Optimal. Leaf size=169 \[ -\frac {\left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a}+\frac {e^d \text {Ei}(e x)}{a} \]
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Rubi [A] time = 0.41, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2270, 2178} \[ -\frac {\left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a}+\frac {e^d \text {Ei}(e x)}{a} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2270
Rubi steps
\begin {align*} \int \frac {e^{d+e x}}{x \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {e^{d+e x}}{a x}+\frac {e^{d+e x} (-b-c x)}{a \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {e^{d+e x}}{x} \, dx}{a}+\frac {\int \frac {e^{d+e x} (-b-c x)}{a+b x+c x^2} \, dx}{a}\\ &=\frac {e^d \text {Ei}(e x)}{a}+\frac {\int \left (\frac {\left (-c-\frac {b c}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (-c+\frac {b c}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{a}\\ &=\frac {e^d \text {Ei}(e x)}{a}-\frac {\left (c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{a}-\frac {\left (c \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{a}\\ &=\frac {e^d \text {Ei}(e x)}{a}-\frac {\left (1+\frac {b}{\sqrt {b^2-4 a 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 )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a 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 )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 163, normalized size = 0.96 \[ \frac {e^d \left (\frac {e^{-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \left (\left (b-\sqrt {b^2-4 a c}\right ) \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )-\left (\sqrt {b^2-4 a c}+b\right ) e^{\frac {e \sqrt {b^2-4 a c}}{c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )\right )}{\sqrt {b^2-4 a c}}+2 \text {Ei}(e x)\right )}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 241, normalized size = 1.43 \[ \frac {2 \, {\left (b^{2} - 4 \, a c\right )} e {\rm Ei}\left (e x\right ) e^{d} - {\left (b c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} + {\left (b^{2} - 4 \, a c\right )} e\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} + {\left (b c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} - {\left (b^{2} - 4 \, a c\right )} e\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e} \]
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 \[ \int \frac {e^{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 369, normalized size = 2.18 \[ -\frac {\Ei \left (1, -e x \right ) {\mathrm e}^{d}}{a}+\frac {b e \Ei \left (1, \frac {-b e +2 c d -2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}-b e \Ei \left (1, -\frac {b e -2 c d +2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \Ei \left (1, \frac {-b e +2 c d -2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \Ei \left (1, -\frac {b e -2 c d +2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, a} \]
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 \[ \int \frac {e^{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} x}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{d+e\,x}}{x\,\left (c\,x^2+b\,x+a\right )} \,d x \]
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 \[ e^{d} \int \frac {e^{e x}}{a x + b x^{2} + c x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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