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

Optimal. Leaf size=126 \[ \frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{c x^4-a}}+\frac{a^{3/4} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{c x^4-a}} \]

[Out]

(a^(3/4)*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3
/4)*Sqrt[-a + c*x^4]) + (a^(3/4)*((Sqrt[c]*d)/Sqrt[a] - e)*Sqrt[1 - (c*x^4)/a]*E
llipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[-a + c*x^4])

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Rubi [A]  time = 0.214027, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{a^{3/4} \sqrt{1-\frac{c x^4}{a}} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{c x^4-a}}+\frac{a^{3/4} e \sqrt{1-\frac{c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt{c x^4-a}} \]

Antiderivative was successfully verified.

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

[Out]

(a^(3/4)*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3
/4)*Sqrt[-a + c*x^4]) + (a^(3/4)*((Sqrt[c]*d)/Sqrt[a] - e)*Sqrt[1 - (c*x^4)/a]*E
llipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[-a + c*x^4])

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Rubi in Sympy [A]  time = 35.8939, size = 112, normalized size = 0.89 \[ \frac{a^{\frac{3}{4}} e \sqrt{1 - \frac{c x^{4}}{a}} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{c^{\frac{3}{4}} \sqrt{- a + c x^{4}}} - \frac{\sqrt [4]{a} \sqrt{1 - \frac{c x^{4}}{a}} \left (\sqrt{a} e - \sqrt{c} d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{c^{\frac{3}{4}} \sqrt{- a + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/4)*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)*(sqrt(a)*e - sqrt(c)*d)*ellip
tic_f(asin(c**(1/4)*x/a**(1/4)), -1)/(c**(3/4)*sqrt(-a + c*x**4))

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Mathematica [C]  time = 0.130873, size = 128, normalized size = 1.02 \[ \frac{i \sqrt{1-\frac{c x^4}{a}} \left (\left (\sqrt{c} d-\sqrt{a} e\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{a} e E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\sqrt{a} \left (-\frac{\sqrt{c}}{\sqrt{a}}\right )^{3/2} \sqrt{c x^4-a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 160, normalized size = 1.3 \[{d\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}}+{e\sqrt{a}\sqrt{1+{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{-{1\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}-a}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d/(-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*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)-E
llipticE(x*(-1/a^(1/2)*c^(1/2))^(1/2),I))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.14346, size = 73, normalized size = 0.58 \[ - \frac{i d x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{c x^{4}}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} - \frac{i e x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4}}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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