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

Optimal. Leaf size=56 \[ \frac{149}{16} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{1}{8} \left (37-6 x^2\right ) \sqrt{x^4+5 x^2+3} \]

[Out]

-((37 - 6*x^2)*Sqrt[3 + 5*x^2 + x^4])/8 + (149*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5
*x^2 + x^4])])/16

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Rubi [A]  time = 0.12406, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{149}{16} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{1}{8} \left (37-6 x^2\right ) \sqrt{x^4+5 x^2+3} \]

Antiderivative was successfully verified.

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

[Out]

-((37 - 6*x^2)*Sqrt[3 + 5*x^2 + x^4])/8 + (149*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5
*x^2 + x^4])])/16

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Rubi in Sympy [A]  time = 14.4906, size = 49, normalized size = 0.88 \[ - \frac{\left (- 3 x^{2} + \frac{37}{2}\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{4} + \frac{149 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-3*x**2 + 37/2)*sqrt(x**4 + 5*x**2 + 3)/4 + 149*atanh((2*x**2 + 5)/(2*sqrt(x**
4 + 5*x**2 + 3)))/16

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Mathematica [A]  time = 0.025046, size = 54, normalized size = 0.96 \[ \frac{1}{8} \sqrt{x^4+5 x^2+3} \left (6 x^2-37\right )+\frac{149}{16} \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-37 + 6*x^2)*Sqrt[3 + 5*x^2 + x^4])/8 + (149*Log[5 + 2*x^2 + 2*Sqrt[3 + 5*x^2
+ x^4]])/16

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Maple [A]  time = 0.017, size = 53, normalized size = 1. \[ -{\frac{37}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{149}{16}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }+{\frac{3\,{x}^{2}}{4}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37/8*(x^4+5*x^2+3)^(1/2)+149/16*ln(x^2+5/2+(x^4+5*x^2+3)^(1/2))+3/4*x^2*(x^4+5*
x^2+3)^(1/2)

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Maxima [A]  time = 0.703414, size = 76, normalized size = 1.36 \[ \frac{3}{4} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} - \frac{37}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{149}{16} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*sqrt(x^4 + 5*x^2 + 3)*x^2 - 37/8*sqrt(x^4 + 5*x^2 + 3) + 149/16*log(2*x^2 +
2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]  time = 0.271298, size = 198, normalized size = 3.54 \[ -\frac{384 \, x^{8} + 512 \, x^{6} - 8248 \, x^{4} - 16024 \, x^{2} + 596 \,{\left (8 \, x^{4} + 40 \, x^{2} - 4 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \, x^{2} + 5\right )} + 37\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) - 4 \,{\left (96 \, x^{6} - 112 \, x^{4} - 1626 \, x^{2} - 513\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - 1295}{64 \,{\left (8 \, x^{4} + 40 \, x^{2} - 4 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \, x^{2} + 5\right )} + 37\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(384*x^8 + 512*x^6 - 8248*x^4 - 16024*x^2 + 596*(8*x^4 + 40*x^2 - 4*sqrt(x
^4 + 5*x^2 + 3)*(2*x^2 + 5) + 37)*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) - 4*
(96*x^6 - 112*x^4 - 1626*x^2 - 513)*sqrt(x^4 + 5*x^2 + 3) - 1295)/(8*x^4 + 40*x^
2 - 4*sqrt(x^4 + 5*x^2 + 3)*(2*x^2 + 5) + 37)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.277405, size = 62, normalized size = 1.11 \[ \frac{1}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (6 \, x^{2} - 37\right )} - \frac{149}{16} \,{\rm ln}\left (2 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(x^4 + 5*x^2 + 3)*(6*x^2 - 37) - 149/16*ln(2*x^2 - 2*sqrt(x^4 + 5*x^2 +
3) + 5)