Integrand size = 17, antiderivative size = 92 \[ \int e^{e (c+d x)^3} (a+b x) \, dx=\frac {(b c-a d) (c+d x) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{3 d^2 \sqrt [3]{-e (c+d x)^3}}-\frac {b (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]
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Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2258, 2239, 2250} \[ \int e^{e (c+d x)^3} (a+b x) \, dx=\frac {(c+d x) (b c-a d) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{3 d^2 \sqrt [3]{-e (c+d x)^3}}-\frac {b (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]
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Rule 2239
Rule 2250
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93 \[ \int e^{e (c+d x)^3} (a+b x) \, dx=-\frac {(c+d x) \left (-\left ((b c-a d) \sqrt [3]{-e (c+d x)^3} \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )\right )+b (c+d x) \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]
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\[\int {\mathrm e}^{e \left (d x +c \right )^{3}} \left (b x +a \right )d x\]
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none
Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20 \[ \int e^{e (c+d x)^3} (a+b x) \, dx=\frac {\left (-d^{3} e\right )^{\frac {1}{3}} b d \Gamma \left (\frac {2}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right ) - \left (-d^{3} e\right )^{\frac {2}{3}} {\left (b c - a d\right )} \Gamma \left (\frac {1}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right )}{3 \, d^{4} e} \]
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\[ \int e^{e (c+d x)^3} (a+b x) \, dx=\left (\int a e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx + \int b x e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx\right ) e^{c^{3} e} \]
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\[ \int e^{e (c+d x)^3} (a+b x) \, dx=\int { {\left (b x + a\right )} e^{\left ({\left (d x + c\right )}^{3} e\right )} \,d x } \]
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\[ \int e^{e (c+d x)^3} (a+b x) \, dx=\int { {\left (b x + a\right )} e^{\left ({\left (d x + c\right )}^{3} e\right )} \,d x } \]
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Timed out. \[ \int e^{e (c+d x)^3} (a+b x) \, dx=\int {\mathrm {e}}^{e\,{\left (c+d\,x\right )}^3}\,\left (a+b\,x\right ) \,d x \]
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