3.120 \(\int \frac{x^2}{a^3+x^3} \, dx\)

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

[Out]

Log[a^3 + x^3]/3

________________________________________________________________________________________

Rubi [A]  time = 0.0026214, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {260} \[ \frac{1}{3} \log \left (a^3+x^3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a^3 + x^3),x]

[Out]

Log[a^3 + x^3]/3

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.0021873, size = 12, normalized size = 1. \[ \frac{1}{3} \log \left (a^3+x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a^3 + x^3),x]

[Out]

Log[a^3 + x^3]/3

________________________________________________________________________________________

Maple [A]  time = 0., size = 11, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({a}^{3}+{x}^{3} \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^3+x^3),x)

[Out]

1/3*ln(a^3+x^3)

________________________________________________________________________________________

Maxima [A]  time = 0.917987, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{3} \, \log \left (a^{3} + x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^3+x^3),x, algorithm="maxima")

[Out]

1/3*log(a^3 + x^3)

________________________________________________________________________________________

Fricas [A]  time = 2.06094, size = 27, normalized size = 2.25 \begin{align*} \frac{1}{3} \, \log \left (a^{3} + x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^3+x^3),x, algorithm="fricas")

[Out]

1/3*log(a^3 + x^3)

________________________________________________________________________________________

Sympy [A]  time = 0.092382, size = 8, normalized size = 0.67 \begin{align*} \frac{\log{\left (a^{3} + x^{3} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a**3+x**3),x)

[Out]

log(a**3 + x**3)/3

________________________________________________________________________________________

Giac [A]  time = 1.05799, size = 15, normalized size = 1.25 \begin{align*} \frac{1}{3} \, \log \left ({\left | a^{3} + x^{3} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a^3+x^3),x, algorithm="giac")

[Out]

1/3*log(abs(a^3 + x^3))