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

Optimal. Leaf size=192 \[ -\frac{6 \sqrt{x^4+5}}{x}+\frac{6 \sqrt{x^4+5} x}{x^2+\sqrt{5}}+\frac{\left (2+9 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{5} \sqrt{x^4+5}}-\frac{6 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}}-\frac{\left (2-9 x^2\right ) \sqrt{x^4+5}}{3 x^3} \]

[Out]

(-6*Sqrt[5 + x^4])/x - ((2 - 9*x^2)*Sqrt[5 + x^4])/(3*x^3) + (6*x*Sqrt[5 + x^4])
/(Sqrt[5] + x^2) - (6*5^(1/4)*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*
EllipticE[2*ArcTan[x/5^(1/4)], 1/2])/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*
5^(1/4)*Sqrt[5 + x^4])

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Rubi [A]  time = 0.202889, antiderivative size = 192, 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 \[ -\frac{6 \sqrt{x^4+5}}{x}+\frac{6 \sqrt{x^4+5} x}{x^2+\sqrt{5}}+\frac{\left (2+9 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{5} \sqrt{x^4+5}}-\frac{6 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}}-\frac{\left (2-9 x^2\right ) \sqrt{x^4+5}}{3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[5 + x^4])/x - ((2 - 9*x^2)*Sqrt[5 + x^4])/(3*x^3) + (6*x*Sqrt[5 + x^4])
/(Sqrt[5] + x^2) - (6*5^(1/4)*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*
EllipticE[2*ArcTan[x/5^(1/4)], 1/2])/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*
5^(1/4)*Sqrt[5 + x^4])

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Rubi in Sympy [A]  time = 19.3798, size = 189, normalized size = 0.98 \[ \frac{6 x \sqrt{x^{4} + 5}}{x^{2} + \sqrt{5}} - \frac{6 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{4} + 5}} + \frac{\sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (2 \sqrt{5} + 45\right ) \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{15 \sqrt{x^{4} + 5}} - \frac{6 \sqrt{x^{4} + 5}}{x} - \frac{\left (- 9 x^{2} + 2\right ) \sqrt{x^{4} + 5}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*x*sqrt(x**4 + 5)/(x**2 + sqrt(5)) - 6*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)/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)/(15*sqrt(x**4 + 5)) - 6*sqrt
(x**4 + 5)/x - (-9*x**2 + 2)*sqrt(x**4 + 5)/(3*x**3)

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Mathematica [C]  time = 0.223878, size = 98, normalized size = 0.51 \[ \frac{1}{15} \left (-\frac{5 \left (9 x^6+2 x^4+45 x^2+10\right )}{x^3 \sqrt{x^4+5}}+2 \sqrt [4]{-5} \left (-2 \sqrt{5}+45 i\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )-90 (-1)^{3/4} \sqrt [4]{5} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-5*(10 + 45*x^2 + 2*x^4 + 9*x^6))/(x^3*Sqrt[5 + x^4]) - 90*(-1)^(3/4)*5^(1/4)*
EllipticE[I*ArcSinh[(-1/5)^(1/4)*x], -1] + 2*(-5)^(1/4)*(45*I - 2*Sqrt[5])*Ellip
ticF[I*ArcSinh[(-1/5)^(1/4)*x], -1])/15

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Maple [C]  time = 0.023, size = 170, normalized size = 0.9 \[ -{\frac{2}{3\,{x}^{3}}\sqrt{{x}^{4}+5}}+{\frac{4\,\sqrt{5}}{75\,\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ){\frac{1}{\sqrt{{x}^{4}+5}}}}-3\,{\frac{\sqrt{{x}^{4}+5}}{x}}+{\frac{{\frac{6\,i}{5}}}{\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(x^4+5)^(1/2)/x^3+4/75*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*(x^4+5)^(1/2)/x+6/5*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. \[ \int \frac{\sqrt{x^{4} + 5}{\left (3 \, x^{2} + 2\right )}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.2087, size = 83, normalized size = 0.43 \[ \frac{3 \sqrt{5} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} + \frac{\sqrt{5} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{2 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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