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

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

[Out]

Log[1 + Sqrt[1 + x^3]]

________________________________________________________________________________________

Rubi [A]  time = 0.0533348, 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, Rules used = {12, 2155, 31} \[ \log \left (\sqrt{x^3+1}+1\right ) \]

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]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2155

Int[(x_)^(m_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/n, Subst[Int
[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c
- a*d, 0] && IntegerQ[(m + 1)/n]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{3 x^2}{2 \left (1+x^3+\sqrt{1+x^3}\right )} \, dx &=\frac{3}{2} \int \frac{x^2}{1+x^3+\sqrt{1+x^3}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x+\sqrt{1+x}} \, dx,x,x^3\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt{1+x^3}\right )\\ &=\log \left (1+\sqrt{1+x^3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0213799, 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 = 0.009, size = 39, normalized size = 3.3 \begin{align*}{\frac{3\,\ln \left ( x \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{2}}+{\frac{\ln \left ({x}^{3}+1 \right ) }{2}}+{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \end{align*}

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

________________________________________________________________________________________

Maxima [B]  time = 1.46015, size = 54, normalized size = 4.5 \begin{align*} -\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 ) \end{align*}

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, 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 [B]  time = 1.63561, size = 95, normalized size = 7.92 \begin{align*} \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 ) \end{align*}

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, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 1.72642, size = 48, normalized size = 4. \begin{align*} - \frac{\log{\left (2 \sqrt{x^{3} + 1} \right )}}{2} + \frac{\log{\left (2 \sqrt{x^{3} + 1} + 2 \right )}}{2} + \frac{\log{\left (3 x^{3} + 3 \sqrt{x^{3} + 1} + 3 \right )}}{2} \end{align*}

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]

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

________________________________________________________________________________________

Giac [A]  time = 1.10482, size = 14, normalized size = 1.17 \begin{align*} \log \left (\sqrt{x^{3} + 1} + 1\right ) \end{align*}

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, algorithm="giac")

[Out]

log(sqrt(x^3 + 1) + 1)