Optimal. Leaf size=125 \[ -\frac {(b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b e e^{\frac {e}{c+d x}} (c+d x)}{2 d^2}+\frac {b e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^2}+\frac {(b c-a d) e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^2}-\frac {b e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2258, 2237,
2241, 2245} \begin {gather*} \frac {e (b c-a d) \text {Ei}\left (\frac {e}{c+d x}\right )}{d^2}-\frac {(c+d x) (b c-a d) e^{\frac {e}{c+d x}}}{d^2}-\frac {b e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^2}+\frac {b e (c+d x) e^{\frac {e}{c+d x}}}{2 d^2}+\frac {b (c+d x)^2 e^{\frac {e}{c+d x}}}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2237
Rule 2241
Rule 2245
Rule 2258
Rubi steps
\begin {align*} \int e^{\frac {e}{c+d x}} (a+b x) \, dx &=\int \left (\frac {(-b c+a d) e^{\frac {e}{c+d x}}}{d}+\frac {b e^{\frac {e}{c+d x}} (c+d x)}{d}\right ) \, dx\\ &=\frac {b \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d}+\frac {(-b c+a d) \int e^{\frac {e}{c+d x}} \, dx}{d}\\ &=-\frac {(b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^2}+\frac {(b e) \int e^{\frac {e}{c+d x}} \, dx}{2 d}+\frac {((-b c+a d) e) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d}\\ &=-\frac {(b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b e e^{\frac {e}{c+d x}} (c+d x)}{2 d^2}+\frac {b e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^2}+\frac {(b c-a d) e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^2}+\frac {\left (b e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 d}\\ &=-\frac {(b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b e e^{\frac {e}{c+d x}} (c+d x)}{2 d^2}+\frac {b e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^2}+\frac {(b c-a d) e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^2}-\frac {b e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 91, normalized size = 0.73 \begin {gather*} \frac {c (2 a d+b (-c+e)) e^{\frac {e}{c+d x}}}{2 d^2}+\frac {d e^{\frac {e}{c+d x}} x (2 a d+b (e+d x))-e (2 a d+b (-2 c+e)) \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 150, normalized size = 1.20
method | result | size |
derivativedivides | \(-\frac {e \left (a \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\expIntegral \left (1, -\frac {e}{d x +c}\right )\right )+\frac {b e \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\expIntegral \left (1, -\frac {e}{d x +c}\right )}{2}\right )}{d}-\frac {b c \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\expIntegral \left (1, -\frac {e}{d x +c}\right )\right )}{d}\right )}{d}\) | \(150\) |
default | \(-\frac {e \left (a \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\expIntegral \left (1, -\frac {e}{d x +c}\right )\right )+\frac {b e \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\expIntegral \left (1, -\frac {e}{d x +c}\right )}{2}\right )}{d}-\frac {b c \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\expIntegral \left (1, -\frac {e}{d x +c}\right )\right )}{d}\right )}{d}\) | \(150\) |
risch | \(a \,{\mathrm e}^{\frac {e}{d x +c}} x +\frac {a \,{\mathrm e}^{\frac {e}{d x +c}} c}{d}+\frac {e a \expIntegral \left (1, -\frac {e}{d x +c}\right )}{d}+\frac {b \,{\mathrm e}^{\frac {e}{d x +c}} x^{2}}{2}-\frac {b \,{\mathrm e}^{\frac {e}{d x +c}} c^{2}}{2 d^{2}}+\frac {e b \,{\mathrm e}^{\frac {e}{d x +c}} x}{2 d}+\frac {e b \,{\mathrm e}^{\frac {e}{d x +c}} c}{2 d^{2}}+\frac {e^{2} b \expIntegral \left (1, -\frac {e}{d x +c}\right )}{2 d^{2}}-\frac {e b c \expIntegral \left (1, -\frac {e}{d x +c}\right )}{d^{2}}\) | \(161\) |
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 = 0.35, size = 86, normalized size = 0.69 \begin {gather*} -\frac {{\left (b e^{2} - 2 \, {\left (b c - a d\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (b d^{2} x^{2} + 2 \, a d^{2} x - b c^{2} + 2 \, a c d + {\left (b d x + b c\right )} e\right )} e^{\left (\frac {e}{d x + c}\right )}}{2 \, d^{2}} \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*} \int \left (a + b x\right ) e^{\frac {e}{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.41, size = 171, normalized size = 1.37 \begin {gather*} \frac {{\left (d x + c\right )}^{2} {\left (\frac {2 \, b c {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{4}}{{\left (d x + c\right )}^{2}} - \frac {2 \, a d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{4}}{{\left (d x + c\right )}^{2}} + b e^{\left (\frac {e}{d x + c} + 3\right )} - \frac {2 \, b c e^{\left (\frac {e}{d x + c} + 3\right )}}{d x + c} + \frac {2 \, a d e^{\left (\frac {e}{d x + c} + 3\right )}}{d x + c} - \frac {b {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{5}}{{\left (d x + c\right )}^{2}} + \frac {b e^{\left (\frac {e}{d x + c} + 4\right )}}{d x + c}\right )} e^{\left (-3\right )}}{2 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.67, size = 153, normalized size = 1.22 \begin {gather*} \frac {\frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (2\,a\,c^2\,d-b\,c^3+b\,c^2\,e\right )}{2\,d^2}+x\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (2\,a\,c-\frac {\frac {b\,c^2}{2}-b\,c\,e}{d}\right )+x^2\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (a\,d+\frac {b\,c}{2}+\frac {b\,e}{2}\right )+\frac {b\,d\,x^3\,{\mathrm {e}}^{\frac {e}{c+d\,x}}}{2}}{c+d\,x}-\frac {\mathrm {ei}\left (\frac {e}{c+d\,x}\right )\,\left (b\,e^2+2\,a\,d\,e-2\,b\,c\,e\right )}{2\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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