Optimal. Leaf size=94 \[ -\frac {f^{a+b x+c x^2}}{x}+\sqrt {c} f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)}+b \log (f) \text {Int}\left (\frac {f^{a+b x+c x^2}}{x},x\right ) \]
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Rubi [A]
time = 0.06, 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 = {}
\begin {gather*} \int \frac {f^{a+b x+c x^2}}{x^2} \, dx \end {gather*}
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 = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{a+b x+c x^2}}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.01, 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 \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 = 0.00, 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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{a + b x + c x^{2}}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {f^{c\,x^2+b\,x+a}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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