Optimal. Leaf size=37 \[ \frac {(c+d x) e^{\frac {e}{c+d x}}}{d}-\frac {e \text {Ei}\left (\frac {e}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2206, 2210} \[ \frac {(c+d x) e^{\frac {e}{c+d x}}}{d}-\frac {e \text {Ei}\left (\frac {e}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2206
Rule 2210
Rubi steps
\begin {align*} \int e^{\frac {e}{c+d x}} \, dx &=\frac {e^{\frac {e}{c+d x}} (c+d x)}{d}+e \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx\\ &=\frac {e^{\frac {e}{c+d x}} (c+d x)}{d}-\frac {e \text {Ei}\left (\frac {e}{c+d x}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ \frac {(c+d x) e^{\frac {e}{c+d x}}}{d}-\frac {e \text {Ei}\left (\frac {e}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 35, normalized size = 0.95 \[ -\frac {e {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (d x + c\right )} e^{\left (\frac {e}{d x + c}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 49, normalized size = 1.32 \[ -\frac {{\left (d x + c\right )} {\left (\frac {{\rm Ei}\left (\frac {e}{d x + c}\right ) e^{3}}{d x + c} - e^{\left (\frac {e}{d x + c} + 2\right )}\right )} e^{\left (-2\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 1.14 \[ -\frac {\left (-\Ei \left (1, -\frac {e}{d x +c}\right )-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}\right ) e}{d} \]
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 \[ d e \int \frac {x e^{\left (\frac {e}{d x + c}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} + x e^{\left (\frac {e}{d x + c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.62, size = 44, normalized size = 1.19 \[ x\,{\mathrm {e}}^{\frac {e}{c+d\,x}}-\frac {e\,\mathrm {ei}\left (\frac {e}{c+d\,x}\right )-c\,{\mathrm {e}}^{\frac {e}{c+d\,x}}}{d} \]
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 \[ \int e^{\frac {e}{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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