Optimal. Leaf size=99 \[ -\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b^3 \sqrt [3]{-(a+b x)^3}}+\frac {e^{(a+b x)^3}}{3 b^3}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{3 b^3 \left (-(a+b x)^3\right )^{2/3}} \]
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Rubi [A] time = 0.12, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2227, 2226, 2208, 2218, 2209} \[ -\frac {a^2 (a+b x) \text {Gamma}\left (\frac {1}{3},-(a+b x)^3\right )}{3 b^3 \sqrt [3]{-(a+b x)^3}}+\frac {2 a (a+b x)^2 \text {Gamma}\left (\frac {2}{3},-(a+b x)^3\right )}{3 b^3 \left (-(a+b x)^3\right )^{2/3}}+\frac {e^{(a+b x)^3}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 2209
Rule 2218
Rule 2226
Rule 2227
Rubi steps
\begin {align*} \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x^2 \, dx &=\int e^{(a+b x)^3} x^2 \, dx\\ &=\int \left (\frac {a^2 e^{(a+b x)^3}}{b^2}-\frac {2 a e^{(a+b x)^3} (a+b x)}{b^2}+\frac {e^{(a+b x)^3} (a+b x)^2}{b^2}\right ) \, dx\\ &=\frac {\int e^{(a+b x)^3} (a+b x)^2 \, dx}{b^2}-\frac {(2 a) \int e^{(a+b x)^3} (a+b x) \, dx}{b^2}+\frac {a^2 \int e^{(a+b x)^3} \, dx}{b^2}\\ &=\frac {e^{(a+b x)^3}}{3 b^3}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b^3 \sqrt [3]{-(a+b x)^3}}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{3 b^3 \left (-(a+b x)^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 89, normalized size = 0.90 \[ \frac {-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{\sqrt [3]{-(a+b x)^3}}+e^{(a+b x)^3}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{\left (-(a+b x)^3\right )^{2/3}}}{3 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 124, normalized size = 1.25 \[ \frac {\left (-b^{3}\right )^{\frac {2}{3}} a^{2} \Gamma \left (\frac {1}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right ) - 2 \, \left (-b^{3}\right )^{\frac {1}{3}} a b \Gamma \left (\frac {2}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right ) + b^{2} e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )}}{3 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int x^{2} {\mathrm e}^{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {e}}^{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \,d x \]
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 \[ e^{a^{3}} \int x^{2} e^{b^{3} x^{3}} e^{3 a b^{2} x^{2}} e^{3 a^{2} b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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