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

Optimal. Leaf size=99 \[ -\frac{\sqrt{x^4+5 x^2+3} \left (23 x^2+6\right )}{12 x^4}+\frac{3}{2} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{77 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{24 \sqrt{3}} \]

[Out]

-((6 + 23*x^2)*Sqrt[3 + 5*x^2 + x^4])/(12*x^4) + (3*ArcTanh[(5 + 2*x^2)/(2*Sqrt[
3 + 5*x^2 + x^4])])/2 - (77*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4]
)])/(24*Sqrt[3])

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Rubi [A]  time = 0.207388, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\sqrt{x^4+5 x^2+3} \left (23 x^2+6\right )}{12 x^4}+\frac{3}{2} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{77 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{24 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-((6 + 23*x^2)*Sqrt[3 + 5*x^2 + x^4])/(12*x^4) + (3*ArcTanh[(5 + 2*x^2)/(2*Sqrt[
3 + 5*x^2 + x^4])])/2 - (77*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4]
)])/(24*Sqrt[3])

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Rubi in Sympy [A]  time = 20.9532, size = 88, normalized size = 0.89 \[ \frac{3 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{2} - \frac{77 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{72} - \frac{\left (23 x^{2} + 6\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*atanh((2*x**2 + 5)/(2*sqrt(x**4 + 5*x**2 + 3)))/2 - 77*sqrt(3)*atanh(sqrt(3)*(
5*x**2 + 6)/(6*sqrt(x**4 + 5*x**2 + 3)))/72 - (23*x**2 + 6)*sqrt(x**4 + 5*x**2 +
3)/(12*x**4)

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Mathematica [A]  time = 0.156872, size = 102, normalized size = 1.03 \[ \frac{1}{12} \left (-\frac{\sqrt{x^4+5 x^2+3} \left (23 x^2+6\right )}{x^4}+18 \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right )+\frac{77 \left (2 \log (x)-\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )\right )}{2 \sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-(((6 + 23*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^4) + 18*Log[5 + 2*x^2 + 2*Sqrt[3 + 5*x
^2 + x^4]] + (77*(2*Log[x] - Log[6 + 5*x^2 + 2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4]]))/
(2*Sqrt[3]))/12

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Maple [A]  time = 0.022, size = 121, normalized size = 1.2 \[ -{\frac{1}{6\,{x}^{4}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{13}{36\,{x}^{2}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{77}{72}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{77\,\sqrt{3}}{72}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }+{\frac{26\,{x}^{2}+65}{72}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{3}{2}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/x^4*(x^4+5*x^2+3)^(3/2)-13/36/x^2*(x^4+5*x^2+3)^(3/2)+77/72*(x^4+5*x^2+3)^(
1/2)-77/72*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)+13/72*(2*x
^2+5)*(x^4+5*x^2+3)^(1/2)+3/2*ln(x^2+5/2+(x^4+5*x^2+3)^(1/2))

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Maxima [A]  time = 0.818556, size = 143, normalized size = 1.44 \[ -\frac{77}{72} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{1}{6} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}}{12 \, x^{2}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{6 \, x^{4}} + \frac{3}{2} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-77/72*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 1/6*sqrt(x
^4 + 5*x^2 + 3) - 13/12*sqrt(x^4 + 5*x^2 + 3)/x^2 - 1/6*(x^4 + 5*x^2 + 3)^(3/2)/
x^4 + 3/2*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]  time = 0.26974, size = 405, normalized size = 4.09 \[ \frac{2 \, \sqrt{3}{\left (508 \, x^{4} + 1091 \, x^{2} + 222\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - 36 \,{\left (4 \, \sqrt{3}{\left (2 \, x^{6} + 5 \, x^{4}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (8 \, x^{8} + 40 \, x^{6} + 37 \, x^{4}\right )}\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) - 77 \,{\left (8 \, x^{8} + 40 \, x^{6} + 37 \, x^{4} - 4 \,{\left (2 \, x^{6} + 5 \, x^{4}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}\right )} \log \left (\frac{6 \, x^{2} + \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2} + 6\right )} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (\sqrt{3} x^{2} + 3\right )}}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - 2 \, \sqrt{3}{\left (508 \, x^{6} + 2361 \, x^{4} + 2124 \, x^{2} + 360\right )}}{24 \,{\left (4 \, \sqrt{3}{\left (2 \, x^{6} + 5 \, x^{4}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (8 \, x^{8} + 40 \, x^{6} + 37 \, x^{4}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(2*sqrt(3)*(508*x^4 + 1091*x^2 + 222)*sqrt(x^4 + 5*x^2 + 3) - 36*(4*sqrt(3)
*(2*x^6 + 5*x^4)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)*(8*x^8 + 40*x^6 + 37*x^4))*log(
-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) - 77*(8*x^8 + 40*x^6 + 37*x^4 - 4*(2*x^6 +
5*x^4)*sqrt(x^4 + 5*x^2 + 3))*log((6*x^2 + sqrt(3)*(2*x^4 + 5*x^2 + 6) - 2*sqrt
(x^4 + 5*x^2 + 3)*(sqrt(3)*x^2 + 3))/(2*x^4 - 2*sqrt(x^4 + 5*x^2 + 3)*x^2 + 5*x^
2)) - 2*sqrt(3)*(508*x^6 + 2361*x^4 + 2124*x^2 + 360))/(4*sqrt(3)*(2*x^6 + 5*x^4
)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)*(8*x^8 + 40*x^6 + 37*x^4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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