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

Optimal. Leaf size=219 \[ \frac{300}{77} \sqrt{x^4+5} x+\frac{40 \sqrt{x^4+5} x}{3 \left (x^2+\sqrt{5}\right )}+\frac{10 \sqrt [4]{5} \left (154-45 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{231 \sqrt{x^4+5}}-\frac{40 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{3 \sqrt{x^4+5}}+\frac{1}{99} \left (27 x^2+22\right ) \left (x^4+5\right )^{3/2} x^3+\frac{2}{231} \left (135 x^2+154\right ) \sqrt{x^4+5} x^3 \]

[Out]

(300*x*Sqrt[5 + x^4])/77 + (40*x*Sqrt[5 + x^4])/(3*(Sqrt[5] + x^2)) + (2*x^3*(15
4 + 135*x^2)*Sqrt[5 + x^4])/231 + (x^3*(22 + 27*x^2)*(5 + x^4)^(3/2))/99 - (40*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])/(3*Sqrt[5 + x^4]) + (10*5^(1/4)*(154 - 45*Sqrt[5])*(Sqrt[5] + x^2)
*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticF[2*ArcTan[x/5^(1/4)], 1/2])/(231*Sqr
t[5 + x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.29666, antiderivative size = 219, 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 \[ \frac{300}{77} \sqrt{x^4+5} x+\frac{40 \sqrt{x^4+5} x}{3 \left (x^2+\sqrt{5}\right )}+\frac{10 \sqrt [4]{5} \left (154-45 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{231 \sqrt{x^4+5}}-\frac{40 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{3 \sqrt{x^4+5}}+\frac{1}{99} \left (27 x^2+22\right ) \left (x^4+5\right )^{3/2} x^3+\frac{2}{231} \left (135 x^2+154\right ) \sqrt{x^4+5} x^3 \]

Antiderivative was successfully verified.

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

[Out]

(300*x*Sqrt[5 + x^4])/77 + (40*x*Sqrt[5 + x^4])/(3*(Sqrt[5] + x^2)) + (2*x^3*(15
4 + 135*x^2)*Sqrt[5 + x^4])/231 + (x^3*(22 + 27*x^2)*(5 + x^4)^(3/2))/99 - (40*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])/(3*Sqrt[5 + x^4]) + (10*5^(1/4)*(154 - 45*Sqrt[5])*(Sqrt[5] + x^2)
*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticF[2*ArcTan[x/5^(1/4)], 1/2])/(231*Sqr
t[5 + x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.8658, size = 219, normalized size = 1. \[ \frac{x^{3} \left (27 x^{2} + 22\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{99} + \frac{2 x^{3} \left (135 x^{2} + 154\right ) \sqrt{x^{4} + 5}}{231} + \frac{300 x \sqrt{x^{4} + 5}}{77} + \frac{40 x \sqrt{x^{4} + 5}}{3 \left (x^{2} + \sqrt{5}\right )} - \frac{40 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{x^{4} + 5}} + \frac{10 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (- 135 \sqrt{5} + 462\right ) \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{693 \sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*(27*x**2 + 22)*(x**4 + 5)**(3/2)/99 + 2*x**3*(135*x**2 + 154)*sqrt(x**4 + 5
)/231 + 300*x*sqrt(x**4 + 5)/77 + 40*x*sqrt(x**4 + 5)/(3*(x**2 + sqrt(5))) - 40*
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)/(3*sqrt(x**4 + 5)) + 10*5**(1/4)*sqrt((x**4 + 5)/(s
qrt(5)*x**2/5 + 1)**2)*(-135*sqrt(5) + 462)*(sqrt(5)*x**2/5 + 1)*elliptic_f(2*at
an(5**(3/4)*x/5), 1/2)/(693*sqrt(x**4 + 5))

_______________________________________________________________________________________

Mathematica [C]  time = 0.230519, size = 110, normalized size = 0.5 \[ \frac{1}{693} \left (\frac{x \left (189 x^{12}+154 x^{10}+2700 x^8+2464 x^6+11475 x^4+8470 x^2+13500\right )}{\sqrt{x^4+5}}+60 \sqrt [4]{-5} \left (45 \sqrt{5}+154 i\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )-9240 (-1)^{3/4} \sqrt [4]{5} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((x*(13500 + 8470*x^2 + 11475*x^4 + 2464*x^6 + 2700*x^8 + 154*x^10 + 189*x^12))/
Sqrt[5 + x^4] - 9240*(-1)^(3/4)*5^(1/4)*EllipticE[I*ArcSinh[(-1/5)^(1/4)*x], -1]
+ 60*(-5)^(1/4)*(154*I + 45*Sqrt[5])*EllipticF[I*ArcSinh[(-1/5)^(1/4)*x], -1])/
693

_______________________________________________________________________________________

Maple [C]  time = 0.017, size = 204, normalized size = 0.9 \[{\frac{2\,{x}^{7}}{9}\sqrt{{x}^{4}+5}}+{\frac{22\,{x}^{3}}{9}\sqrt{{x}^{4}+5}}+{\frac{{\frac{8\,i}{3}}}{\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{3\,{x}^{9}}{11}\sqrt{{x}^{4}+5}}+{\frac{195\,{x}^{5}}{77}\sqrt{{x}^{4}+5}}+{\frac{300\,x}{77}\sqrt{{x}^{4}+5}}-{\frac{60\,\sqrt{5}}{77\,\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ){\frac{1}{\sqrt{{x}^{4}+5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*x^7*(x^4+5)^(1/2)+22/9*x^3*(x^4+5)^(1/2)+8/3*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))+3/11*x^9*(x
^4+5)^(1/2)+195/77*x^5*(x^4+5)^(1/2)+300/77*x*(x^4+5)^(1/2)-60/77*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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (3 \, x^{8} + 2 \, x^{6} + 15 \, x^{4} + 10 \, x^{2}\right )} \sqrt{x^{4} + 5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((3*x^8 + 2*x^6 + 15*x^4 + 10*x^2)*sqrt(x^4 + 5), x)

_______________________________________________________________________________________

Sympy [A]  time = 8.01579, size = 160, normalized size = 0.73 \[ \frac{3 \sqrt{5} x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} + \frac{\sqrt{5} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{2 \Gamma \left (\frac{11}{4}\right )} + \frac{15 \sqrt{5} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{5 \sqrt{5} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{2 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(5)*x**9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), x**4*exp_polar(I*pi)/5)/(4
*gamma(13/4)) + sqrt(5)*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), x**4*exp_pol
ar(I*pi)/5)/(2*gamma(11/4)) + 15*sqrt(5)*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4
,), x**4*exp_polar(I*pi)/5)/(4*gamma(9/4)) + 5*sqrt(5)*x**3*gamma(3/4)*hyper((-1
/2, 3/4), (7/4,), x**4*exp_polar(I*pi)/5)/(2*gamma(7/4))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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