3.53 \(\int \frac{2+3 x^2}{x^2 \left (5+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=196 \[ -\frac{3 \sqrt{x^4+5}}{25 x}+\frac{3 \sqrt{x^4+5} x}{25 \left (x^2+\sqrt{5}\right )}+\frac{3 x^2+2}{10 \sqrt{x^4+5} x}+\frac{3 \left (2+\sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{20\ 5^{3/4} \sqrt{x^4+5}}-\frac{3 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{5\ 5^{3/4} \sqrt{x^4+5}} \]

[Out]

(2 + 3*x^2)/(10*x*Sqrt[5 + x^4]) - (3*Sqrt[5 + x^4])/(25*x) + (3*x*Sqrt[5 + x^4]
)/(25*(Sqrt[5] + x^2)) - (3*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*El
lipticE[2*ArcTan[x/5^(1/4)], 1/2])/(5*5^(3/4)*Sqrt[5 + x^4]) + (3*(2 + Sqrt[5])*
(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticF[2*ArcTan[x/5^(1/4)],
1/2])/(20*5^(3/4)*Sqrt[5 + x^4])

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Rubi [A]  time = 0.211718, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 \sqrt{x^4+5}}{25 x}+\frac{3 \sqrt{x^4+5} x}{25 \left (x^2+\sqrt{5}\right )}+\frac{3 x^2+2}{10 \sqrt{x^4+5} x}+\frac{3 \left (2+\sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{20\ 5^{3/4} \sqrt{x^4+5}}-\frac{3 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{5\ 5^{3/4} \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x^2)/(x^2*(5 + x^4)^(3/2)),x]

[Out]

(2 + 3*x^2)/(10*x*Sqrt[5 + x^4]) - (3*Sqrt[5 + x^4])/(25*x) + (3*x*Sqrt[5 + x^4]
)/(25*(Sqrt[5] + x^2)) - (3*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*El
lipticE[2*ArcTan[x/5^(1/4)], 1/2])/(5*5^(3/4)*Sqrt[5 + x^4]) + (3*(2 + Sqrt[5])*
(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticF[2*ArcTan[x/5^(1/4)],
1/2])/(20*5^(3/4)*Sqrt[5 + x^4])

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Rubi in Sympy [A]  time = 18.6915, size = 194, normalized size = 0.99 \[ \frac{3 x \sqrt{x^{4} + 5}}{25 \left (x^{2} + \sqrt{5}\right )} - \frac{3 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{25 \sqrt{x^{4} + 5}} + \frac{\sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (6 + 3 \sqrt{5}\right ) \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{100 \sqrt{x^{4} + 5}} + \frac{3 x^{2} + 2}{10 x \sqrt{x^{4} + 5}} - \frac{3 \sqrt{x^{4} + 5}}{25 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)/x**2/(x**4+5)**(3/2),x)

[Out]

3*x*sqrt(x**4 + 5)/(25*(x**2 + sqrt(5))) - 3*5**(1/4)*sqrt((x**4 + 5)/(sqrt(5)*x
**2/5 + 1)**2)*(sqrt(5)*x**2/5 + 1)*elliptic_e(2*atan(5**(3/4)*x/5), 1/2)/(25*sq
rt(x**4 + 5)) + 5**(1/4)*sqrt((x**4 + 5)/(sqrt(5)*x**2/5 + 1)**2)*(6 + 3*sqrt(5)
)*(sqrt(5)*x**2/5 + 1)*elliptic_f(2*atan(5**(3/4)*x/5), 1/2)/(100*sqrt(x**4 + 5)
) + (3*x**2 + 2)/(10*x*sqrt(x**4 + 5)) - 3*sqrt(x**4 + 5)/(25*x)

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Mathematica [C]  time = 0.204498, size = 108, normalized size = 0.55 \[ -\frac{6 x^4+3 \sqrt [4]{-5} \left (\sqrt{5}-2 i\right ) \sqrt{x^4+5} x F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )+6 (-1)^{3/4} \sqrt [4]{5} \sqrt{x^4+5} x E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )-15 x^2+20}{50 x \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x^2)/(x^2*(5 + x^4)^(3/2)),x]

[Out]

-(20 - 15*x^2 + 6*x^4 + 6*(-1)^(3/4)*5^(1/4)*x*Sqrt[5 + x^4]*EllipticE[I*ArcSinh
[(-1/5)^(1/4)*x], -1] + 3*(-5)^(1/4)*(-2*I + Sqrt[5])*x*Sqrt[5 + x^4]*EllipticF[
I*ArcSinh[(-1/5)^(1/4)*x], -1])/(50*x*Sqrt[5 + x^4])

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Maple [C]  time = 0.027, size = 180, normalized size = 0.9 \[{\frac{3\,x}{10}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{3\,\sqrt{5}}{250\,\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ){\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{{x}^{3}}{25}{\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{2}{25\,x}\sqrt{{x}^{4}+5}}+{\frac{{\frac{3\,i}{125}}}{\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)/x^2/(x^4+5)^(3/2),x)

[Out]

3/10*x/(x^4+5)^(1/2)+3/250*5^(1/2)/(I*5^(1/2))^(1/2)*(25-5*I*5^(1/2)*x^2)^(1/2)*
(25+5*I*5^(1/2)*x^2)^(1/2)/(x^4+5)^(1/2)*EllipticF(1/5*x*5^(1/2)*(I*5^(1/2))^(1/
2),I)-1/25*x^3/(x^4+5)^(1/2)-2/25*(x^4+5)^(1/2)/x+3/125*I/(I*5^(1/2))^(1/2)*(25-
5*I*5^(1/2)*x^2)^(1/2)*(25+5*I*5^(1/2)*x^2)^(1/2)/(x^4+5)^(1/2)*(EllipticF(1/5*x
*5^(1/2)*(I*5^(1/2))^(1/2),I)-EllipticE(1/5*x*5^(1/2)*(I*5^(1/2))^(1/2),I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{3 \, x^{2} + 2}{{\left (x^{4} + 5\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/((x^4 + 5)^(3/2)*x^2),x, algorithm="maxima")

[Out]

integrate((3*x^2 + 2)/((x^4 + 5)^(3/2)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{3 \, x^{2} + 2}{{\left (x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/((x^4 + 5)^(3/2)*x^2),x, algorithm="fricas")

[Out]

integral((3*x^2 + 2)/((x^6 + 5*x^2)*sqrt(x^4 + 5)), x)

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Sympy [A]  time = 17.7922, size = 75, normalized size = 0.38 \[ \frac{3 \sqrt{5} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{100 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{5} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{50 x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2)/x**2/(x**4+5)**(3/2),x)

[Out]

3*sqrt(5)*x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), x**4*exp_polar(I*pi)/5)/(100*ga
mma(5/4)) + sqrt(5)*gamma(-1/4)*hyper((-1/4, 3/2), (3/4,), x**4*exp_polar(I*pi)/
5)/(50*x*gamma(3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{3 \, x^{2} + 2}{{\left (x^{4} + 5\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/((x^4 + 5)^(3/2)*x^2),x, algorithm="giac")

[Out]

integrate((3*x^2 + 2)/((x^4 + 5)^(3/2)*x^2), x)