Optimal. Leaf size=158 \[ \frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c}+\frac {\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c} \]
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Rubi [A]
time = 0.15, 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 = {2302, 2209}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2302
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 = 0.17, size = 153, normalized size = 0.97 \begin {gather*} \frac {e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \left (\left (-b+\sqrt {b^2-4 a c}\right ) e^{\frac {\sqrt {b^2-4 a c} e}{c}} \text {Ei}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )+\left (b+\sqrt {b^2-4 a c}\right ) \text {Ei}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )\right )}{2 c \sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(684\) vs.
\(2(132)=264\).
time = 0.10, size = 685, normalized size = 4.34
method | result | size |
risch | \(-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b}{2 c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b}{2 c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )}{2 c}-\frac {{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )}{2 c}\) | \(350\) |
derivativedivides | \(\frac {-\frac {e^{2} \left (-{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e +2 \,{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) c d +{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e -2 \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) c d +{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}{2 c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {d \,e^{2} \left ({\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )-{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}}{e^{2}}\) | \(685\) |
default | \(\frac {-\frac {e^{2} \left (-{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e +2 \,{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) c d +{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b e -2 \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) c d +{\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}{2 c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {d \,e^{2} \left ({\mathrm e}^{\frac {-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, \frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )-{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \expIntegral \left (1, -\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right )\right )}{\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}}{e^{2}}\) | \(685\) |
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 = 0.36, size = 221, normalized size = 1.40 \begin {gather*} -\frac {{\left ({\left (b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e - {\left (b^{2} - 4 \, a c\right )} e\right )} {\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}\right )} - {\left (b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} e + {\left (b^{2} - 4 \, a c\right )} e\right )} {\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}\right )}\right )} e^{\left (-1\right )}}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}} \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 {x 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 {x\,{\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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