Optimal. Leaf size=12 \[ -\frac {2}{3 x^3}+\log (1+x) \]
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Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1607, 1634}
\begin {gather*} \log (x+1)-\frac {2}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 1607
Rule 1634
Rubi steps
\begin {align*} \int \frac {2+2 x+x^4}{x^4+x^5} \, dx &=\int \frac {2+2 x+x^4}{x^4 (1+x)} \, dx\\ &=\int \left (\frac {2}{x^4}+\frac {1}{1+x}\right ) \, dx\\ &=-\frac {2}{3 x^3}+\log (1+x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {2}{3 x^3}+\log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 11, normalized size = 0.92
method | result | size |
default | \(-\frac {2}{3 x^{3}}+\ln \left (1+x \right )\) | \(11\) |
norman | \(-\frac {2}{3 x^{3}}+\ln \left (1+x \right )\) | \(11\) |
meijerg | \(-\frac {2}{3 x^{3}}+\ln \left (1+x \right )\) | \(11\) |
risch | \(-\frac {2}{3 x^{3}}+\ln \left (1+x \right )\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 10, normalized size = 0.83 \begin {gather*} -\frac {2}{3 \, x^{3}} + \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 16, normalized size = 1.33 \begin {gather*} \frac {3 \, x^{3} \log \left (x + 1\right ) - 2}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 10, normalized size = 0.83 \begin {gather*} \log {\left (x + 1 \right )} - \frac {2}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.44, size = 11, normalized size = 0.92 \begin {gather*} -\frac {2}{3 \, x^{3}} + \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.12, size = 10, normalized size = 0.83 \begin {gather*} \ln \left (x+1\right )-\frac {2}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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