3.402 \(\int \frac{1}{\left (d+e x^2\right )^2 \sqrt{2+3 x^2+x^4}} \, dx\)

Optimal. Leaf size=316 \[ \frac{e^2 x \sqrt{x^4+3 x^2+2}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}-\frac{e x \left (x^2+2\right )}{2 d \sqrt{x^4+3 x^2+2} \left (d^2-3 d e+2 e^2\right )}-\frac{e \left (x^2+2\right ) \left (3 d^2-6 d e+2 e^2\right ) \Pi \left (1-\frac{e}{d};\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} d^2 \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)^2}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{2 x^2+2}} (2 d-e) F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2 d \sqrt{x^4+3 x^2+2} (d-e)^2}+\frac{e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} d \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)} \]

[Out]

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

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Rubi [A]  time = 0.491622, antiderivative size = 399, normalized size of antiderivative = 1.26, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{e^2 x \sqrt{x^4+3 x^2+2}}{2 d \left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}-\frac{e x \left (x^2+2\right )}{2 d \sqrt{x^4+3 x^2+2} \left (d^2-3 d e+2 e^2\right )}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (3 d^2-6 d e+2 e^2\right ) F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} d \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)^2}-\frac{e \left (x^2+2\right ) \left (3 d^2-6 d e+2 e^2\right ) \Pi \left (1-\frac{e}{d};\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} d^2 \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)^2}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{2} \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)}+\frac{e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} d \sqrt{x^4+3 x^2+2} (d-2 e) (d-e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 77.5317, size = 332, normalized size = 1.05 \[ - \frac{\sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{16 \left (d - 2 e\right ) \left (d - e\right ) \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{e^{2} x \sqrt{x^{4} + 3 x^{2} + 2}}{2 d \left (d - 2 e\right ) \left (d - e\right ) \left (d + e x^{2}\right )} - \frac{e x \left (2 x^{2} + 4\right )}{4 d \left (d - 2 e\right ) \left (d - e\right ) \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{e \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{8 d \left (d - 2 e\right ) \left (d - e\right ) \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{\sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) \left (3 d^{2} - 6 d e + 2 e^{2}\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{16 d \left (d - 2 e\right ) \left (d - e\right )^{2} \sqrt{x^{4} + 3 x^{2} + 2}} - \frac{\sqrt{2} e \left (3 d^{2} - 6 d e + 2 e^{2}\right ) \sqrt{x^{4} + 3 x^{2} + 2} \Pi \left (1 - \frac{e}{d}; \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{4 d^{2} \sqrt{\frac{x^{2} + 2}{x^{2} + 1}} \left (d - 2 e\right ) \left (d - e\right )^{2} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_f(atan(x), 1/2)/(16*(d - 2*
e)*(d - e)*sqrt(x**4 + 3*x**2 + 2)) + e**2*x*sqrt(x**4 + 3*x**2 + 2)/(2*d*(d - 2
*e)*(d - e)*(d + e*x**2)) - e*x*(2*x**2 + 4)/(4*d*(d - 2*e)*(d - e)*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)/(8*d*(d - 2*e)*(d - e)*sqrt(x**4 + 3*x**2 + 2)) + sqrt((2*x**2 + 4)/(x**2 +
1))*(4*x**2 + 4)*(3*d**2 - 6*d*e + 2*e**2)*elliptic_f(atan(x), 1/2)/(16*d*(d -
2*e)*(d - e)**2*sqrt(x**4 + 3*x**2 + 2)) - sqrt(2)*e*(3*d**2 - 6*d*e + 2*e**2)*s
qrt(x**4 + 3*x**2 + 2)*elliptic_pi(1 - e/d, atan(x), 1/2)/(4*d**2*sqrt((x**2 + 2
)/(x**2 + 1))*(d - 2*e)*(d - e)**2*(x**2 + 1))

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Mathematica [C]  time = 0.759612, size = 175, normalized size = 0.55 \[ \frac{\frac{e^2 x \left (x^4+3 x^2+2\right )}{\left (d^2-3 d e+2 e^2\right ) \left (d+e x^2\right )}+\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \left (\left (-3 d^2+6 d e-2 e^2\right ) \Pi \left (\frac{2 e}{d};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+d (d-e) F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+d e E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )\right )}{d (d-2 e) (d-e)}}{2 d \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.037, size = 443, normalized size = 1.4 \[{\frac{{e}^{2}x}{2\, \left ({d}^{2}-3\,de+2\,{e}^{2} \right ) d \left ( e{x}^{2}+d \right ) }\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{{\frac{i}{4}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) }{{d}^{2}-3\,de+2\,{e}^{2}}\sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{\frac{{\frac{i}{4}}e\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) }{ \left ({d}^{2}-3\,de+2\,{e}^{2} \right ) d}\sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{\frac{{\frac{i}{4}}e\sqrt{2}{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) }{ \left ({d}^{2}-3\,de+2\,{e}^{2} \right ) d}\sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{\frac{{\frac{3\,i}{2}}\sqrt{2}}{{d}^{2}-3\,de+2\,{e}^{2}}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}\sqrt{2}x,2\,{\frac{e}{d}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{\frac{3\,ie\sqrt{2}}{ \left ({d}^{2}-3\,de+2\,{e}^{2} \right ) d}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}\sqrt{2}x,2\,{\frac{e}{d}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{\frac{i{e}^{2}\sqrt{2}}{ \left ({d}^{2}-3\,de+2\,{e}^{2} \right ){d}^{2}}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}\sqrt{2}x,2\,{\frac{e}{d}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} + 3 \, x^{2} + 2}{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 3*x^2 + 2)*(e*x^2 + d)^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^4 + 3*x^2 + 2)*(e*x^2 + d)^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt{x^{4} + 3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 3*x^2 + 2)*(e*x^2 + d)^2),x, algorithm="fricas")

[Out]

integral(1/((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. \[ \int \frac{1}{\sqrt{\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (d + e x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} + 3 \, x^{2} + 2}{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 3*x^2 + 2)*(e*x^2 + d)^2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^4 + 3*x^2 + 2)*(e*x^2 + d)^2), x)