Optimal. Leaf size=39 \[ \frac {f^{a-\frac {b^2}{4 c}} \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c} \]
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Rubi [A] time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2238} \[ \frac {f^{a-\frac {b^2}{4 c}} \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c} \]
Antiderivative was successfully verified.
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Rule 2238
Rubi steps
\begin {align*} \int \frac {f^{a+b x+c x^2}}{b+2 c x} \, dx &=\frac {f^{a-\frac {b^2}{4 c}} \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 1.00 \[ \frac {f^{a-\frac {b^2}{4 c}} \text {Ei}\left (\frac {(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 47, normalized size = 1.21 \[ \frac {{\rm Ei}\left (\frac {{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \relax (f)}{4 \, c}\right )}{4 \, c f^{\frac {b^{2} - 4 \, a c}{4 \, c}}} \]
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 \[ \int \frac {f^{c x^{2} + b x + a}}{2 \, c x + b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 1.03 \[ -\frac {f^{\frac {4 a c -b^{2}}{4 c}} \Ei \left (1, -\frac {\left (2 c x +b \right )^{2} \ln \relax (f )}{4 c}\right )}{4 c} \]
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 \[ \int \frac {f^{c x^{2} + b x + a}}{2 \, c x + b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {f^{c\,x^2+b\,x+a}}{b+2\,c\,x} \,d x \]
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 \[ \int \frac {f^{a + b x + c x^{2}}}{b + 2 c x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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