Optimal. Leaf size=19 \[ -8 \sqrt {x}+x+32 \log \left (4+\sqrt {x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1598, 272, 45}
\begin {gather*} x-8 \sqrt {x}+32 \log \left (\sqrt {x}+4\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {x}{4 \sqrt {x}+x} \, dx &=\int \frac {\sqrt {x}}{4+\sqrt {x}} \, dx\\ &=2 \text {Subst}\left (\int \frac {x^2}{4+x} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (-4+x+\frac {16}{4+x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-8 \sqrt {x}+x+32 \log \left (4+\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} -8 \sqrt {x}+x+32 \log \left (4+\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 16, normalized size = 0.84
method | result | size |
derivativedivides | \(x +32 \ln \left (4+\sqrt {x}\right )-8 \sqrt {x}\) | \(16\) |
default | \(x +32 \ln \left (4+\sqrt {x}\right )-8 \sqrt {x}\) | \(16\) |
trager | \(-1+x -8 \sqrt {x}+16 \ln \left (8 \sqrt {x}+16+x \right )\) | \(20\) |
meijerg | \(-\frac {4 \sqrt {x}\, \left (-\frac {3 \sqrt {x}}{4}+6\right )}{3}+32 \ln \left (1+\frac {\sqrt {x}}{4}\right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 15, normalized size = 0.79 \begin {gather*} x - 8 \, \sqrt {x} + 32 \, \log \left (\sqrt {x} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 15, normalized size = 0.79 \begin {gather*} x - 8 \, \sqrt {x} + 32 \, \log \left (\sqrt {x} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 17, normalized size = 0.89 \begin {gather*} - 8 \sqrt {x} + x + 32 \log {\left (\sqrt {x} + 4 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.30, size = 15, normalized size = 0.79 \begin {gather*} x - 8 \, \sqrt {x} + 32 \, \log \left (\sqrt {x} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 15, normalized size = 0.79 \begin {gather*} x+32\,\ln \left (\sqrt {x}+4\right )-8\,\sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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