Integrand size = 31, antiderivative size = 80 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x \, dx=\frac {a (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b^2 \sqrt [3]{-(a+b x)^3}}-\frac {(a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{3 b^2 \left (-(a+b x)^3\right )^{2/3}} \]
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Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2259, 2258, 2239, 2250} \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x \, dx=\frac {a (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b^2 \sqrt [3]{-(a+b x)^3}}-\frac {(a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{3 b^2 \left (-(a+b x)^3\right )^{2/3}} \]
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Rule 2239
Rule 2250
Rule 2258
Rule 2259
Rubi steps \begin{align*} \text {integral}& = \int e^{(a+b x)^3} x \, dx \\ & = \int \left (-\frac {a e^{(a+b x)^3}}{b}+\frac {e^{(a+b x)^3} (a+b x)}{b}\right ) \, dx \\ & = \frac {\int e^{(a+b x)^3} (a+b x) \, dx}{b}-\frac {a \int e^{(a+b x)^3} \, dx}{b} \\ & = \frac {a (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b^2 \sqrt [3]{-(a+b x)^3}}-\frac {(a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{3 b^2 \left (-(a+b x)^3\right )^{2/3}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x \, dx=\frac {(a+b x) \left (a \sqrt [3]{-(a+b x)^3} \Gamma \left (\frac {1}{3},-(a+b x)^3\right )-(a+b x) \Gamma \left (\frac {2}{3},-(a+b x)^3\right )\right )}{3 b^2 \left (-(a+b x)^3\right )^{2/3}} \]
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\[\int {\mathrm e}^{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}} x d x\]
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none
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.11 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x \, dx=-\frac {\left (-b^{3}\right )^{\frac {2}{3}} a \Gamma \left (\frac {1}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right ) - \left (-b^{3}\right )^{\frac {1}{3}} b \Gamma \left (\frac {2}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right )}{3 \, b^{4}} \]
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\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x \, dx=e^{a^{3}} \int x e^{b^{3} x^{3}} e^{3 a b^{2} x^{2}} e^{3 a^{2} b x}\, dx \]
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\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x \, dx=\int { x e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \,d x } \]
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\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x \, dx=\int { x e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \,d x } \]
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Timed out. \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x \, dx=\int x\,{\mathrm {e}}^{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \,d x \]
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