Optimal. Leaf size=94 \[ b \log (f) \text {Int}\left (\frac {f^{a+b x+c x^2}}{x},x\right )+\sqrt {\pi } \sqrt {c} \sqrt {\log (f)} f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )-\frac {f^{a+b x+c x^2}}{x} \]
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Rubi [A] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {f^{a+b x+c x^2}}{x^2} \, dx &=-\frac {f^{a+b x+c x^2}}{x}+(b \log (f)) \int \frac {f^{a+b x+c x^2}}{x} \, dx+(2 c \log (f)) \int f^{a+b x+c x^2} \, dx\\ &=-\frac {f^{a+b x+c x^2}}{x}+(b \log (f)) \int \frac {f^{a+b x+c x^2}}{x} \, dx+\left (2 c f^{a-\frac {b^2}{4 c}} \log (f)\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx\\ &=-\frac {f^{a+b x+c x^2}}{x}+\sqrt {c} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)}+(b \log (f)) \int \frac {f^{a+b x+c x^2}}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {f^{a+b x+c x^2}}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {f^{c x^{2} + b x + a}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{c x^{2} + b x + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {f^{c \,x^{2}+b x +a}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{c x^{2} + b x + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {f^{c\,x^2+b\,x+a}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + b x + c x^{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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