3.13.97 \(\int \frac {x^4}{1+x^5} \, dx\) [1297]

Optimal. Leaf size=10 \[ \frac {1}{5} \log \left (1+x^5\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {266} \begin {gather*} \frac {1}{5} \log \left (x^5+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(1 + x^5),x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{5} \log \left (1+x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(1 + x^5),x]

[Out]

Log[1 + x^5]/5

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Maple [A]
time = 0.16, size = 9, normalized size = 0.90

method	result	size
derivativedivides	\(\frac {\ln \left (x^{5}+1\right )}{5}\)	\(9\)
default	\(\frac {\ln \left (x^{5}+1\right )}{5}\)	\(9\)
meijerg	\(\frac {\ln \left (x^{5}+1\right )}{5}\)	\(9\)
risch	\(\frac {\ln \left (x^{5}+1\right )}{5}\)	\(9\)
norman	\(\frac {\ln \left (x +1\right )}{5}+\frac {\ln \left (x^{4}-x^{3}+x^{2}-x +1\right )}{5}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^5+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{5} \, \log \left (x^{5} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^5+1),x, algorithm="maxima")

[Out]

1/5*log(x^5 + 1)

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Fricas [A]
time = 0.35, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{5} \, \log \left (x^{5} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^5+1),x, algorithm="fricas")

[Out]

1/5*log(x^5 + 1)

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Sympy [A]
time = 0.02, size = 7, normalized size = 0.70 \begin {gather*} \frac {\log {\left (x^{5} + 1 \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(x**5+1),x)

[Out]

log(x**5 + 1)/5

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Giac [A]
time = 2.36, size = 9, normalized size = 0.90 \begin {gather*} \frac {1}{5} \, \log \left ({\left | x^{5} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^5+1),x, algorithm="giac")

[Out]

1/5*log(abs(x^5 + 1))

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Mupad [B]
time = 0.03, size = 8, normalized size = 0.80 \begin {gather*} \frac {\ln \left (x^5+1\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^5 + 1),x)

[Out]

log(x^5 + 1)/5

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