3.204 \(\int \frac{3 x^2}{2 \left (1+x^3+\sqrt{1+x^3}\right )} \, dx\)

Optimal. Leaf size=12 \[ \log \left (\sqrt{x^3+1}+1\right ) \]

[Out]

Log[1 + Sqrt[1 + x^3]]

_______________________________________________________________________________________

Rubi [A]  time = 0.0817713, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \log \left (\sqrt{x^3+1}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3*x^2)/(2*(1 + x^3 + Sqrt[1 + x^3])),x]

[Out]

Log[1 + Sqrt[1 + x^3]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.21734, size = 10, normalized size = 0.83 \[ \log{\left (\sqrt{x^{3} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sqrt(x**3 + 1) + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00919247, size = 12, normalized size = 1. \[ \log \left (\sqrt{x^3+1}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(3*x^2)/(2*(1 + x^3 + Sqrt[1 + x^3])),x]

[Out]

Log[1 + Sqrt[1 + x^3]]

_______________________________________________________________________________________

Maple [B]  time = 0.058, size = 39, normalized size = 3.3 \[{\frac{3\,\ln \left ( x \right ) }{2}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}}+{\frac{\ln \left ({x}^{3}+1 \right ) }{2}}+{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*ln(x)-1/2*ln(x^2-x+1)-1/2*ln(1+x)+1/2*ln(x^3+1)+arctanh((x^3+1)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 1.48647, size = 54, normalized size = 4.5 \[ -\frac{1}{2} \, \log \left (x^{2} - x + 1\right ) + \log \left (\frac{x^{3} + \sqrt{x^{2} - x + 1} \sqrt{x + 1} + 1}{\sqrt{x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(3/2*x^2/(x^3 + sqrt(x^3 + 1) + 1),x, algorithm="maxima")

[Out]

-1/2*log(x^2 - x + 1) + log((x^3 + sqrt(x^2 - x + 1)*sqrt(x + 1) + 1)/sqrt(x + 1
))

_______________________________________________________________________________________

Fricas [A]  time = 0.218551, size = 39, normalized size = 3.25 \[ \frac{3}{2} \, \log \left (x\right ) + \frac{1}{2} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(3/2*x^2/(x^3 + sqrt(x^3 + 1) + 1),x, algorithm="fricas")

[Out]

3/2*log(x) + 1/2*log(sqrt(x^3 + 1) + 1) - 1/2*log(sqrt(x^3 + 1) - 1)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 \int \frac{x^{2}}{x^{3} + \sqrt{x^{3} + 1} + 1}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*Integral(x**2/(x**3 + sqrt(x**3 + 1) + 1), x)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.198971, size = 14, normalized size = 1.17 \[{\rm ln}\left (\sqrt{x^{3} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(3/2*x^2/(x^3 + sqrt(x^3 + 1) + 1),x, algorithm="giac")

[Out]

ln(sqrt(x^3 + 1) + 1)