Optimal. Leaf size=92 \[ \frac{(c+d x) (b c-a d) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d^2 \sqrt [3]{-e (c+d x)^3}}-\frac{b (c+d x)^2 \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]
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Rubi [A] time = 0.0602533, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2226, 2208, 2218} \[ \frac{(c+d x) (b c-a d) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d^2 \sqrt [3]{-e (c+d x)^3}}-\frac{b (c+d x)^2 \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2226
Rule 2208
Rule 2218
Rubi steps
\begin{align*} \int e^{e (c+d x)^3} (a+b x) \, dx &=\int \left (\frac{(-b c+a d) e^{e (c+d x)^3}}{d}+\frac{b e^{e (c+d x)^3} (c+d x)}{d}\right ) \, dx\\ &=\frac{b \int e^{e (c+d x)^3} (c+d x) \, dx}{d}+\frac{(-b c+a d) \int e^{e (c+d x)^3} \, dx}{d}\\ &=\frac{(b c-a d) (c+d x) \Gamma \left (\frac{1}{3},-e (c+d x)^3\right )}{3 d^2 \sqrt [3]{-e (c+d x)^3}}-\frac{b (c+d x)^2 \Gamma \left (\frac{2}{3},-e (c+d x)^3\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0598877, size = 86, normalized size = 0.93 \[ -\frac{(c+d x) \left (b (c+d x) \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )-(b c-a d) \sqrt [3]{-e (c+d x)^3} \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{e \left ( dx+c \right ) ^{3}}} \left ( bx+a \right ) \, 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{\left (b x + a\right )} 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 = 1.49636, size = 250, normalized size = 2.72 \begin{align*} \frac{\left (-d^{3} e\right )^{\frac{1}{3}} b d \Gamma \left (\frac{2}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right ) - \left (-d^{3} e\right )^{\frac{2}{3}}{\left (b c - a d\right )} \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^{4} 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*} \left (\int a e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx + \int b x e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx\right ) e^{c^{3} e} \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{\left (b x + a\right )} 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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