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

Optimal. Leaf size=162 \[ \frac{2 a^{3/4} d e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{a-c x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \left (-6 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 c^{5/4} \sqrt{a-c x^4}}-\frac{e^2 x \sqrt{a-c x^4}}{3 c} \]

[Out]

-(e^2*x*Sqrt[a - c*x^4])/(3*c) + (2*a^(3/4)*d*e*Sqrt[1 - (c*x^4)/a]*EllipticE[Ar
cSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*(3*c*d^2 -
6*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/
a^(1/4)], -1])/(3*c^(5/4)*Sqrt[a - c*x^4])

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Rubi [A]  time = 0.339464, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{2 a^{3/4} d e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{a-c x^4}}+\frac{\sqrt [4]{a} \sqrt{1-\frac{c x^4}{a}} \left (-6 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 c^{5/4} \sqrt{a-c x^4}}-\frac{e^2 x \sqrt{a-c x^4}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

-(e^2*x*Sqrt[a - c*x^4])/(3*c) + (2*a^(3/4)*d*e*Sqrt[1 - (c*x^4)/a]*EllipticE[Ar
cSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*(3*c*d^2 -
6*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/
a^(1/4)], -1])/(3*c^(5/4)*Sqrt[a - c*x^4])

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Rubi in Sympy [A]  time = 56.9213, size = 150, normalized size = 0.93 \[ \frac{2 a^{\frac{3}{4}} d e \sqrt{1 - \frac{c x^{4}}{a}} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{c^{\frac{3}{4}} \sqrt{a - c x^{4}}} + \frac{\sqrt [4]{a} \sqrt{1 - \frac{c x^{4}}{a}} \left (- 6 \sqrt{a} \sqrt{c} d e + a e^{2} + 3 c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{3 c^{\frac{5}{4}} \sqrt{a - c x^{4}}} - \frac{e^{2} x \sqrt{a - c x^{4}}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**(3/4)*d*e*sqrt(1 - c*x**4/a)*elliptic_e(asin(c**(1/4)*x/a**(1/4)), -1)/(c**
(3/4)*sqrt(a - c*x**4)) + a**(1/4)*sqrt(1 - c*x**4/a)*(-6*sqrt(a)*sqrt(c)*d*e +
a*e**2 + 3*c*d**2)*elliptic_f(asin(c**(1/4)*x/a**(1/4)), -1)/(3*c**(5/4)*sqrt(a
- c*x**4)) - e**2*x*sqrt(a - c*x**4)/(3*c)

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Mathematica [C]  time = 0.439196, size = 192, normalized size = 1.19 \[ \frac{-i \sqrt{1-\frac{c x^4}{a}} \left (-6 \sqrt{a} \sqrt{c} d e+a e^2+3 c d^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-6 i \sqrt{a} \sqrt{c} d e \sqrt{1-\frac{c x^4}{a}} E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+e^2 x \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \left (c x^4-a\right )}{3 c \sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} \sqrt{a-c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-(Sqrt[c]/Sqrt[a])]*e^2*x*(-a + c*x^4) - (6*I)*Sqrt[a]*Sqrt[c]*d*e*Sqrt[1
- (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - I*(3*c*d^2 -
6*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(S
qrt[c]/Sqrt[a])]*x], -1])/(3*Sqrt[-(Sqrt[c]/Sqrt[a])]*c*Sqrt[a - c*x^4])

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Maple [A]  time = 0.01, size = 246, normalized size = 1.5 \[{{d}^{2}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}}+{e}^{2} \left ( -{\frac{x}{3\,c}\sqrt{-c{x}^{4}+a}}+{\frac{a}{3\,c}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-c{x}^{4}+a}}}} \right ) -2\,{\frac{de\sqrt{a}}{\sqrt{-c{x}^{4}+a}\sqrt{c}}\sqrt{1-{\frac{{x}^{2}\sqrt{c}}{\sqrt{a}}}}\sqrt{1+{\frac{{x}^{2}\sqrt{c}}{\sqrt{a}}}} \left ({\it EllipticF} \left ( x\sqrt{{\frac{\sqrt{c}}{\sqrt{a}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{\frac{\sqrt{c}}{\sqrt{a}}}},i \right ) \right ){\frac{1}{\sqrt{{\frac{\sqrt{c}}{\sqrt{a}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.25555, size = 129, normalized size = 0.8 \[ \frac{d^{2} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} + \frac{d e x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4} e^{2 i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} + \frac{e^{2} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(
a)*gamma(5/4)) + d*e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x**4*exp_polar(
2*I*pi)/a)/(2*sqrt(a)*gamma(7/4)) + e**2*x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,
), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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