Optimal. Leaf size=84 \[ \frac {\sqrt {\pi } \sqrt {\log (f)} f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {f^{a+b x+c x^2}}{2 c (b+2 c x)} \]
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Rubi [A] time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2239, 2234, 2204} \[ \frac {\sqrt {\pi } \sqrt {\log (f)} f^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {f^{a+b x+c x^2}}{2 c (b+2 c x)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2239
Rubi steps
\begin {align*} \int \frac {f^{a+b x+c x^2}}{(b+2 c x)^2} \, dx &=-\frac {f^{a+b x+c x^2}}{2 c (b+2 c x)}+\frac {\log (f) \int f^{a+b x+c x^2} \, dx}{2 c}\\ &=-\frac {f^{a+b x+c x^2}}{2 c (b+2 c x)}+\frac {\left (f^{a-\frac {b^2}{4 c}} \log (f)\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{2 c}\\ &=-\frac {f^{a+b x+c x^2}}{2 c (b+2 c x)}+\frac {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)}}{4 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 96, normalized size = 1.14 \[ \frac {f^{a-\frac {b^2}{4 c}} \left (\sqrt {\pi } \sqrt {\log (f)} (b+2 c x) \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )-2 \sqrt {c} f^{\frac {(b+2 c x)^2}{4 c}}\right )}{4 c^{3/2} (b+2 c x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 85, normalized size = 1.01 \[ -\frac {2 \, c f^{c x^{2} + b x + a} + \frac {\sqrt {\pi } {\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{4 \, {\left (2 \, c^{3} x + b c^{2}\right )}} \]
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 \[ \int \frac {f^{c x^{2} + b x + a}}{{\left (2 \, c x + b\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 101, normalized size = 1.20 \[ -\frac {f^{\frac {4 a c -b^{2}}{4 c}} f^{\frac {\left (2 c x +b \right )^{2}}{4 c}}}{2 \left (2 c x +b \right ) c}+\frac {\sqrt {\pi }\, f^{\frac {4 a c -b^{2}}{4 c}} \erf \left (\frac {\sqrt {-\frac {\ln \relax (f )}{c}}\, \left (2 c x +b \right )}{2}\right ) \ln \relax (f )}{4 \sqrt {-\frac {\ln \relax (f )}{c}}\, c^{2}} \]
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 \[ \int \frac {f^{c x^{2} + b x + a}}{{\left (2 \, c x + b\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.14, size = 76, normalized size = 0.90 \[ \frac {f^{a-\frac {b^2}{4\,c}}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \relax (f)}{2}+c\,x\,\ln \relax (f)}{\sqrt {c\,\ln \relax (f)}}\right )\,\ln \relax (f)}{4\,c\,\sqrt {c\,\ln \relax (f)}}-\frac {f^a\,f^{c\,x^2}\,f^{b\,x}}{2\,c\,\left (b+2\,c\,x\right )} \]
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 \[ \int \frac {f^{a + b x + c x^{2}}}{\left (b + 2 c x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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