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

Optimal. Leaf size=122 \[ -\frac{\left (2-x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^2}+\frac{3}{16} \left (18 x^2+109\right ) \sqrt{x^4+5 x^2+3}+\frac{609}{32} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-12 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]

[Out]

(3*(109 + 18*x^2)*Sqrt[3 + 5*x^2 + x^4])/16 - ((2 - x^2)*(3 + 5*x^2 + x^4)^(3/2)
)/(2*x^2) + (609*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/32 - 12*Sqrt[3]
*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]

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Rubi [A]  time = 0.266192, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{\left (2-x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{2 x^2}+\frac{3}{16} \left (18 x^2+109\right ) \sqrt{x^4+5 x^2+3}+\frac{609}{32} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-12 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*(109 + 18*x^2)*Sqrt[3 + 5*x^2 + x^4])/16 - ((2 - x^2)*(3 + 5*x^2 + x^4)^(3/2)
)/(2*x^2) + (609*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/32 - 12*Sqrt[3]
*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]

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Rubi in Sympy [A]  time = 26.5, size = 110, normalized size = 0.9 \[ \frac{\left (27 x^{2} + \frac{327}{2}\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{8} + \frac{609 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{32} - 12 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )} - \frac{\left (- 3 x^{2} + 6\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{6 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(27*x**2 + 327/2)*sqrt(x**4 + 5*x**2 + 3)/8 + 609*atanh((2*x**2 + 5)/(2*sqrt(x**
4 + 5*x**2 + 3)))/32 - 12*sqrt(3)*atanh(sqrt(3)*(5*x**2 + 6)/(6*sqrt(x**4 + 5*x*
*2 + 3))) - (-3*x**2 + 6)*(x**4 + 5*x**2 + 3)**(3/2)/(6*x**2)

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Mathematica [A]  time = 0.161482, size = 107, normalized size = 0.88 \[ \frac{1}{16} \sqrt{x^4+5 x^2+3} \left (8 x^4+78 x^2-\frac{48}{x^2}+271\right )+\frac{609}{32} \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right )+12 \sqrt{3} \left (2 \log (x)-\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 5*x^2 + x^4]*(271 - 48/x^2 + 78*x^2 + 8*x^4))/16 + (609*Log[5 + 2*x^2
+ 2*Sqrt[3 + 5*x^2 + x^4]])/32 + 12*Sqrt[3]*(2*Log[x] - Log[6 + 5*x^2 + 2*Sqrt[3
]*Sqrt[3 + 5*x^2 + x^4]])

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Maple [A]  time = 0.025, size = 117, normalized size = 1. \[{\frac{39\,{x}^{2}}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{271}{16}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{609}{32}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }-3\,{\frac{\sqrt{{x}^{4}+5\,{x}^{2}+3}}{{x}^{2}}}-12\,{\it Artanh} \left ( 1/6\,{\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}} \right ) \sqrt{3}+{\frac{{x}^{4}}{2}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

39/8*x^2*(x^4+5*x^2+3)^(1/2)+271/16*(x^4+5*x^2+3)^(1/2)+609/32*ln(x^2+5/2+(x^4+5
*x^2+3)^(1/2))-3*(x^4+5*x^2+3)^(1/2)/x^2-12*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5
*x^2+3)^(1/2))*3^(1/2)+1/2*x^4*(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.819874, size = 162, normalized size = 1.33 \[ \frac{27}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + \frac{1}{2} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} - 12 \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{327}{16} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{x^{2}} + \frac{609}{32} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/8*sqrt(x^4 + 5*x^2 + 3)*x^2 + 1/2*(x^4 + 5*x^2 + 3)^(3/2) - 12*sqrt(3)*log(2*
sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 327/16*sqrt(x^4 + 5*x^2 + 3) -
(x^4 + 5*x^2 + 3)^(3/2)/x^2 + 609/32*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]  time = 0.278944, size = 524, normalized size = 4.3 \[ -\frac{8192 \, x^{16} + 182272 \, x^{14} + 1747968 \, x^{12} + 8805504 \, x^{10} + 23858432 \, x^{8} + 32630816 \, x^{6} + 18064928 \, x^{4} + 300307 \, x^{2} + 2436 \,{\left (128 \, x^{10} + 1280 \, x^{8} + 4384 \, x^{6} + 5920 \, x^{4} + 2569 \, x^{2} - 8 \,{\left (16 \, x^{8} + 120 \, x^{6} + 274 \, x^{4} + 185 \, x^{2}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) + 1536 \,{\left (8 \, \sqrt{3}{\left (16 \, x^{8} + 120 \, x^{6} + 274 \, x^{4} + 185 \, x^{2}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (128 \, x^{10} + 1280 \, x^{8} + 4384 \, x^{6} + 5920 \, x^{4} + 2569 \, x^{2}\right )}\right )} \log \left (\frac{2 \, x^{4} + 2 \, \sqrt{3} x^{2} + 5 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (x^{2} + \sqrt{3}\right )} + 6}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - 16 \,{\left (512 \, x^{14} + 10112 \, x^{12} + 84800 \, x^{10} + 352696 \, x^{8} + 712008 \, x^{6} + 586018 \, x^{4} + 63477 \, x^{2} - 61656\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - 1704960}{128 \,{\left (128 \, x^{10} + 1280 \, x^{8} + 4384 \, x^{6} + 5920 \, x^{4} + 2569 \, x^{2} - 8 \,{\left (16 \, x^{8} + 120 \, x^{6} + 274 \, x^{4} + 185 \, x^{2}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*(8192*x^16 + 182272*x^14 + 1747968*x^12 + 8805504*x^10 + 23858432*x^8 + 3
2630816*x^6 + 18064928*x^4 + 300307*x^2 + 2436*(128*x^10 + 1280*x^8 + 4384*x^6 +
5920*x^4 + 2569*x^2 - 8*(16*x^8 + 120*x^6 + 274*x^4 + 185*x^2)*sqrt(x^4 + 5*x^2
+ 3))*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) + 1536*(8*sqrt(3)*(16*x^8 + 120
*x^6 + 274*x^4 + 185*x^2)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)*(128*x^10 + 1280*x^8 +
4384*x^6 + 5920*x^4 + 2569*x^2))*log((2*x^4 + 2*sqrt(3)*x^2 + 5*x^2 - 2*sqrt(x^
4 + 5*x^2 + 3)*(x^2 + sqrt(3)) + 6)/(2*x^4 - 2*sqrt(x^4 + 5*x^2 + 3)*x^2 + 5*x^2
)) - 16*(512*x^14 + 10112*x^12 + 84800*x^10 + 352696*x^8 + 712008*x^6 + 586018*x
^4 + 63477*x^2 - 61656)*sqrt(x^4 + 5*x^2 + 3) - 1704960)/(128*x^10 + 1280*x^8 +
4384*x^6 + 5920*x^4 + 2569*x^2 - 8*(16*x^8 + 120*x^6 + 274*x^4 + 185*x^2)*sqrt(x
^4 + 5*x^2 + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x**2 + 2)*(x**4 + 5*x**2 + 3)**(3/2)/x**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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