Optimal. Leaf size=13 \[ \sqrt {x^2+2 x^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1602}
\begin {gather*} \sqrt {2 x^3+x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 1602
Rubi steps
\begin {align*} \int \frac {x+3 x^2}{\sqrt {x^2+2 x^3}} \, dx &=\sqrt {x^2+2 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \sqrt {x^2 (1+2 x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 21, normalized size = 1.62
method | result | size |
trager | \(\sqrt {2 x^{3}+x^{2}}\) | \(12\) |
gosper | \(\frac {x^{2} \left (2 x +1\right )}{\sqrt {2 x^{3}+x^{2}}}\) | \(21\) |
default | \(\frac {x^{2} \left (2 x +1\right )}{\sqrt {2 x^{3}+x^{2}}}\) | \(21\) |
risch | \(\frac {x^{2} \left (2 x +1\right )}{\sqrt {x^{2} \left (2 x +1\right )}}\) | \(21\) |
meijerg | \(\frac {\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (-8 x +8\right ) \sqrt {2 x +1}}{8}}{\sqrt {\pi }}+\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {2 x +1}}{2 \sqrt {\pi }}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 11, normalized size = 0.85 \begin {gather*} \sqrt {2 \, x^{3} + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 11, normalized size = 0.85 \begin {gather*} \sqrt {2 \, x^{3} + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 10, normalized size = 0.77 \begin {gather*} \sqrt {2 x^{3} + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs.
\(2 (11) = 22\).
time = 3.33, size = 23, normalized size = 1.77 \begin {gather*} \frac {{\left (2 \, x + 1\right )}^{\frac {3}{2}} - \sqrt {2 \, x + 1}}{2 \, \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.23, size = 10, normalized size = 0.77 \begin {gather*} \left |x\right |\,\sqrt {2\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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