Optimal. Leaf size=40 \[ -\frac{(c+d x) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d \sqrt [3]{-e (c+d x)^3}} \]
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Rubi [A] time = 0.0051903, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2208} \[ -\frac{(c+d x) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d \sqrt [3]{-e (c+d x)^3}} \]
Antiderivative was successfully verified.
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Rule 2208
Rubi steps
\begin{align*} \int e^{e (c+d x)^3} \, dx &=-\frac{(c+d x) \Gamma \left (\frac{1}{3},-e (c+d x)^3\right )}{3 d \sqrt [3]{-e (c+d x)^3}}\\ \end{align*}
Mathematica [A] time = 0.0061005, size = 40, normalized size = 1. \[ -\frac{(c+d x) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d \sqrt [3]{-e (c+d x)^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.012, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{e \left ( dx+c \right ) ^{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 ({\left (d x + c\right )}^{3} e\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53867, size = 120, normalized size = 3. \begin{align*} \frac{\left (-d^{3} e\right )^{\frac{2}{3}} \Gamma \left (\frac{1}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right )}{3 \, d^{3} e} \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^{c^{3} e} \int e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e 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 ({\left (d x + c\right )}^{3} e\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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