Optimal. Leaf size=40 \[ \frac {(c+d x) \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2208} \[ \frac {(c+d x) \sqrt [3]{-\frac {e}{(c+d x)^3}} \text {Gamma}\left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 2208
Rubi steps
\begin {align*} \int e^{\frac {e}{(c+d x)^3}} \, dx &=\frac {\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}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ \frac {(c+d x) \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 89, normalized size = 2.22 \[ -\frac {d \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (d x + c\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{d} \]
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 \[ \int e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )}\,{d x} \]
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 \[ \int {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}}\, dx \]
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 \[ 3 \, d e \int \frac {x e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} + x e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.95, size = 61, normalized size = 1.52 \[ \frac {\left (c+d\,x\right )\,\left ({\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}+\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {e}{{\left (c+d\,x\right )}^3}\right )}^{1/3}-{\left (-\frac {e}{{\left (c+d\,x\right )}^3}\right )}^{1/3}\,\Gamma \left (\frac {2}{3},-\frac {e}{{\left (c+d\,x\right )}^3}\right )\right )}{d} \]
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 \[ \int e^{\frac {e}{\left (c + d x\right )^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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