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

Optimal. Leaf size=235 \[ \frac{200}{77} \sqrt{x^4+5} x+\frac{20}{13} \sqrt{x^4+5} x^3-\frac{300 \sqrt{x^4+5} x}{13 \left (x^2+\sqrt{5}\right )}-\frac{50 \sqrt [4]{5} \left (231+26 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{1001 \sqrt{x^4+5}}+\frac{300 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{13 \sqrt{x^4+5}}+\frac{1}{143} \left (33 x^2+26\right ) \left (x^4+5\right )^{3/2} x^5+\frac{10 \left (77 x^2+78\right ) \sqrt{x^4+5} x^5}{1001} \]

[Out]

(200*x*Sqrt[5 + x^4])/77 + (20*x^3*Sqrt[5 + x^4])/13 - (300*x*Sqrt[5 + x^4])/(13
*(Sqrt[5] + x^2)) + (10*x^5*(78 + 77*x^2)*Sqrt[5 + x^4])/1001 + (x^5*(26 + 33*x^
2)*(5 + x^4)^(3/2))/143 + (300*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])/(13*Sqrt[5 + x^4]) - (50*5^(1/4)*(
231 + 26*Sqrt[5])*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticF[2*
ArcTan[x/5^(1/4)], 1/2])/(1001*Sqrt[5 + x^4])

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Rubi [A]  time = 0.337565, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{200}{77} \sqrt{x^4+5} x+\frac{20}{13} \sqrt{x^4+5} x^3-\frac{300 \sqrt{x^4+5} x}{13 \left (x^2+\sqrt{5}\right )}-\frac{50 \sqrt [4]{5} \left (231+26 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{1001 \sqrt{x^4+5}}+\frac{300 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{13 \sqrt{x^4+5}}+\frac{1}{143} \left (33 x^2+26\right ) \left (x^4+5\right )^{3/2} x^5+\frac{10 \left (77 x^2+78\right ) \sqrt{x^4+5} x^5}{1001} \]

Antiderivative was successfully verified.

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

[Out]

(200*x*Sqrt[5 + x^4])/77 + (20*x^3*Sqrt[5 + x^4])/13 - (300*x*Sqrt[5 + x^4])/(13
*(Sqrt[5] + x^2)) + (10*x^5*(78 + 77*x^2)*Sqrt[5 + x^4])/1001 + (x^5*(26 + 33*x^
2)*(5 + x^4)^(3/2))/143 + (300*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])/(13*Sqrt[5 + x^4]) - (50*5^(1/4)*(
231 + 26*Sqrt[5])*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticF[2*
ArcTan[x/5^(1/4)], 1/2])/(1001*Sqrt[5 + x^4])

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Rubi in Sympy [A]  time = 31.0299, size = 235, normalized size = 1. \[ \frac{x^{5} \left (33 x^{2} + 26\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{143} + \frac{10 x^{5} \left (231 x^{2} + 234\right ) \sqrt{x^{4} + 5}}{3003} + \frac{20 x^{3} \sqrt{x^{4} + 5}}{13} + \frac{200 x \sqrt{x^{4} + 5}}{77} - \frac{300 x \sqrt{x^{4} + 5}}{13 \left (x^{2} + \sqrt{5}\right )} + \frac{300 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{13 \sqrt{x^{4} + 5}} - \frac{10 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (1170 \sqrt{5} + 10395\right ) \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{9009 \sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*(33*x**2 + 26)*(x**4 + 5)**(3/2)/143 + 10*x**5*(231*x**2 + 234)*sqrt(x**4 +
5)/3003 + 20*x**3*sqrt(x**4 + 5)/13 + 200*x*sqrt(x**4 + 5)/77 - 300*x*sqrt(x**4
+ 5)/(13*(x**2 + sqrt(5))) + 300*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)/(13*sqrt(x**4 + 5
)) - 10*5**(1/4)*sqrt((x**4 + 5)/(sqrt(5)*x**2/5 + 1)**2)*(1170*sqrt(5) + 10395)
*(sqrt(5)*x**2/5 + 1)*elliptic_f(2*atan(5**(3/4)*x/5), 1/2)/(9009*sqrt(x**4 + 5)
)

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Mathematica [C]  time = 0.229889, size = 115, normalized size = 0.49 \[ \frac{\frac{x \left (231 x^{14}+182 x^{12}+3080 x^{10}+2600 x^8+11165 x^6+11050 x^4+7700 x^2+13000\right )}{\sqrt{x^4+5}}+100 \sqrt [4]{-5} \left (26 \sqrt{5}-231 i\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )+23100 (-1)^{3/4} \sqrt [4]{5} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )}{1001} \]

Antiderivative was successfully verified.

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

[Out]

((x*(13000 + 7700*x^2 + 11050*x^4 + 11165*x^6 + 2600*x^8 + 3080*x^10 + 182*x^12
+ 231*x^14))/Sqrt[5 + x^4] + 23100*(-1)^(3/4)*5^(1/4)*EllipticE[I*ArcSinh[(-1/5)
^(1/4)*x], -1] + 100*(-5)^(1/4)*(-231*I + 26*Sqrt[5])*EllipticF[I*ArcSinh[(-1/5)
^(1/4)*x], -1])/1001

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Maple [C]  time = 0.025, size = 216, normalized size = 0.9 \[{\frac{2\,{x}^{9}}{11}\sqrt{{x}^{4}+5}}+{\frac{130\,{x}^{5}}{77}\sqrt{{x}^{4}+5}}+{\frac{200\,x}{77}\sqrt{{x}^{4}+5}}-{\frac{40\,\sqrt{5}}{77\,\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ){\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{3\,{x}^{11}}{13}\sqrt{{x}^{4}+5}}+{\frac{25\,{x}^{7}}{13}\sqrt{{x}^{4}+5}}+{\frac{20\,{x}^{3}}{13}\sqrt{{x}^{4}+5}}-{\frac{{\frac{60\,i}{13}}}{\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*x^9*(x^4+5)^(1/2)+130/77*x^5*(x^4+5)^(1/2)+200/77*x*(x^4+5)^(1/2)-40/77*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)+3/13*x^11*(x^4+5)^(1/2)+
25/13*x^7*(x^4+5)^(1/2)+20/13*x^3*(x^4+5)^(1/2)-60/13*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. \[ \int{\left (x^{4} + 5\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.9589, size = 160, normalized size = 0.68 \[ \frac{3 \sqrt{5} x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} + \frac{\sqrt{5} x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{2 \Gamma \left (\frac{13}{4}\right )} + \frac{15 \sqrt{5} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} + \frac{5 \sqrt{5} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{2 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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