Optimal. Leaf size=138 \[ \frac {e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}} \]
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Rubi [A]
time = 0.14, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2300, 2209}
\begin {gather*} \frac {e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{\sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2300
Rubi steps
\begin {align*} \int \frac {e^{d+e x}}{a+b x+c x^2} \, dx &=\int \left (\frac {2 c e^{d+e x}}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}-\frac {2 c e^{d+e x}}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \, dx\\ &=\frac {(2 c) \int \frac {e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}-\frac {e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 127, normalized size = 0.92 \begin {gather*} \frac {e^{d+\frac {\left (-b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )-e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{\sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 169, normalized size = 1.22
method | result | size |
derivativedivides | \(-\frac {e \left ({\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )-{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\) | \(169\) |
default | \(-\frac {e \left ({\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )-{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\) | \(169\) |
risch | \(\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}\) | \(186\) |
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 = 0.38, size = 187, normalized size = 1.36 \begin {gather*} \frac {{\left (c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} {\rm Ei}\left (-\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e - {\left (2 \, c x + b\right )} e}{2 \, c}\right ) e^{\left (\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e + 2 \, c d - b e}{2 \, c} + 1\right )} - c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} {\rm Ei}\left (\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e + {\left (2 \, c x + b\right )} e}{2 \, c}\right ) e^{\left (-\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e - 2 \, c d + b e}{2 \, c} + 1\right )}\right )} e^{\left (-1\right )}}{b^{2} - 4 \, a c} \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*} e^{d} \int \frac {e^{e x}}{a + b x + c x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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