Optimal. Leaf size=28 \[ \log (x)-\log \left (-x-2 x^2+2 \sqrt {x^3+x^4}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.64, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2054, 212}
\begin {gather*} 2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2054
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {x^3+x^4}} \, dx &=2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x^3+x^4}}\right )\\ &=2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^3+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 39, normalized size = 1.39 \begin {gather*} \frac {2 x^{3/2} \sqrt {1+x} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {1+x}}\right )}{\sqrt {x^3 (1+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 30, normalized size = 1.07
method | result | size |
meijerg | \(2 \arcsinh \left (\sqrt {x}\right )\) | \(7\) |
trager | \(\ln \left (\frac {2 x^{2}+2 \sqrt {x^{4}+x^{3}}+x}{x}\right )\) | \(24\) |
default | \(\frac {x \sqrt {x \left (1+x \right )}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{\sqrt {x^{4}+x^{3}}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 26, normalized size = 0.93 \begin {gather*} -\log \left (-\frac {2 \, x^{2} + x - 2 \, \sqrt {x^{4} + x^{3}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{3} \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 22, normalized size = 0.79 \begin {gather*} -\frac {\log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right )}{\mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x}{\sqrt {x^4+x^3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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