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

Optimal. Leaf size=67 \[ \frac{3}{10} \left (x^4+5\right )^{3/2} x^4-\frac{25}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{5}{8} \sqrt{x^4+5} x^2-\frac{1}{4} \left (4-x^2\right ) \left (x^4+5\right )^{3/2} \]

[Out]

(-5*x^2*Sqrt[5 + x^4])/8 + (3*x^4*(5 + x^4)^(3/2))/10 - ((4 - x^2)*(5 + x^4)^(3/
2))/4 - (25*ArcSinh[x^2/Sqrt[5]])/8

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Rubi [A]  time = 0.144855, antiderivative size = 67, 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 \[ \frac{3}{10} \left (x^4+5\right )^{3/2} x^4-\frac{25}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{5}{8} \sqrt{x^4+5} x^2-\frac{1}{4} \left (4-x^2\right ) \left (x^4+5\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-5*x^2*Sqrt[5 + x^4])/8 + (3*x^4*(5 + x^4)^(3/2))/10 - ((4 - x^2)*(5 + x^4)^(3/
2))/4 - (25*ArcSinh[x^2/Sqrt[5]])/8

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Rubi in Sympy [A]  time = 11.3758, size = 61, normalized size = 0.91 \[ \frac{3 x^{4} \left (x^{4} + 5\right )^{\frac{3}{2}}}{10} - \frac{5 x^{2} \sqrt{x^{4} + 5}}{8} - \frac{\left (- 30 x^{2} + 120\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{120} - \frac{25 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**4*(x**4 + 5)**(3/2)/10 - 5*x**2*sqrt(x**4 + 5)/8 - (-30*x**2 + 120)*(x**4 +
5)**(3/2)/120 - 25*asinh(sqrt(5)*x**2/5)/8

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Mathematica [A]  time = 0.0345895, size = 54, normalized size = 0.81 \[ \frac{1}{2} \sqrt{x^4+5} \left (\frac{3 x^8}{5}+\frac{x^6}{2}+x^4+\frac{5 x^2}{4}-10\right )-\frac{25}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[5 + x^4]*(-10 + (5*x^2)/4 + x^4 + x^6/2 + (3*x^8)/5))/2 - (25*ArcSinh[x^2/
Sqrt[5]])/8

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Maple [A]  time = 0.028, size = 53, normalized size = 0.8 \[{\frac{{x}^{2}}{4} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{x}^{2}}{8}\sqrt{{x}^{4}+5}}-{\frac{25}{8}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) }+{\frac{3\,{x}^{4}-10}{10} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^2*(x^4+5)^(3/2)-5/8*x^2*(x^4+5)^(1/2)-25/8*arcsinh(1/5*5^(1/2)*x^2)+1/10*(
x^4+5)^(3/2)*(3*x^4-10)

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Maxima [A]  time = 0.779942, size = 138, normalized size = 2.06 \[ \frac{3}{10} \,{\left (x^{4} + 5\right )}^{\frac{5}{2}} - \frac{5}{2} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} - \frac{25 \,{\left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + \frac{{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}}{8 \,{\left (\frac{2 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} - \frac{25}{16} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) + \frac{25}{16} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/10*(x^4 + 5)^(5/2) - 5/2*(x^4 + 5)^(3/2) - 25/8*(sqrt(x^4 + 5)/x^2 + (x^4 + 5)
^(3/2)/x^6)/(2*(x^4 + 5)/x^4 - (x^4 + 5)^2/x^8 - 1) - 25/16*log(sqrt(x^4 + 5)/x^
2 + 1) + 25/16*log(sqrt(x^4 + 5)/x^2 - 1)

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Fricas [A]  time = 0.262343, size = 271, normalized size = 4.04 \[ -\frac{192 \, x^{20} + 160 \, x^{18} + 2000 \, x^{16} + 1800 \, x^{14} + 3500 \, x^{12} + 6750 \, x^{10} - 20000 \, x^{8} + 9375 \, x^{6} - 62500 \, x^{4} + 3125 \, x^{2} - 125 \,{\left (16 \, x^{10} + 100 \, x^{6} + 125 \, x^{2} -{\left (16 \, x^{8} + 60 \, x^{4} + 25\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (192 \, x^{18} + 160 \, x^{16} + 1520 \, x^{14} + 1400 \, x^{12} + 300 \, x^{10} + 3750 \, x^{8} - 17500 \, x^{6} + 3125 \, x^{4} - 25000 \, x^{2}\right )} \sqrt{x^{4} + 5} - 25000}{40 \,{\left (16 \, x^{10} + 100 \, x^{6} + 125 \, x^{2} -{\left (16 \, x^{8} + 60 \, x^{4} + 25\right )} \sqrt{x^{4} + 5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40*(192*x^20 + 160*x^18 + 2000*x^16 + 1800*x^14 + 3500*x^12 + 6750*x^10 - 200
00*x^8 + 9375*x^6 - 62500*x^4 + 3125*x^2 - 125*(16*x^10 + 100*x^6 + 125*x^2 - (1
6*x^8 + 60*x^4 + 25)*sqrt(x^4 + 5))*log(-x^2 + sqrt(x^4 + 5)) - (192*x^18 + 160*
x^16 + 1520*x^14 + 1400*x^12 + 300*x^10 + 3750*x^8 - 17500*x^6 + 3125*x^4 - 2500
0*x^2)*sqrt(x^4 + 5) - 25000)/(16*x^10 + 100*x^6 + 125*x^2 - (16*x^8 + 60*x^4 +
25)*sqrt(x^4 + 5))

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Sympy [A]  time = 13.4333, size = 97, normalized size = 1.45 \[ \frac{x^{10}}{4 \sqrt{x^{4} + 5}} + \frac{3 x^{8} \sqrt{x^{4} + 5}}{10} + \frac{15 x^{6}}{8 \sqrt{x^{4} + 5}} + \frac{x^{4} \sqrt{x^{4} + 5}}{2} + \frac{25 x^{2}}{8 \sqrt{x^{4} + 5}} - 5 \sqrt{x^{4} + 5} - \frac{25 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**10/(4*sqrt(x**4 + 5)) + 3*x**8*sqrt(x**4 + 5)/10 + 15*x**6/(8*sqrt(x**4 + 5))
+ x**4*sqrt(x**4 + 5)/2 + 25*x**2/(8*sqrt(x**4 + 5)) - 5*sqrt(x**4 + 5) - 25*as
inh(sqrt(5)*x**2/5)/8

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GIAC/XCAS [A]  time = 0.264544, size = 70, normalized size = 1.04 \[ \frac{1}{40} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left ({\left (6 \, x^{2} + 5\right )} x^{2} + 10\right )} x^{2} + 25\right )} x^{2} - 200\right )} + \frac{25}{8} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40*sqrt(x^4 + 5)*((2*((6*x^2 + 5)*x^2 + 10)*x^2 + 25)*x^2 - 200) + 25/8*ln(-x^
2 + sqrt(x^4 + 5))