Optimal. Leaf size=99 \[ -\frac{a^2 (a+b x) \text{Gamma}\left (\frac{1}{3},-(a+b x)^3\right )}{3 b^3 \sqrt [3]{-(a+b x)^3}}+\frac{2 a (a+b x)^2 \text{Gamma}\left (\frac{2}{3},-(a+b x)^3\right )}{3 b^3 \left (-(a+b x)^3\right )^{2/3}}+\frac{e^{(a+b x)^3}}{3 b^3} \]
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Rubi [A] time = 0.120953, antiderivative size = 99, normalized size of antiderivative = 1., 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 2227
Rule 2226
Rule 2208
Rule 2218
Rule 2209
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*}
Mathematica [A] time = 0.0809934, size = 89, normalized size = 0.9 \[ \frac{-\frac{a^2 (a+b x) \text{Gamma}\left (\frac{1}{3},-(a+b x)^3\right )}{\sqrt [3]{-(a+b x)^3}}+\frac{2 a (a+b x)^2 \text{Gamma}\left (\frac{2}{3},-(a+b x)^3\right )}{\left (-(a+b x)^3\right )^{2/3}}+e^{(a+b x)^3}}{3 b^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{b}^{3}{x}^{3}+3\,a{b}^{2}{x}^{2}+3\,{a}^{2}bx+{a}^{3}}}{x}^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55053, size = 277, normalized size = 2.8 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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