Integrand size = 23, antiderivative size = 186 \[ \int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx=\frac {e^{d+e x}}{c e}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^2}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )}{2 c^2} \]
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Time = 0.26 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2302, 2225, 2209} \[ \int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx=-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^2}-\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^2}+\frac {e^{d+e x}}{c e} \]
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Rule 2209
Rule 2225
Rule 2302
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^{d+e x}}{c}-\frac {e^{d+e x} (a+b x)}{c \left (a+b x+c x^2\right )}\right ) \, dx \\ & = \frac {\int e^{d+e x} \, dx}{c}-\frac {\int \frac {e^{d+e x} (a+b x)}{a+b x+c x^2} \, dx}{c} \\ & = \frac {e^{d+e x}}{c e}-\frac {\int \left (\frac {\left (b+\frac {-b^2+2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (b-\frac {-b^2+2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{c} \\ & = \frac {e^{d+e x}}{c e}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {e^{d+e x}}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{c}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {e^{d+e x}}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{c} \\ & = \frac {e^{d+e x}}{c e}-\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 c^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 c^2} \\ \end{align*}
Time = 0.89 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.17 \[ \int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx=-\frac {e^{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}} \left (-2 c \sqrt {b^2-4 a c} e^{\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}}+\left (-b^2+2 a c+b \sqrt {b^2-4 a c}\right ) e e^{\frac {\sqrt {b^2-4 a c} e}{c}} \operatorname {ExpIntegralEi}\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )+\left (b^2-2 a c+b \sqrt {b^2-4 a c}\right ) e \operatorname {ExpIntegralEi}\left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}\right )\right )}{2 c^2 \sqrt {b^2-4 a c} e} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(560\) vs. \(2(160)=320\).
Time = 0.61 (sec) , antiderivative size = 561, normalized size of antiderivative = 3.02
method | result | size |
risch | \(\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) a}{c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2}}{2 c^{2} \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}-\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) a}{c \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {e \,{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b^{2}}{2 c^{2} \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}+\frac {{\mathrm e}^{-\frac {b e -2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (\frac {-b e +2 c d -2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b}{2 c^{2}}+\frac {{\mathrm e}^{-\frac {b e -2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}} \operatorname {Ei}_{1}\left (-\frac {b e -2 c d +2 c \left (e x +d \right )+\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2 c}\right ) b}{2 c^{2}}+\frac {{\mathrm e}^{e x +d}}{c e}\) | \(561\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1730\) |
default | \(\text {Expression too large to display}\) | \(1730\) |
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Time = 0.27 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.44 \[ \int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx=-\frac {{\left ({\left (b^{3} - 4 \, a b c\right )} e - {\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} + {\left ({\left (b^{3} - 4 \, a b c\right )} e + {\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e^{\left (e x + d\right )}}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e} \]
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\[ \int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx=e^{d} \int \frac {x^{2} e^{e x}}{a + b x + c x^{2}}\, dx \]
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\[ \int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx=\int { \frac {x^{2} e^{\left (e x + d\right )}}{c x^{2} + b x + a} \,d x } \]
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\[ \int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx=\int { \frac {x^{2} e^{\left (e x + d\right )}}{c x^{2} + b x + a} \,d x } \]
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Timed out. \[ \int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx=\int \frac {x^2\,{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \]
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