Optimal. Leaf size=177 \[ -\frac{b (c+d x)^2 (b c-a d)^2 \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{d^4 \left (-e (c+d x)^3\right )^{2/3}}+\frac{(c+d x) (b c-a d)^3 \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d^4 \sqrt [3]{-e (c+d x)^3}}-\frac{b^3 (c+d x)^4 \text{Gamma}\left (\frac{4}{3},-e (c+d x)^3\right )}{3 d^4 \left (-e (c+d x)^3\right )^{4/3}}-\frac{b^2 (b c-a d) e^{e (c+d x)^3}}{d^4 e} \]
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Rubi [A] time = 0.155975, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2226, 2208, 2218, 2209} \[ -\frac{b (c+d x)^2 (b c-a d)^2 \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{d^4 \left (-e (c+d x)^3\right )^{2/3}}+\frac{(c+d x) (b c-a d)^3 \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{3 d^4 \sqrt [3]{-e (c+d x)^3}}-\frac{b^3 (c+d x)^4 \text{Gamma}\left (\frac{4}{3},-e (c+d x)^3\right )}{3 d^4 \left (-e (c+d x)^3\right )^{4/3}}-\frac{b^2 (b c-a d) e^{e (c+d x)^3}}{d^4 e} \]
Antiderivative was successfully verified.
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Rule 2226
Rule 2208
Rule 2218
Rule 2209
Rubi steps
\begin{align*} \int e^{e (c+d x)^3} (a+b x)^3 \, dx &=\int \left (\frac{(-b c+a d)^3 e^{e (c+d x)^3}}{d^3}+\frac{3 b (b c-a d)^2 e^{e (c+d x)^3} (c+d x)}{d^3}-\frac{3 b^2 (b c-a d) e^{e (c+d x)^3} (c+d x)^2}{d^3}+\frac{b^3 e^{e (c+d x)^3} (c+d x)^3}{d^3}\right ) \, dx\\ &=\frac{b^3 \int e^{e (c+d x)^3} (c+d x)^3 \, dx}{d^3}-\frac{\left (3 b^2 (b c-a d)\right ) \int e^{e (c+d x)^3} (c+d x)^2 \, dx}{d^3}+\frac{\left (3 b (b c-a d)^2\right ) \int e^{e (c+d x)^3} (c+d x) \, dx}{d^3}-\frac{(b c-a d)^3 \int e^{e (c+d x)^3} \, dx}{d^3}\\ &=-\frac{b^2 (b c-a d) e^{e (c+d x)^3}}{d^4 e}+\frac{(b c-a d)^3 (c+d x) \Gamma \left (\frac{1}{3},-e (c+d x)^3\right )}{3 d^4 \sqrt [3]{-e (c+d x)^3}}-\frac{b (b c-a d)^2 (c+d x)^2 \Gamma \left (\frac{2}{3},-e (c+d x)^3\right )}{d^4 \left (-e (c+d x)^3\right )^{2/3}}-\frac{b^3 (c+d x)^4 \Gamma \left (\frac{4}{3},-e (c+d x)^3\right )}{3 d^4 \left (-e (c+d x)^3\right )^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.213214, size = 167, normalized size = 0.94 \[ \frac{-\frac{3 b (c+d x)^2 (b c-a d)^2 \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{\left (-e (c+d x)^3\right )^{2/3}}+\frac{(c+d x) (b c-a d)^3 \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{\sqrt [3]{-e (c+d x)^3}}+\frac{b^3 (c+d x) \text{Gamma}\left (\frac{4}{3},-e (c+d x)^3\right )}{e \sqrt [3]{-e (c+d x)^3}}-\frac{3 b^2 (b c-a d) e^{e (c+d x)^3}}{e}}{3 d^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{e \left ( dx+c \right ) ^{3}}} \left ( bx+a \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{3} e^{\left ({\left (d x + c\right )}^{3} e\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53813, size = 514, normalized size = 2.9 \begin{align*} \frac{9 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} \left (-d^{3} e\right )^{\frac{1}{3}} e \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^{3} + 3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e\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 \,{\left (b^{3} d^{3} e x -{\left (2 \, b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e\right )} e^{\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + 3 \, c^{2} d e x + c^{3} e\right )}}{9 \, d^{6} e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{3} e^{\left ({\left (d x + c\right )}^{3} e\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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