3.43 \(\int \frac{2+3 x^2}{x^4 \sqrt{5+x^4}} \, dx\)

Optimal. Leaf size=189 \[ -\frac{3 \sqrt{x^4+5}}{5 x}-\frac{2 \sqrt{x^4+5}}{15 x^3}+\frac{3 \sqrt{x^4+5} x}{5 \left (x^2+\sqrt{5}\right )}-\frac{\left (2-9 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{30 \sqrt [4]{5} \sqrt{x^4+5}}-\frac{3 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{5^{3/4} \sqrt{x^4+5}} \]

[Out]

(-2*Sqrt[5 + x^4])/(15*x^3) - (3*Sqrt[5 + x^4])/(5*x) + (3*x*Sqrt[5 + x^4])/(5*(
Sqrt[5] + x^2)) - (3*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticE
[2*ArcTan[x/5^(1/4)], 1/2])/(5^(3/4)*Sqrt[5 + x^4]) - ((2 - 9*Sqrt[5])*(Sqrt[5]
+ x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticF[2*ArcTan[x/5^(1/4)], 1/2])/(3
0*5^(1/4)*Sqrt[5 + x^4])

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Rubi [A]  time = 0.214632, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 \sqrt{x^4+5}}{5 x}-\frac{2 \sqrt{x^4+5}}{15 x^3}+\frac{3 \sqrt{x^4+5} x}{5 \left (x^2+\sqrt{5}\right )}-\frac{\left (2-9 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{30 \sqrt [4]{5} \sqrt{x^4+5}}-\frac{3 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{5^{3/4} \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[5 + x^4])/(15*x^3) - (3*Sqrt[5 + x^4])/(5*x) + (3*x*Sqrt[5 + x^4])/(5*(
Sqrt[5] + x^2)) - (3*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticE
[2*ArcTan[x/5^(1/4)], 1/2])/(5^(3/4)*Sqrt[5 + x^4]) - ((2 - 9*Sqrt[5])*(Sqrt[5]
+ x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticF[2*ArcTan[x/5^(1/4)], 1/2])/(3
0*5^(1/4)*Sqrt[5 + x^4])

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Rubi in Sympy [A]  time = 17.9886, size = 190, normalized size = 1.01 \[ \frac{3 x \sqrt{x^{4} + 5}}{5 \left (x^{2} + \sqrt{5}\right )} - \frac{3 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{5 \sqrt{x^{4} + 5}} + \frac{\sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (- 2 \sqrt{5} + 45\right ) \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{150 \sqrt{x^{4} + 5}} - \frac{3 \sqrt{x^{4} + 5}}{5 x} - \frac{2 \sqrt{x^{4} + 5}}{15 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)/x**4/(x**4+5)**(1/2),x)

[Out]

3*x*sqrt(x**4 + 5)/(5*(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)/(5*sqrt
(x**4 + 5)) + 5**(1/4)*sqrt((x**4 + 5)/(sqrt(5)*x**2/5 + 1)**2)*(-2*sqrt(5) + 45
)*(sqrt(5)*x**2/5 + 1)*elliptic_f(2*atan(5**(3/4)*x/5), 1/2)/(150*sqrt(x**4 + 5)
) - 3*sqrt(x**4 + 5)/(5*x) - 2*sqrt(x**4 + 5)/(15*x**3)

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Mathematica [C]  time = 0.204651, size = 97, normalized size = 0.51 \[ \frac{1}{75} \left (-\frac{5 \left (9 x^6+2 x^4+45 x^2+10\right )}{x^3 \sqrt{x^4+5}}+\sqrt [4]{-5} \left (2 \sqrt{5}+45 i\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )-45 (-1)^{3/4} \sqrt [4]{5} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-5*(10 + 45*x^2 + 2*x^4 + 9*x^6))/(x^3*Sqrt[5 + x^4]) - 45*(-1)^(3/4)*5^(1/4)*
EllipticE[I*ArcSinh[(-1/5)^(1/4)*x], -1] + (-5)^(1/4)*(45*I + 2*Sqrt[5])*Ellipti
cF[I*ArcSinh[(-1/5)^(1/4)*x], -1])/75

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Maple [C]  time = 0.024, size = 170, normalized size = 0.9 \[ -{\frac{2}{15\,{x}^{3}}\sqrt{{x}^{4}+5}}-{\frac{2\,\sqrt{5}}{375\,\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ){\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{3}{5\,x}\sqrt{{x}^{4}+5}}+{\frac{{\frac{3\,i}{25}}}{\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)/x^4/(x^4+5)^(1/2),x)

[Out]

-2/15*(x^4+5)^(1/2)/x^3-2/375*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)-3/5*(x^4+5)^(1/2)/x+3/25*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. \[ \int \frac{3 \, x^{2} + 2}{\sqrt{x^{4} + 5} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.17723, size = 80, normalized size = 0.42 \[ \frac{3 \sqrt{5} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{20 x \Gamma \left (\frac{3}{4}\right )} + \frac{\sqrt{5} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{10 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2)/x**4/(x**4+5)**(1/2),x)

[Out]

3*sqrt(5)*gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), x**4*exp_polar(I*pi)/5)/(20*x*g
amma(3/4)) + sqrt(5)*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), x**4*exp_polar(I*pi)
/5)/(10*x**3*gamma(1/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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