Optimal. Leaf size=38 \[ -\frac{(a+b x) \text{Gamma}\left (\frac{1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]
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Rubi [A] time = 0.0085331, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2227, 2208} \[ -\frac{(a+b x) \text{Gamma}\left (\frac{1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]
Antiderivative was successfully verified.
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Rule 2227
Rule 2208
Rubi steps
\begin{align*} \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx &=\int e^{(a+b x)^3} \, dx\\ &=-\frac{(a+b x) \Gamma \left (\frac{1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}}\\ \end{align*}
Mathematica [A] time = 0.0054023, size = 38, normalized size = 1. \[ -\frac{(a+b x) \text{Gamma}\left (\frac{1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.011, 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}}}\, 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 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 = 1.47753, size = 101, normalized size = 2.66 \begin{align*} \frac{\left (-b^{3}\right )^{\frac{2}{3}} \Gamma \left (\frac{1}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right )}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{a^{3}} \int e^{b^{3} x^{3}} e^{3 a b^{2} x^{2}} e^{3 a^{2} b x}\, dx \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 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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