3.2.40 \(\int \frac {x^4}{a^5+x^5} \, dx\) [140]

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

[Out]

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

________________________________________________________________________________________

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 = {266} \begin {gather*} \frac {1}{5} \log \left (a^5+x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a^5 + x^5),x]

[Out]

Log[a^5 + x^5]/5

Rule 266

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 = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{5} \log \left (a^5+x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a^5 + x^5),x]

[Out]

Log[a^5 + x^5]/5

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 11, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\ln \left (a^{5}+x^{5}\right )}{5}\)	\(11\)
default	\(\frac {\ln \left (a^{5}+x^{5}\right )}{5}\)	\(11\)
risch	\(\frac {\ln \left (a^{5}+x^{5}\right )}{5}\)	\(11\)
norman	\(\frac {\ln \left (a +x \right )}{5}+\frac {\ln \left (a^{4}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}\right )}{5}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a^5+x^5),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 2.39, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{5} \, \log \left (a^{5} + x^{5}\right ) \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.76, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{5} \, \log \left (a^{5} + x^{5}\right ) \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.03, size = 8, normalized size = 0.67 \begin {gather*} \frac {\log {\left (a^{5} + x^{5} \right )}}{5} \end {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 0.55, size = 11, normalized size = 0.92 \begin {gather*} \frac {1}{5} \, \log \left ({\left | a^{5} + x^{5} \right |}\right ) \end {gather*}

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))

________________________________________________________________________________________

Mupad [B]
time = 0.18, size = 10, normalized size = 0.83 \begin {gather*} \frac {\ln \left (a^5+x^5\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a^5 + x^5),x)

[Out]

log(a^5 + x^5)/5

________________________________________________________________________________________