Optimal. Leaf size=186 \[ -\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^2}-\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^2}+\frac {e^{d+e x}}{c e} \]
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Rubi [A] time = 0.41, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2270, 2194, 2178} \[ -\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) e^{d-\frac {e \left (b-\sqrt {b^2-4 a c}\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^2}-\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )}{2 c^2}+\frac {e^{d+e x}}{c e} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2194
Rule 2270
Rubi steps
\begin {align*} \int \frac {e^{d+e x} x^2}{a+b x+c x^2} \, dx &=\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*}
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Mathematica [A] time = 0.53, size = 217, normalized size = 1.17 \[ -\frac {e^{d-\frac {e \left (\sqrt {b^2-4 a c}+b\right )}{2 c}} \left (e \left (b \sqrt {b^2-4 a c}+2 a c-b^2\right ) e^{\frac {e \sqrt {b^2-4 a c}}{c}} \text {Ei}\left (\frac {e \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{2 c}\right )+e \left (b \sqrt {b^2-4 a c}-2 a c+b^2\right ) \text {Ei}\left (\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c}\right )-2 c \sqrt {b^2-4 a c} e^{\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c}}\right )}{2 c^2 e \sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 267, normalized size = 1.44 \[ -\frac {{\left ({\left (b^{3} - 4 \, a b c\right )} e - {\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )} {\rm Ei}\left (\frac {2 \, c e x + b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} + {\left ({\left (b^{3} - 4 \, a b c\right )} e + {\left (b^{2} c - 2 \, a c^{2}\right )} \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}\right )} {\rm Ei}\left (\frac {2 \, c e x + b e + c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right ) e^{\left (\frac {2 \, c d - b e - c \sqrt {\frac {{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{2 \, c}\right )} - 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e^{\left (e x + d\right )}}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e} \]
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 \[ \int \frac {x^{2} e^{\left (e x + d\right )}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1730, normalized size = 9.30 \[ \text {result too large to display} \]
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 \[ \frac {x^{2} e^{\left (e x + d\right )}}{c e x^{2} + b e x + a e} - \int \frac {{\left (b x^{2} e^{d} + 2 \, a x e^{d}\right )} e^{\left (e x\right )}}{c^{2} e x^{4} + 2 \, b c e x^{3} + 2 \, a b e x + a^{2} e + {\left (b^{2} e + 2 \, a c e\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {x^2\,{\mathrm {e}}^{d+e\,x}}{c\,x^2+b\,x+a} \,d x \]
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 \[ e^{d} \int \frac {x^{2} e^{e x}}{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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