Optimal. Leaf size=92 \[ \frac {(c+d x) (b c-a d) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{3 d^2 \sqrt [3]{-e (c+d x)^3}}-\frac {b (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]
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Rubi [A] time = 0.06, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2226, 2208, 2218} \[ \frac {(c+d x) (b c-a d) \text {Gamma}\left (\frac {1}{3},-e (c+d x)^3\right )}{3 d^2 \sqrt [3]{-e (c+d x)^3}}-\frac {b (c+d x)^2 \text {Gamma}\left (\frac {2}{3},-e (c+d x)^3\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 2218
Rule 2226
Rubi steps
\begin {align*} \int e^{e (c+d x)^3} (a+b x) \, dx &=\int \left (\frac {(-b c+a d) e^{e (c+d x)^3}}{d}+\frac {b e^{e (c+d x)^3} (c+d x)}{d}\right ) \, dx\\ &=\frac {b \int e^{e (c+d x)^3} (c+d x) \, dx}{d}+\frac {(-b c+a d) \int e^{e (c+d x)^3} \, dx}{d}\\ &=\frac {(b c-a d) (c+d x) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{3 d^2 \sqrt [3]{-e (c+d x)^3}}-\frac {b (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 86, normalized size = 0.93 \[ -\frac {(c+d x) \left (b (c+d x) \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )-(b c-a d) \sqrt [3]{-e (c+d x)^3} \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )\right )}{3 d^2 \left (-e (c+d x)^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 110, normalized size = 1.20 \[ \frac {\left (-d^{3} e\right )^{\frac {1}{3}} b d \Gamma \left (\frac {2}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right ) - \left (-d^{3} e\right )^{\frac {2}{3}} {\left (b c - a d\right )} \Gamma \left (\frac {1}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right )}{3 \, d^{4} e} \]
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 \[ \int {\left (b x + a\right )} e^{\left ({\left (d x + c\right )}^{3} e\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right ) {\mathrm e}^{\left (d x +c \right )^{3} e}\, dx \]
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 \[ \int {\left (b x + a\right )} e^{\left ({\left (d x + c\right )}^{3} e\right )}\,{d x} \]
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 \[ \int {\mathrm {e}}^{e\,{\left (c+d\,x\right )}^3}\,\left (a+b\,x\right ) \,d x \]
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 \[ \left (\int a e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx + \int b x e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx\right ) e^{c^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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