Optimal. Leaf size=28 \[ -\frac {2}{3} e^{x^{3/2}}+\frac {2}{3} e^{x^{3/2}} x^{3/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2247, 2243,
2240} \begin {gather*} \frac {2}{3} e^{x^{3/2}} x^{3/2}-\frac {2 e^{x^{3/2}}}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2243
Rule 2247
Rubi steps
\begin {align*} \int e^{x^{3/2}} x^2 \, dx &=2 \text {Subst}\left (\int e^{x^3} x^5 \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} e^{x^{3/2}} x^{3/2}-2 \text {Subst}\left (\int e^{x^3} x^2 \, dx,x,\sqrt {x}\right )\\ &=-\frac {2}{3} e^{x^{3/2}}+\frac {2}{3} e^{x^{3/2}} x^{3/2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.00, size = 13, normalized size = 0.46 \begin {gather*} -\frac {2}{3} \Gamma \left (2,-x^{3/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 17, normalized size = 0.61
method | result | size |
meijerg | \(\frac {2}{3}-\frac {\left (2-2 x^{\frac {3}{2}}\right ) {\mathrm e}^{x^{\frac {3}{2}}}}{3}\) | \(16\) |
derivativedivides | \(-\frac {2 \,{\mathrm e}^{x^{\frac {3}{2}}}}{3}+\frac {2 \,{\mathrm e}^{x^{\frac {3}{2}}} x^{\frac {3}{2}}}{3}\) | \(17\) |
default | \(-\frac {2 \,{\mathrm e}^{x^{\frac {3}{2}}}}{3}+\frac {2 \,{\mathrm e}^{x^{\frac {3}{2}}} x^{\frac {3}{2}}}{3}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 11, normalized size = 0.39 \begin {gather*} \frac {2}{3} \, {\left (x^{\frac {3}{2}} - 1\right )} e^{\left (x^{\frac {3}{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 11, normalized size = 0.39 \begin {gather*} \frac {2}{3} \, {\left (x^{\frac {3}{2}} - 1\right )} e^{\left (x^{\frac {3}{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.48, size = 24, normalized size = 0.86 \begin {gather*} \frac {2 x^{\frac {3}{2}} e^{x^{\frac {3}{2}}}}{3} - \frac {2 e^{x^{\frac {3}{2}}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.72, size = 11, normalized size = 0.39 \begin {gather*} \frac {2}{3} \, {\left (x^{\frac {3}{2}} - 1\right )} e^{\left (x^{\frac {3}{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.57, size = 16, normalized size = 0.57 \begin {gather*} \frac {2\,x^{3/2}\,{\mathrm {e}}^{x^{3/2}}}{3}-\frac {2\,{\mathrm {e}}^{x^{3/2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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