Optimal. Leaf size=23 \[ \frac {1}{12} (2+3 x) \left (4+12 x+9 x^2\right )^{3/2} \]
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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {623}
\begin {gather*} \frac {1}{12} (3 x+2) \left (9 x^2+12 x+4\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
Rubi steps
\begin {align*} \int \left (4+12 x+9 x^2\right )^{3/2} \, dx &=\frac {1}{12} (2+3 x) \left (4+12 x+9 x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 0.87 \begin {gather*} \frac {1}{12} (2+3 x) \left ((2+3 x)^2\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 17, normalized size = 0.74
method | result | size |
default | \(\frac {\left (2+3 x \right ) \left (\left (2+3 x \right )^{2}\right )^{\frac {3}{2}}}{12}\) | \(17\) |
risch | \(\frac {\sqrt {\left (2+3 x \right )^{2}}\, \left (2+3 x \right )^{3}}{12}\) | \(19\) |
gosper | \(\frac {x \left (27 x^{3}+72 x^{2}+72 x +32\right ) \left (\left (2+3 x \right )^{2}\right )^{\frac {3}{2}}}{4 \left (2+3 x \right )^{3}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 30, normalized size = 1.30 \begin {gather*} \frac {1}{4} \, {\left (9 \, x^{2} + 12 \, x + 4\right )}^{\frac {3}{2}} x + \frac {1}{6} \, {\left (9 \, x^{2} + 12 \, x + 4\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.15, size = 19, normalized size = 0.83 \begin {gather*} \frac {27}{4} \, x^{4} + 18 \, x^{3} + 18 \, x^{2} + 8 \, x \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 \left (9 x^{2} + 12 x + 4\right )^{\frac {3}{2}}\, dx \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. 45 vs.
\(2 (19) = 38\).
time = 2.44, size = 45, normalized size = 1.96 \begin {gather*} \frac {3}{4} \, {\left (3 \, x^{2} + 4 \, x\right )}^{2} \mathrm {sgn}\left (3 \, x + 2\right ) + 2 \, {\left (3 \, x^{2} + 4 \, x\right )} \mathrm {sgn}\left (3 \, x + 2\right ) + \frac {4}{3} \, \mathrm {sgn}\left (3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 19, normalized size = 0.83 \begin {gather*} \frac {\left (9\,x+6\right )\,{\left (9\,x^2+12\,x+4\right )}^{3/2}}{36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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