3.140 \(\int \frac{x^4}{a^5+x^5} \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{5} \log \left (a^5+x^5\right ) \]

[Out]

Log[a^5 + x^5]/5

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Rubi [A]  time = 0.002512, 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} \[ \frac{1}{5} \log \left (a^5+x^5\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a^5 + x^5),x]

[Out]

Log[a^5 + x^5]/5

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^4}{a^5+x^5} \, dx &=\frac{1}{5} \log \left (a^5+x^5\right )\\ \end{align*}

Mathematica [A]  time = 0.0023469, size = 12, normalized size = 1. \[ \frac{1}{5} \log \left (a^5+x^5\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a^5 + x^5),x]

[Out]

Log[a^5 + x^5]/5

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Maple [A]  time = 0.001, size = 11, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({a}^{5}+{x}^{5} \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a^5+x^5),x)

[Out]

1/5*ln(a^5+x^5)

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Maxima [A]  time = 0.929379, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{5} \, \log \left (a^{5} + x^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^5+x^5),x, algorithm="maxima")

[Out]

1/5*log(a^5 + x^5)

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Fricas [A]  time = 1.79827, size = 27, normalized size = 2.25 \begin{align*} \frac{1}{5} \, \log \left (a^{5} + x^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^5+x^5),x, algorithm="fricas")

[Out]

1/5*log(a^5 + x^5)

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Sympy [A]  time = 0.101514, size = 8, normalized size = 0.67 \begin{align*} \frac{\log{\left (a^{5} + x^{5} \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(a**5+x**5),x)

[Out]

log(a**5 + x**5)/5

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Giac [A]  time = 1.0589, size = 15, normalized size = 1.25 \begin{align*} \frac{1}{5} \, \log \left ({\left | a^{5} + x^{5} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^5+x^5),x, algorithm="giac")

[Out]

1/5*log(abs(a^5 + x^5))