Optimal. Leaf size=212 \[ \frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a^2}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a^2}-\frac {b e^d \text {Ei}(e x)}{a^2}+\frac {e^d e \text {Ei}(e x)}{a}-\frac {e^{d+e x}}{a x} \]
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Rubi [A] time = 0.62, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2270, 2177, 2178} \[ \frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a^2}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 a^2}-\frac {b e^d \text {Ei}(e x)}{a^2}+\frac {e^d e \text {Ei}(e x)}{a}-\frac {e^{d+e x}}{a x} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
Rule 2270
Rubi steps
\begin {align*} \int \frac {e^{d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {e^{d+e x}}{a x^2}-\frac {b e^{d+e x}}{a^2 x}+\frac {e^{d+e x} \left (b^2-a c+b c x\right )}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {e^{d+e x} \left (b^2-a c+b c x\right )}{a+b x+c x^2} \, dx}{a^2}+\frac {\int \frac {e^{d+e x}}{x^2} \, dx}{a}-\frac {b \int \frac {e^{d+e x}}{x} \, dx}{a^2}\\ &=-\frac {e^{d+e x}}{a x}-\frac {b e^d \text {Ei}(e x)}{a^2}+\frac {\int \left (\frac {\left (b c+\frac {c \left (b^2-2 a c\right )}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (b c-\frac {c \left (b^2-2 a c\right )}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{a^2}+\frac {e \int \frac {e^{d+e x}}{x} \, dx}{a}\\ &=-\frac {e^{d+e x}}{a x}-\frac {b e^d \text {Ei}(e x)}{a^2}+\frac {e e^d \text {Ei}(e x)}{a}+\frac {\left (c \left (b-\frac {b^2-2 a c}{\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^2}+\frac {\left (c \left (b+\frac {b^2-2 a c}{\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^2}\\ &=-\frac {e^{d+e x}}{a x}-\frac {b e^d \text {Ei}(e x)}{a^2}+\frac {e e^d \text {Ei}(e x)}{a}+\frac {\left (b+\frac {b^2-2 a c}{\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^2}+\frac {\left (b-\frac {b^2-2 a c}{\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^2}\\ \end {align*}
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Mathematica [A] time = 1.14, size = 232, normalized size = 1.09 \[ \frac {e^d \left (\frac {e^{-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \left (x \left (b \sqrt {b^2-4 a c}-2 a c+b^2\right ) e^{\frac {e \sqrt {b^2-4 a c}}{c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )+x \left (b \sqrt {b^2-4 a c}+2 a c-b^2\right ) \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )-2 a \sqrt {b^2-4 a c} e^{\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c}}\right )}{x \sqrt {b^2-4 a c}}-2 (b-a e) \text {Ei}(e x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 313, normalized size = 1.48 \[ \frac {2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} e^{2} - {\left (b^{3} - 4 \, a b c\right )} e\right )} x {\rm Ei}\left (e x\right ) e^{d} - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e e^{\left (e x + d\right )} + {\left ({\left (b^{3} - 4 \, a b c\right )} e x + {\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} x\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} + {\left ({\left (b^{3} - 4 \, a b c\right )} e x - {\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}} x\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )}}{2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e x} \]
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 \[ \int \frac {e^{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 561, normalized size = 2.65 \[ \left (-\frac {\left (a e -b \right ) \Ei \left (1, -e x \right ) {\mathrm e}^{d}}{a^{2} e}-\frac {{\mathrm e}^{e x +d}}{a e x}-\frac {-2 a c e \Ei \left (1, \frac {-b e +2 c d -2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}+2 a c e \Ei \left (1, -\frac {b e -2 c d +2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}+b^{2} e \Ei \left (1, \frac {-b e +2 c d -2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}-b^{2} e \Ei \left (1, -\frac {b e -2 c d +2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, b \Ei \left (1, \frac {-b e +2 c d -2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, b \Ei \left (1, -\frac {b e -2 c d +2 \left (e x +d \right ) c +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) {\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}}}{2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, a^{2} e}\right ) e \]
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 \[ \int \frac {e^{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{d+e\,x}}{x^2\,\left (c\,x^2+b\,x+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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