Optimal. Leaf size=158 \[ \frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c}+\frac {\left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c} \]
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Rubi [A] time = 0.21, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2270, 2178} \[ \frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c}+\frac {\left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2270
Rubi steps
\begin {align*} \int \frac {e^{d+e x} x}{a+b x+c x^2} \, dx &=\int \left (\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx\\ &=\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x} \, dx+\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x} \, dx\\ &=\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 c}+\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 c}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 153, normalized size = 0.97 \[ \frac {e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \left (\left (\sqrt {b^2-4 a c}-b\right ) e^{\frac {e \sqrt {b^2-4 a c}}{c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )+\left (\sqrt {b^2-4 a c}+b\right ) \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )\right )}{2 c \sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 224, normalized size = 1.42 \[ -\frac {{\left (b c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} - {\left (b^{2} - 4 \, a c\right )} e\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - {\left (b c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} + {\left (b^{2} - 4 \, a c\right )} e\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e} \]
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 \[ \int \frac {x e^{\left (e x + d\right )}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 685, normalized size = 4.34 \[ \frac {\frac {\left (\Ei \left (1, \frac {-b e +2 c d -2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}-\Ei \left (1, -\frac {b e -2 c d +2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}\right ) d \,e^{2}}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {\left (-b e \Ei \left (1, \frac {-b e +2 c d -2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}+b e \Ei \left (1, -\frac {b e -2 c d +2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}+2 c d \Ei \left (1, \frac {-b e +2 c d -2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}-2 c d \Ei \left (1, -\frac {b e -2 c d +2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \Ei \left (1, \frac {-b e +2 c d -2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \Ei \left (1, -\frac {b e -2 c d +2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}\right ) e^{2}}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c}}{e^{2}} \]
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 \[ \frac {x e^{\left (e x + d\right )}}{c e x^{2} + b e x + a e} + \int \frac {{\left (c x^{2} e^{d} - a e^{d}\right )} e^{\left (e x\right )}}{c^{2} e x^{4} + 2 \, b c e x^{3} + 2 \, a b e x + a^{2} e + {\left (b^{2} e + 2 \, a c e\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {x\,{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \]
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 \[ e^{d} \int \frac {x e^{e x}}{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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