a*e^2-c*d^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)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1/2),-e*a^(1/2)/d/c^(1/2),(-1/a^(1/2)*c
^(1/2))^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2))*c

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-c*x^4 + 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 + 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. \[ \int \frac{1}{\sqrt{a - c x^{4}} \left (d + e x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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