Integrand size = 33, antiderivative size = 99 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x^2 \, dx=\frac {e^{(a+b x)^3}}{3 b^3}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b^3 \sqrt [3]{-(a+b x)^3}}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{3 b^3 \left (-(a+b x)^3\right )^{2/3}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2259, 2258, 2239, 2250, 2240} \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x^2 \, dx=-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b^3 \sqrt [3]{-(a+b x)^3}}+\frac {e^{(a+b x)^3}}{3 b^3}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{3 b^3 \left (-(a+b x)^3\right )^{2/3}} \]
[In]
[Out]
Rule 2239
Rule 2240
Rule 2250
Rule 2258
Rule 2259
Rubi steps \begin{align*} \text {integral}& = \int e^{(a+b x)^3} x^2 \, dx \\ & = \int \left (\frac {a^2 e^{(a+b x)^3}}{b^2}-\frac {2 a e^{(a+b x)^3} (a+b x)}{b^2}+\frac {e^{(a+b x)^3} (a+b x)^2}{b^2}\right ) \, dx \\ & = \frac {\int e^{(a+b x)^3} (a+b x)^2 \, dx}{b^2}-\frac {(2 a) \int e^{(a+b x)^3} (a+b x) \, dx}{b^2}+\frac {a^2 \int e^{(a+b x)^3} \, dx}{b^2} \\ & = \frac {e^{(a+b x)^3}}{3 b^3}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b^3 \sqrt [3]{-(a+b x)^3}}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{3 b^3 \left (-(a+b x)^3\right )^{2/3}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.90 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x^2 \, dx=\frac {e^{(a+b x)^3}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{\sqrt [3]{-(a+b x)^3}}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{\left (-(a+b x)^3\right )^{2/3}}}{3 b^3} \]
[In]
[Out]
\[\int {\mathrm e}^{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}} x^{2}d x\]
[In]
[Out]
none
Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.25 \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x^2 \, dx=\frac {\left (-b^{3}\right )^{\frac {2}{3}} a^{2} \Gamma \left (\frac {1}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right ) - 2 \, \left (-b^{3}\right )^{\frac {1}{3}} a b \Gamma \left (\frac {2}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right ) + b^{2} e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )}}{3 \, b^{5}} \]
[In]
[Out]
\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x^2 \, dx=e^{a^{3}} \int x^{2} e^{b^{3} x^{3}} e^{3 a b^{2} x^{2}} e^{3 a^{2} b x}\, dx \]
[In]
[Out]
\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x^2 \, dx=\int { x^{2} e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \,d x } \]
[In]
[Out]
\[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x^2 \, dx=\int { x^{2} e^{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3} x^2 \, dx=\int x^2\,{\mathrm {e}}^{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \,d x \]
[In]
[Out]