Optimal. Leaf size=62 \[ -\frac {\text {Ei}\left (\frac {e}{c+d x}\right )}{b}+\frac {e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b} \]
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Rubi [A]
time = 0.14, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2254, 2241,
2260, 2209} \begin {gather*} \frac {e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b}-\frac {\text {Ei}\left (\frac {e}{c+d x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2241
Rule 2254
Rule 2260
Rubi steps
\begin {align*} \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx &=\frac {d \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{b}-\frac {(-b c+a d) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x) (c+d x)} \, dx}{b}\\ &=-\frac {\text {Ei}\left (\frac {e}{c+d x}\right )}{b}+\frac {\text {Subst}\left (\int \frac {\exp \left (-\frac {b e}{-b c+a d}+\frac {d e x}{-b c+a d}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b}\\ &=-\frac {\text {Ei}\left (\frac {e}{c+d x}\right )}{b}+\frac {e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 56, normalized size = 0.90 \begin {gather*} \frac {-\text {Ei}\left (\frac {e}{c+d x}\right )+e^{\frac {b e}{b c-a d}} \text {Ei}\left (e \left (\frac {b}{-b c+a d}+\frac {1}{c+d x}\right )\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 79, normalized size = 1.27
method | result | size |
risch | \(\frac {\expIntegral \left (1, -\frac {e}{d x +c}\right )}{b}-\frac {{\mathrm e}^{-\frac {b e}{a d -c b}} \expIntegral \left (1, -\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{b}\) | \(65\) |
derivativedivides | \(-\frac {e \left (-\frac {d \expIntegral \left (1, -\frac {e}{d x +c}\right )}{b e}+\frac {d \,{\mathrm e}^{-\frac {b e}{a d -c b}} \expIntegral \left (1, -\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{b e}\right )}{d}\) | \(79\) |
default | \(-\frac {e \left (-\frac {d \expIntegral \left (1, -\frac {e}{d x +c}\right )}{b e}+\frac {d \,{\mathrm e}^{-\frac {b e}{a d -c b}} \expIntegral \left (1, -\frac {e}{d x +c}-\frac {b e}{a d -c b}\right )}{b e}\right )}{d}\) | \(79\) |
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 = 0.39, size = 73, normalized size = 1.18 \begin {gather*} \frac {{\rm Ei}\left (-\frac {{\left (b d x + a d\right )} e}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac {b e}{b c - a d}\right )} - {\rm Ei}\left (\frac {e}{d x + c}\right )}{b} \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 \frac {e^{\frac {e}{c + d x}}}{a + b x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 492 vs.
\(2 (64) = 128\).
time = 2.03, size = 492, normalized size = 7.94 \begin {gather*} \frac {{\left (\frac {2 \, b^{2} c^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{3}}{{\left (d x + c\right )}^{2}} - \frac {4 \, a b c d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{3}}{{\left (d x + c\right )}^{2}} + \frac {2 \, a^{2} d^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{3}}{{\left (d x + c\right )}^{2}} - \frac {2 \, b^{2} c^{2} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 3\right )}}{{\left (d x + c\right )}^{2}} + \frac {4 \, a b c d {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 3\right )}}{{\left (d x + c\right )}^{2}} - \frac {2 \, a^{2} d^{2} {\rm Ei}\left (-\frac {b e - \frac {b c e}{d x + c} + \frac {a d e}{d x + c}}{b c - a d}\right ) e^{\left (\frac {b e}{b c - a d} + 3\right )}}{{\left (d x + c\right )}^{2}} + \frac {2 \, b^{2} c {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{4}}{{\left (d x + c\right )}^{2}} - \frac {2 \, a b d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{4}}{{\left (d x + c\right )}^{2}} - b^{2} e^{\left (\frac {e}{d x + c} + 3\right )} - \frac {2 \, b^{2} c e^{\left (\frac {e}{d x + c} + 3\right )}}{d x + c} + \frac {2 \, a b d e^{\left (\frac {e}{d x + c} + 3\right )}}{d x + c} + \frac {b^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{5}}{{\left (d x + c\right )}^{2}} - \frac {b^{2} e^{\left (\frac {e}{d x + c} + 4\right )}}{d x + c}\right )} {\left (d x + c\right )}^{2} e^{\left (-4\right )}}{2 \, b^{3} d} \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 = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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