Optimal. Leaf size=177 \[ -\frac {b^2 (b c-a d) e^{e (c+d x)^3}}{d^4 e}+\frac {(b c-a d)^3 (c+d x) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{3 d^4 \sqrt [3]{-e (c+d x)^3}}-\frac {b (b c-a d)^2 (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{d^4 \left (-e (c+d x)^3\right )^{2/3}}-\frac {b^3 (c+d x)^4 \Gamma \left (\frac {4}{3},-e (c+d x)^3\right )}{3 d^4 \left (-e (c+d x)^3\right )^{4/3}} \]
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Rubi [A]
time = 0.10, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2258, 2239,
2250, 2240} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 2240
Rule 2250
Rule 2258
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*}
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Mathematica [A]
time = 0.45, size = 167, normalized size = 0.94 \begin {gather*} \frac {-\frac {3 b^2 (b c-a d) e^{e (c+d x)^3}}{e}+\frac {(b c-a d)^3 (c+d x) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{\sqrt [3]{-e (c+d x)^3}}-\frac {3 b (b c-a d)^2 (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{\left (-e (c+d x)^3\right )^{2/3}}+\frac {b^3 (c+d x) \Gamma \left (\frac {4}{3},-e (c+d x)^3\right )}{e \sqrt [3]{-e (c+d x)^3}}}{3 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{e \left (d x +c \right )^{3}} \left (b x +a \right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.08, size = 231, normalized size = 1.31 \begin {gather*} \frac {{\left (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}, -{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} 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}, -{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} e\right ) + 3 \, {\left (b^{3} d^{3} x - 2 \, b^{3} c d^{2} + 3 \, a b^{2} d^{3}\right )} e^{\left ({\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} e + 1\right )}\right )} e^{\left (-2\right )}}{9 \, d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \left (\int a^{3} e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx + \int b^{3} x^{3} e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx + \int 3 a b^{2} x^{2} e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx + \int 3 a^{2} b x e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx\right ) e^{c^{3} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {e}}^{e\,{\left (c+d\,x\right )}^3}\,{\left (a+b\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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