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

Optimal. Leaf size=78 \[ -5 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{24} \left (9 x^2+8\right ) \left (x^4+5\right )^{3/2}+\frac{5}{16} \left (9 x^2+16\right ) \sqrt{x^4+5} \]

[Out]

(5*(16 + 9*x^2)*Sqrt[5 + x^4])/16 + ((8 + 9*x^2)*(5 + x^4)^(3/2))/24 + (225*ArcS
inh[x^2/Sqrt[5]])/16 - 5*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]]

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Rubi [A]  time = 0.197324, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -5 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{24} \left (9 x^2+8\right ) \left (x^4+5\right )^{3/2}+\frac{5}{16} \left (9 x^2+16\right ) \sqrt{x^4+5} \]

Antiderivative was successfully verified.

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

[Out]

(5*(16 + 9*x^2)*Sqrt[5 + x^4])/16 + ((8 + 9*x^2)*(5 + x^4)^(3/2))/24 + (225*ArcS
inh[x^2/Sqrt[5]])/16 - 5*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]]

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Rubi in Sympy [A]  time = 16.1055, size = 71, normalized size = 0.91 \[ \frac{\left (9 x^{2} + 8\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{24} + \frac{\left (45 x^{2} + 80\right ) \sqrt{x^{4} + 5}}{16} + \frac{225 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} - 5 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(9*x**2 + 8)*(x**4 + 5)**(3/2)/24 + (45*x**2 + 80)*sqrt(x**4 + 5)/16 + 225*asinh
(sqrt(5)*x**2/5)/16 - 5*sqrt(5)*atanh(sqrt(5)*sqrt(x**4 + 5)/5)

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Mathematica [A]  time = 0.100323, size = 68, normalized size = 0.87 \[ -5 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{48} \sqrt{x^4+5} \left (18 x^6+16 x^4+225 x^2+320\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[5 + x^4]*(320 + 225*x^2 + 16*x^4 + 18*x^6))/48 + (225*ArcSinh[x^2/Sqrt[5]]
)/16 - 5*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]]

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Maple [A]  time = 0.021, size = 75, normalized size = 1. \[{\frac{{x}^{4}}{3}\sqrt{{x}^{4}+5}}+{\frac{20}{3}\sqrt{{x}^{4}+5}}-5\,\sqrt{5}{\it Artanh} \left ({\frac{\sqrt{5}}{\sqrt{{x}^{4}+5}}} \right ) +{\frac{3\,{x}^{6}}{8}\sqrt{{x}^{4}+5}}+{\frac{75\,{x}^{2}}{16}\sqrt{{x}^{4}+5}}+{\frac{225}{16}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^4*(x^4+5)^(1/2)+20/3*(x^4+5)^(1/2)-5*5^(1/2)*arctanh(5^(1/2)/(x^4+5)^(1/2)
)+3/8*x^6*(x^4+5)^(1/2)+75/16*x^2*(x^4+5)^(1/2)+225/16*arcsinh(1/5*5^(1/2)*x^2)

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Maxima [A]  time = 0.785344, size = 186, normalized size = 2.38 \[ \frac{1}{3} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} + \frac{5}{2} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) + 5 \, \sqrt{x^{4} + 5} + \frac{75 \,{\left (\frac{3 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{5 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}}{16 \,{\left (\frac{2 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} + \frac{225}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{225}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x^4 + 5)^(3/2) + 5/2*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5) + sqrt
(x^4 + 5))) + 5*sqrt(x^4 + 5) + 75/16*(3*sqrt(x^4 + 5)/x^2 - 5*(x^4 + 5)^(3/2)/x
^6)/(2*(x^4 + 5)/x^4 - (x^4 + 5)^2/x^8 - 1) + 225/32*log(sqrt(x^4 + 5)/x^2 + 1)
- 225/32*log(sqrt(x^4 + 5)/x^2 - 1)

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Fricas [A]  time = 0.289188, size = 354, normalized size = 4.54 \[ -\frac{144 \, x^{16} + 128 \, x^{14} + 2880 \, x^{12} + 3520 \, x^{10} + 15300 \, x^{8} + 20800 \, x^{6} + 22500 \, x^{4} + 32000 \, x^{2} + 675 \,{\left (8 \, x^{8} + 40 \, x^{4} - 4 \,{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} + 25\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) + 240 \,{\left (4 \, \sqrt{5}{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} - \sqrt{5}{\left (8 \, x^{8} + 40 \, x^{4} + 25\right )}\right )} \log \left (\frac{x^{4} + \sqrt{5} x^{2} - \sqrt{x^{4} + 5}{\left (x^{2} + \sqrt{5}\right )} + 5}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) -{\left (144 \, x^{14} + 128 \, x^{12} + 2520 \, x^{10} + 3200 \, x^{8} + 9450 \, x^{6} + 13200 \, x^{4} + 5625 \, x^{2} + 8000\right )} \sqrt{x^{4} + 5}}{48 \,{\left (8 \, x^{8} + 40 \, x^{4} - 4 \,{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} + 25\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(144*x^16 + 128*x^14 + 2880*x^12 + 3520*x^10 + 15300*x^8 + 20800*x^6 + 225
00*x^4 + 32000*x^2 + 675*(8*x^8 + 40*x^4 - 4*(2*x^6 + 5*x^2)*sqrt(x^4 + 5) + 25)
*log(-x^2 + sqrt(x^4 + 5)) + 240*(4*sqrt(5)*(2*x^6 + 5*x^2)*sqrt(x^4 + 5) - sqrt
(5)*(8*x^8 + 40*x^4 + 25))*log((x^4 + sqrt(5)*x^2 - sqrt(x^4 + 5)*(x^2 + sqrt(5)
) + 5)/(x^4 - sqrt(x^4 + 5)*x^2)) - (144*x^14 + 128*x^12 + 2520*x^10 + 3200*x^8
+ 9450*x^6 + 13200*x^4 + 5625*x^2 + 8000)*sqrt(x^4 + 5))/(8*x^8 + 40*x^4 - 4*(2*
x^6 + 5*x^2)*sqrt(x^4 + 5) + 25)

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Sympy [A]  time = 22.7554, size = 114, normalized size = 1.46 \[ \frac{3 x^{10}}{8 \sqrt{x^{4} + 5}} + \frac{105 x^{6}}{16 \sqrt{x^{4} + 5}} + \frac{375 x^{2}}{16 \sqrt{x^{4} + 5}} + \frac{\left (x^{4} + 5\right )^{\frac{3}{2}}}{3} + 5 \sqrt{x^{4} + 5} + \frac{5 \sqrt{5} \log{\left (x^{4} \right )}}{2} - 5 \sqrt{5} \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )} + \frac{225 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**10/(8*sqrt(x**4 + 5)) + 105*x**6/(16*sqrt(x**4 + 5)) + 375*x**2/(16*sqrt(x*
*4 + 5)) + (x**4 + 5)**(3/2)/3 + 5*sqrt(x**4 + 5) + 5*sqrt(5)*log(x**4)/2 - 5*sq
rt(5)*log(sqrt(x**4/5 + 1) + 1) + 225*asinh(sqrt(5)*x**2/5)/16

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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