3.400 \(\int \frac{d+e x^2}{\sqrt{2+3 x^2+x^4}} \, dx\)

Optimal. Leaf size=122 \[ \frac{d \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{e x \left (x^2+2\right )}{\sqrt{x^4+3 x^2+2}}-\frac{\sqrt{2} e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}} \]

[Out]

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

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Rubi [A]  time = 0.0831953, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{d \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{e x \left (x^2+2\right )}{\sqrt{x^4+3 x^2+2}}-\frac{\sqrt{2} e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2)/Sqrt[2 + 3*x^2 + x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 13.8699, size = 116, normalized size = 0.95 \[ \frac{d \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{8 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{e x \left (2 x^{2} + 4\right )}{2 \sqrt{x^{4} + 3 x^{2} + 2}} - \frac{e \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{4 \sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d)/(x**4+3*x**2+2)**(1/2),x)

[Out]

d*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_f(atan(x), 1/2)/(8*sqrt(x*
*4 + 3*x**2 + 2)) + e*x*(2*x**2 + 4)/(2*sqrt(x**4 + 3*x**2 + 2)) - e*sqrt((2*x**
2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_e(atan(x), 1/2)/(4*sqrt(x**4 + 3*x**2 +
2))

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Mathematica [C]  time = 0.081516, size = 73, normalized size = 0.6 \[ -\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \left ((d-e) F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+e E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )\right )}{\sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2)/Sqrt[2 + 3*x^2 + x^4],x]

[Out]

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

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Maple [C]  time = 0.008, size = 108, normalized size = 0.9 \[{-{\frac{i}{2}}d\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{i}{2}}e\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)/(x^4+3*x^2+2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{2} + d}{\sqrt{x^{4} + 3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)/sqrt(x^4 + 3*x^2 + 2),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)/sqrt(x^4 + 3*x^2 + 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e x^{2} + d}{\sqrt{x^{4} + 3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)/sqrt(x^4 + 3*x^2 + 2),x, algorithm="fricas")

[Out]

integral((e*x^2 + d)/sqrt(x^4 + 3*x^2 + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d)/(x**4+3*x**2+2)**(1/2),x)

[Out]

Integral((d + e*x**2)/sqrt((x**2 + 1)*(x**2 + 2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{2} + d}{\sqrt{x^{4} + 3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)/sqrt(x^4 + 3*x^2 + 2),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)/sqrt(x^4 + 3*x^2 + 2), x)