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

Optimal. Leaf size=168 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (3 d^2-2 e^2\right ) F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{2 e x \left (x^2+2\right ) (d-e)}{\sqrt{x^4+3 x^2+2}}-\frac{2 \sqrt{2} e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} (d-e) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}}+\frac{1}{3} e^2 x \sqrt{x^4+3 x^2+2} \]

[Out]

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

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Rubi [A]  time = 0.174077, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (3 d^2-2 e^2\right ) F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{2 e x \left (x^2+2\right ) (d-e)}{\sqrt{x^4+3 x^2+2}}-\frac{2 \sqrt{2} e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} (d-e) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}}+\frac{1}{3} e^2 x \sqrt{x^4+3 x^2+2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.162, size = 148, normalized size = 0.88 \[ \frac{e^{2} x \sqrt{x^{4} + 3 x^{2} + 2}}{3} + \frac{e x \left (d - e\right ) \left (2 x^{2} + 4\right )}{\sqrt{x^{4} + 3 x^{2} + 2}} - \frac{e \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (d - e\right ) \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{\sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (\frac{d^{2}}{8} - \frac{e^{2}}{12}\right ) \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.201345, size = 127, normalized size = 0.76 \[ \frac{-i \sqrt{x^2+1} \sqrt{x^2+2} \left (3 d^2-6 d e+4 e^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-6 i e \sqrt{x^2+1} \sqrt{x^2+2} (d-e) E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+e^2 x \left (x^4+3 x^2+2\right )}{3 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.014, size = 235, normalized size = 1.4 \[{-{\frac{i}{2}}{d}^{2}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{e}^{2} \left ({\frac{x}{3}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{{\frac{i}{3}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{i\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \right ) +{ide\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*I*d^2*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))+e^2*(1/3*x*(x^4+3*x^2+2)^(1/2)+1/3*I*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))-I*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))))+I*d*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. \[ \int \frac{{\left (e x^{2} + d\right )}^{2}}{\sqrt{x^{4} + 3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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