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

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = 0.00, size = 12, normalized size = 1.00 \[ \frac {1}{3} \log \left (a^3+x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a^3 + x^3),x]

[Out]

Log[a^3 + x^3]/3

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.00, size = 12, normalized size = 1.00 \[ \frac {1}{3} \log \left (a^3+x^3\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2/(a^3 + x^3),x]

[Out]

Log[a^3 + x^3]/3

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 10, normalized size = 0.83 \[ \frac {1}{3} \, \log \left (a^{3} + x^{3}\right ) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.93, size = 11, normalized size = 0.92 \[ \frac {1}{3} \, \log \left ({\left | a^{3} + x^{3} \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.26, size = 11, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\ln \left (a^{3}+x^{3}\right )}{3}\)	\(11\)
default	\(\frac {\ln \left (a^{3}+x^{3}\right )}{3}\)	\(11\)
risch	\(\frac {\ln \left (a^{3}+x^{3}\right )}{3}\)	\(11\)
norman	\(\frac {\ln \left (a +x \right )}{3}+\frac {\ln \left (a^{2}-a x +x^{2}\right )}{3}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^3+x^3),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 10, normalized size = 0.83 \[ \frac {1}{3} \, \log \left (a^{3} + x^{3}\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 10, normalized size = 0.83 \[ \frac {\ln \left (a^3+x^3\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a^3 + x^3),x)

[Out]

log(a^3 + x^3)/3

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 8, normalized size = 0.67 \[ \frac {\log {\left (a^{3} + x^{3} \right )}}{3} \]

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

________________________________________________________________________________________