Integrand size = 19, antiderivative size = 206 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=-\frac {b^2 (b c-a d) e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{d^4}+\frac {b^2 (b c-a d) e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )}{d^4}+\frac {b^3 \left (-\frac {e}{(c+d x)^3}\right )^{4/3} (c+d x)^4 \Gamma \left (-\frac {4}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4}+\frac {b (b c-a d)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{d^4}-\frac {(b c-a d)^3 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4} \]
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Time = 0.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2258, 2239, 2250, 2245, 2241} \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\frac {b^2 e (b c-a d) \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )}{d^4}-\frac {b^2 (c+d x)^3 (b c-a d) e^{\frac {e}{(c+d x)^3}}}{d^4}+\frac {b (c+d x)^2 (b c-a d)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{d^4}-\frac {(c+d x) (b c-a d)^3 \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4}+\frac {b^3 (c+d x)^4 \left (-\frac {e}{(c+d x)^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4} \]
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Rule 2239
Rule 2241
Rule 2245
Rule 2250
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^3 e^{\frac {e}{(c+d x)^3}}}{d^3}+\frac {3 b (b c-a d)^2 e^{\frac {e}{(c+d x)^3}} (c+d x)}{d^3}-\frac {3 b^2 (b c-a d) e^{\frac {e}{(c+d x)^3}} (c+d x)^2}{d^3}+\frac {b^3 e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{d^3}\right ) \, dx \\ & = \frac {b^3 \int e^{\frac {e}{(c+d x)^3}} (c+d x)^3 \, dx}{d^3}-\frac {\left (3 b^2 (b c-a d)\right ) \int e^{\frac {e}{(c+d x)^3}} (c+d x)^2 \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2\right ) \int e^{\frac {e}{(c+d x)^3}} (c+d x) \, dx}{d^3}-\frac {(b c-a d)^3 \int e^{\frac {e}{(c+d x)^3}} \, dx}{d^3} \\ & = -\frac {b^2 (b c-a d) e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{d^4}+\frac {b^3 \left (-\frac {e}{(c+d x)^3}\right )^{4/3} (c+d x)^4 \Gamma \left (-\frac {4}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4}+\frac {b (b c-a d)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{d^4}-\frac {(b c-a d)^3 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4}-\frac {\left (3 b^2 (b c-a d) e\right ) \int \frac {e^{\frac {e}{(c+d x)^3}}}{c+d x} \, dx}{d^3} \\ & = -\frac {b^2 (b c-a d) e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{d^4}+\frac {b^2 (b c-a d) e \text {Ei}\left (\frac {e}{(c+d x)^3}\right )}{d^4}+\frac {b^3 \left (-\frac {e}{(c+d x)^3}\right )^{4/3} (c+d x)^4 \Gamma \left (-\frac {4}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4}+\frac {b (b c-a d)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{d^4}-\frac {(b c-a d)^3 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.95 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\frac {-3 b^2 (b c-a d) e^{\frac {e}{(c+d x)^3}} (c+d x)^3+3 b^2 (b c-a d) e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )+b^3 \left (-\frac {e}{(c+d x)^3}\right )^{4/3} (c+d x)^4 \Gamma \left (-\frac {4}{3},-\frac {e}{(c+d x)^3}\right )+3 b (b c-a d)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )-(b c-a d)^3 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^4} \]
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\[\int {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}} \left (b x +a \right )^{3}d x\]
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Time = 0.10 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.69 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\frac {4 \, {\left (b^{3} c - a b^{2} d\right )} e {\rm Ei}\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 6 \, {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + {\left (4 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 12 \, a^{2} b c d^{3} - 4 \, a^{3} d^{4} - 3 \, b^{3} d e\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + {\left (b^{3} d^{4} x^{4} + 4 \, a b^{2} d^{4} x^{3} + 6 \, a^{2} b d^{4} x^{2} - b^{3} c^{4} + 4 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} + 4 \, a^{3} c d^{3} + 3 \, b^{3} c e + {\left (4 \, a^{3} d^{4} + 3 \, b^{3} d e\right )} x\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{4 \, d^{4}} \]
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\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\int \left (a + b x\right )^{3} e^{\frac {e}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}}\, dx \]
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\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\int { {\left (b x + a\right )}^{3} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \]
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\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\int { {\left (b x + a\right )}^{3} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \]
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Timed out. \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^3 \, dx=\int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}\,{\left (a+b\,x\right )}^3 \,d x \]
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