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

Optimal. Leaf size=214 \[ -\frac{9 \sqrt{x^4+5}}{50 x}-\frac{\sqrt{x^4+5}}{15 x^3}+\frac{9 \sqrt{x^4+5} x}{50 \left (x^2+\sqrt{5}\right )}+\frac{\left (27-2 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{60\ 5^{3/4} \sqrt{x^4+5}}-\frac{9 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{10\ 5^{3/4} \sqrt{x^4+5}}+\frac{3 x^2+2}{10 \sqrt{x^4+5} x^3} \]

[Out]

(2 + 3*x^2)/(10*x^3*Sqrt[5 + x^4]) - Sqrt[5 + x^4]/(15*x^3) - (9*Sqrt[5 + x^4])/
(50*x) + (9*x*Sqrt[5 + x^4])/(50*(Sqrt[5] + x^2)) - (9*(Sqrt[5] + x^2)*Sqrt[(5 +
x^4)/(Sqrt[5] + x^2)^2]*EllipticE[2*ArcTan[x/5^(1/4)], 1/2])/(10*5^(3/4)*Sqrt[5
+ x^4]) + ((27 - 2*Sqrt[5])*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*E
llipticF[2*ArcTan[x/5^(1/4)], 1/2])/(60*5^(3/4)*Sqrt[5 + x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.270152, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{9 \sqrt{x^4+5}}{50 x}-\frac{\sqrt{x^4+5}}{15 x^3}+\frac{9 \sqrt{x^4+5} x}{50 \left (x^2+\sqrt{5}\right )}+\frac{\left (27-2 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{60\ 5^{3/4} \sqrt{x^4+5}}-\frac{9 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{10\ 5^{3/4} \sqrt{x^4+5}}+\frac{3 x^2+2}{10 \sqrt{x^4+5} x^3} \]

Antiderivative was successfully verified.

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

[Out]

(2 + 3*x^2)/(10*x^3*Sqrt[5 + x^4]) - Sqrt[5 + x^4]/(15*x^3) - (9*Sqrt[5 + x^4])/
(50*x) + (9*x*Sqrt[5 + x^4])/(50*(Sqrt[5] + x^2)) - (9*(Sqrt[5] + x^2)*Sqrt[(5 +
x^4)/(Sqrt[5] + x^2)^2]*EllipticE[2*ArcTan[x/5^(1/4)], 1/2])/(10*5^(3/4)*Sqrt[5
+ x^4]) + ((27 - 2*Sqrt[5])*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*E
llipticF[2*ArcTan[x/5^(1/4)], 1/2])/(60*5^(3/4)*Sqrt[5 + x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.4258, size = 209, normalized size = 0.98 \[ \frac{9 x \sqrt{x^{4} + 5}}{50 \left (x^{2} + \sqrt{5}\right )} - \frac{9 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{50 \sqrt{x^{4} + 5}} + \frac{\sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (- 10 \sqrt{5} + 135\right ) \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{1500 \sqrt{x^{4} + 5}} - \frac{9 \sqrt{x^{4} + 5}}{50 x} + \frac{3 x^{2} + 2}{10 x^{3} \sqrt{x^{4} + 5}} - \frac{\sqrt{x^{4} + 5}}{15 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*x*sqrt(x**4 + 5)/(50*(x**2 + sqrt(5))) - 9*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)/(50*sq
rt(x**4 + 5)) + 5**(1/4)*sqrt((x**4 + 5)/(sqrt(5)*x**2/5 + 1)**2)*(-10*sqrt(5) +
135)*(sqrt(5)*x**2/5 + 1)*elliptic_f(2*atan(5**(3/4)*x/5), 1/2)/(1500*sqrt(x**4
+ 5)) - 9*sqrt(x**4 + 5)/(50*x) + (3*x**2 + 2)/(10*x**3*sqrt(x**4 + 5)) - sqrt(
x**4 + 5)/(15*x**3)

_______________________________________________________________________________________

Mathematica [C]  time = 0.236388, size = 119, normalized size = 0.56 \[ -\frac{27 x^6+10 x^4+90 x^2-\sqrt [4]{-5} \left (2 \sqrt{5}+27 i\right ) \sqrt{x^4+5} x^3 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )+27 (-1)^{3/4} \sqrt [4]{5} \sqrt{x^4+5} x^3 E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )+20}{150 x^3 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

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

[Out]

-(20 + 90*x^2 + 10*x^4 + 27*x^6 + 27*(-1)^(3/4)*5^(1/4)*x^3*Sqrt[5 + x^4]*Ellipt
icE[I*ArcSinh[(-1/5)^(1/4)*x], -1] - (-5)^(1/4)*(27*I + 2*Sqrt[5])*x^3*Sqrt[5 +
x^4]*EllipticF[I*ArcSinh[(-1/5)^(1/4)*x], -1])/(150*x^3*Sqrt[5 + x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.028, size = 192, normalized size = 0.9 \[ -{\frac{2}{75\,{x}^{3}}\sqrt{{x}^{4}+5}}-{\frac{x}{25}{\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{\sqrt{5}}{375\,\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ){\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{3\,{x}^{3}}{50}{\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{3}{25\,x}\sqrt{{x}^{4}+5}}+{\frac{{\frac{9\,i}{250}}}{\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/75*(x^4+5)^(1/2)/x^3-1/25*x/(x^4+5)^(1/2)-1/375*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)-3/50*x^3/(x^4+5)^(1/2)-3/25*(x^4+5)^(1/2)/x+9/250*
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. \[ \int \frac{3 \, x^{2} + 2}{{\left (x^{4} + 5\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{3 \, x^{2} + 2}{{\left (x^{8} + 5 \, x^{4}\right )} \sqrt{x^{4} + 5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((3*x^2 + 2)/((x^8 + 5*x^4)*sqrt(x^4 + 5)), x)

_______________________________________________________________________________________

Sympy [A]  time = 26.5395, size = 80, normalized size = 0.37 \[ \frac{3 \sqrt{5} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{100 x \Gamma \left (\frac{3}{4}\right )} + \frac{\sqrt{5} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{50 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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