Optimal. Leaf size=15 \[ -x+\frac {2}{3} (1+x)^{3/2} \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {378}
\begin {gather*} \frac {2}{3} (x+1)^{3/2}-x \end {gather*}
Antiderivative was successfully verified.
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Rule 378
Rubi steps
\begin {align*} \int \frac {x}{1+\sqrt {1+x}} \, dx &=\text {Subst}\left (\int \left (-1+\sqrt {x}\right ) \, dx,x,1+x\right )\\ &=-x+\frac {2}{3} (1+x)^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.20 \begin {gather*} \frac {1}{3} (1+x) \left (-3+2 \sqrt {1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 13, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-1-x\) | \(13\) |
default | \(\frac {2 \left (1+x \right )^{\frac {3}{2}}}{3}-1-x\) | \(13\) |
trager | \(-x +\left (\frac {2}{3}+\frac {2 x}{3}\right ) \sqrt {1+x}\) | \(16\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, \left (12 x +8\right )}{6}+\frac {\sqrt {\pi }\, \left (8+8 x \right ) \sqrt {1+x}}{6}}{2 \sqrt {\pi }}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.88, size = 12, normalized size = 0.80 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x - 1 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 11, normalized size = 0.73 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs.
\(2 (10) = 20\).
time = 0.45, size = 22, normalized size = 1.47 \begin {gather*} \frac {2 x \sqrt {x + 1}}{3} - x + \frac {2 \sqrt {x + 1}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 12, normalized size = 0.80 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - x - 1 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 11, normalized size = 0.73 \begin {gather*} \frac {2\,{\left (x+1\right )}^{3/2}}{3}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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