Integrand size = 29, antiderivative size = 38 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=-\frac {(a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2259, 2239} \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=-\frac {(a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]
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Rule 2239
Rule 2259
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=-\frac {(a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b \sqrt [3]{-(a+b x)^3}} \]
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\[\int {\mathrm e}^{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x\]
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none
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=\frac {\left (-b^{3}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right )}{3 \, b^{3}} \]
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\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=e^{a^{3}} \int e^{b^{3} x^{3}} e^{3 a b^{2} x^{2}} e^{3 a^{2} b x}\, dx \]
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\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=\int { e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \,d x } \]
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\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=\int { e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \,d x } \]
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Timed out. \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} \, dx=\int {\mathrm {e}}^{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \,d x \]
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