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

Optimal. Leaf size=180 \[ -\frac{3 \sqrt{x^4+5} x}{10 \left (x^2+\sqrt{5}\right )}+\frac{\left (3 x^2+2\right ) x}{10 \sqrt{x^4+5}}+\frac{\left (2-3 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{20 \sqrt [4]{5} \sqrt{x^4+5}}+\frac{3 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{2\ 5^{3/4} \sqrt{x^4+5}} \]

[Out]

(x*(2 + 3*x^2))/(10*Sqrt[5 + x^4]) - (3*x*Sqrt[5 + x^4])/(10*(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])/(2*5^(3/4)*Sqrt[5 + x^4]) + ((2 - 3*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^(1/4)*Sqrt[5
+ x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.13435, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{3 \sqrt{x^4+5} x}{10 \left (x^2+\sqrt{5}\right )}+\frac{\left (3 x^2+2\right ) x}{10 \sqrt{x^4+5}}+\frac{\left (2-3 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{20 \sqrt [4]{5} \sqrt{x^4+5}}+\frac{3 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{2\ 5^{3/4} \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(2 + 3*x^2))/(10*Sqrt[5 + x^4]) - (3*x*Sqrt[5 + x^4])/(10*(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])/(2*5^(3/4)*Sqrt[5 + x^4]) + ((2 - 3*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^(1/4)*Sqrt[5
+ x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.7199, size = 182, normalized size = 1.01 \[ \frac{x \left (3 x^{2} + 2\right )}{10 \sqrt{x^{4} + 5}} - \frac{3 x \sqrt{x^{4} + 5}}{10 \left (x^{2} + \sqrt{5}\right )} + \frac{3 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{10 \sqrt{x^{4} + 5}} - \frac{\sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (- \frac{2 \sqrt{5}}{5} + 3\right ) \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{20 \sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(3*x**2 + 2)/(10*sqrt(x**4 + 5)) - 3*x*sqrt(x**4 + 5)/(10*(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)*ellipti
c_e(2*atan(5**(3/4)*x/5), 1/2)/(10*sqrt(x**4 + 5)) - 5**(1/4)*sqrt((x**4 + 5)/(s
qrt(5)*x**2/5 + 1)**2)*(-2*sqrt(5)/5 + 3)*(sqrt(5)*x**2/5 + 1)*elliptic_f(2*atan
(5**(3/4)*x/5), 1/2)/(20*sqrt(x**4 + 5))

_______________________________________________________________________________________

Mathematica [C]  time = 0.172831, size = 86, normalized size = 0.48 \[ \frac{1}{50} \left (\frac{5 x \left (3 x^2+2\right )}{\sqrt{x^4+5}}-\sqrt [4]{-5} \left (2 \sqrt{5}+15 i\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )+15 (-1)^{3/4} \sqrt [4]{5} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.018, size = 168, normalized size = 0.9 \[{\frac{x}{5}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{\sqrt{5}}{125\,\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ){\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{3\,{x}^{3}}{10}{\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{{\frac{3\,i}{50}}}{\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)/(x^4+5)^(3/2),x)

[Out]

1/5*x/(x^4+5)^(1/2)+1/125*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)+3/10*x^3/(x^4+5)^(1/2)-3/50*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))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 10.6093, size = 73, normalized size = 0.41 \[ \frac{3 \sqrt{5} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{100 \Gamma \left (\frac{7}{4}\right )} + \frac{\sqrt{5} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{50 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(5)*x**3*gamma(3/4)*hyper((3/4, 3/2), (7/4,), x**4*exp_polar(I*pi)/5)/(100
*gamma(7/4)) + sqrt(5)*x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), x**4*exp_polar(I*p
i)/5)/(50*gamma(5/4))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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