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

Optimal. Leaf size=81 \[ -\frac{15}{2} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{15}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\left (2-x^2\right ) \left (x^4+5\right )^{3/2}}{2 x^2}+\frac{3}{2} \left (x^2+5\right ) \sqrt{x^4+5} \]

[Out]

(3*(5 + x^2)*Sqrt[5 + x^4])/2 - ((2 - x^2)*(5 + x^4)^(3/2))/(2*x^2) + (15*ArcSin
h[x^2/Sqrt[5]])/2 - (15*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/2

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Rubi [A]  time = 0.190416, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{15}{2} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{15}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\left (2-x^2\right ) \left (x^4+5\right )^{3/2}}{2 x^2}+\frac{3}{2} \left (x^2+5\right ) \sqrt{x^4+5} \]

Antiderivative was successfully verified.

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

[Out]

(3*(5 + x^2)*Sqrt[5 + x^4])/2 - ((2 - x^2)*(5 + x^4)^(3/2))/(2*x^2) + (15*ArcSin
h[x^2/Sqrt[5]])/2 - (15*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/2

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Rubi in Sympy [A]  time = 16.7501, size = 76, normalized size = 0.94 \[ \frac{\left (12 x^{2} + 60\right ) \sqrt{x^{4} + 5}}{8} + \frac{15 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} - \frac{15 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )}}{2} - \frac{\left (- 3 x^{2} + 6\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{6 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(12*x**2 + 60)*sqrt(x**4 + 5)/8 + 15*asinh(sqrt(5)*x**2/5)/2 - 15*sqrt(5)*atanh(
sqrt(5)*sqrt(x**4 + 5)/5)/2 - (-3*x**2 + 6)*(x**4 + 5)**(3/2)/(6*x**2)

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Mathematica [A]  time = 0.122688, size = 66, normalized size = 0.81 \[ \frac{1}{2} \left (-15 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+15 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{\sqrt{x^4+5} \left (x^6+x^4+20 x^2-10\right )}{x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[5 + x^4]*(-10 + 20*x^2 + x^4 + x^6))/x^2 + 15*ArcSinh[x^2/Sqrt[5]] - 15*S
qrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/2

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Maple [A]  time = 0.023, size = 75, normalized size = 0.9 \[{\frac{{x}^{2}}{2}\sqrt{{x}^{4}+5}}+{\frac{15}{2}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) }-5\,{\frac{\sqrt{{x}^{4}+5}}{{x}^{2}}}+{\frac{{x}^{4}}{2}\sqrt{{x}^{4}+5}}+10\,\sqrt{{x}^{4}+5}-{\frac{15\,\sqrt{5}}{2}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2*(x^4+5)^(1/2)+15/2*arcsinh(1/5*5^(1/2)*x^2)-5*(x^4+5)^(1/2)/x^2+1/2*x^4*
(x^4+5)^(1/2)+10*(x^4+5)^(1/2)-15/2*5^(1/2)*arctanh(5^(1/2)/(x^4+5)^(1/2))

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Maxima [A]  time = 0.7891, size = 165, normalized size = 2.04 \[ \frac{1}{2} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} + \frac{15}{4} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) + \frac{15}{2} \, \sqrt{x^{4} + 5} - \frac{5 \, \sqrt{x^{4} + 5}}{x^{2}} + \frac{5 \, \sqrt{x^{4} + 5}}{2 \, x^{2}{\left (\frac{x^{4} + 5}{x^{4}} - 1\right )}} + \frac{15}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{15}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^4 + 5)^(3/2) + 15/4*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5) + sqr
t(x^4 + 5))) + 15/2*sqrt(x^4 + 5) - 5*sqrt(x^4 + 5)/x^2 + 5/2*sqrt(x^4 + 5)/(x^2
*((x^4 + 5)/x^4 - 1)) + 15/4*log(sqrt(x^4 + 5)/x^2 + 1) - 15/4*log(sqrt(x^4 + 5)
/x^2 - 1)

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Fricas [A]  time = 0.299306, size = 370, normalized size = 4.57 \[ -\frac{8 \, x^{16} + 8 \, x^{14} + 220 \, x^{12} + 60 \, x^{10} + 1300 \, x^{8} - 100 \, x^{6} + 2000 \, x^{4} - 750 \, x^{2} + 15 \,{\left (8 \, x^{10} + 40 \, x^{6} + 25 \, x^{2} - 4 \,{\left (2 \, x^{8} + 5 \, x^{4}\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) + 15 \,{\left (4 \, \sqrt{5}{\left (2 \, x^{8} + 5 \, x^{4}\right )} \sqrt{x^{4} + 5} - \sqrt{5}{\left (8 \, x^{10} + 40 \, x^{6} + 25 \, x^{2}\right )}\right )} \log \left (\frac{x^{4} + \sqrt{5} x^{2} - \sqrt{x^{4} + 5}{\left (x^{2} + \sqrt{5}\right )} + 5}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) -{\left (8 \, x^{14} + 8 \, x^{12} + 200 \, x^{10} + 40 \, x^{8} + 825 \, x^{6} - 175 \, x^{4} + 500 \, x^{2} - 250\right )} \sqrt{x^{4} + 5}}{2 \,{\left (8 \, x^{10} + 40 \, x^{6} + 25 \, x^{2} - 4 \,{\left (2 \, x^{8} + 5 \, x^{4}\right )} \sqrt{x^{4} + 5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(8*x^16 + 8*x^14 + 220*x^12 + 60*x^10 + 1300*x^8 - 100*x^6 + 2000*x^4 - 750
*x^2 + 15*(8*x^10 + 40*x^6 + 25*x^2 - 4*(2*x^8 + 5*x^4)*sqrt(x^4 + 5))*log(-x^2
+ sqrt(x^4 + 5)) + 15*(4*sqrt(5)*(2*x^8 + 5*x^4)*sqrt(x^4 + 5) - sqrt(5)*(8*x^10
+ 40*x^6 + 25*x^2))*log((x^4 + sqrt(5)*x^2 - sqrt(x^4 + 5)*(x^2 + sqrt(5)) + 5)
/(x^4 - sqrt(x^4 + 5)*x^2)) - (8*x^14 + 8*x^12 + 200*x^10 + 40*x^8 + 825*x^6 - 1
75*x^4 + 500*x^2 - 250)*sqrt(x^4 + 5))/(8*x^10 + 40*x^6 + 25*x^2 - 4*(2*x^8 + 5*
x^4)*sqrt(x^4 + 5))

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Sympy [A]  time = 18.9371, size = 114, normalized size = 1.41 \[ \frac{x^{6}}{2 \sqrt{x^{4} + 5}} - \frac{5 x^{2}}{2 \sqrt{x^{4} + 5}} + \frac{\left (x^{4} + 5\right )^{\frac{3}{2}}}{2} + \frac{15 \sqrt{x^{4} + 5}}{2} + \frac{15 \sqrt{5} \log{\left (x^{4} \right )}}{4} - \frac{15 \sqrt{5} \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )}}{2} + \frac{15 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} - \frac{25}{x^{2} \sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**6/(2*sqrt(x**4 + 5)) - 5*x**2/(2*sqrt(x**4 + 5)) + (x**4 + 5)**(3/2)/2 + 15*s
qrt(x**4 + 5)/2 + 15*sqrt(5)*log(x**4)/4 - 15*sqrt(5)*log(sqrt(x**4/5 + 1) + 1)/
2 + 15*asinh(sqrt(5)*x**2/5)/2 - 25/(x**2*sqrt(x**4 + 5))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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