Optimal. Leaf size=44 \[ \frac {\sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2204} \[ \frac {\sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^2} \, dx &=\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d \sqrt {\log (F)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 1.00 \[ \frac {\sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 48, normalized size = 1.09 \[ -\frac {\sqrt {\pi } \sqrt {-b d^{2} \log \relax (F)} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right )}{2 \, b d^{2} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 36, normalized size = 0.82 \[ -\frac {\sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right )}{2 \, \sqrt {-b \log \relax (F)} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 58, normalized size = 1.32 \[ -\frac {\sqrt {\pi }\, F^{-b \,c^{2}} F^{b \,c^{2}+a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 58, normalized size = 1.32 \[ \frac {\sqrt {\pi } F^{b c^{2} + a} \operatorname {erf}\left (\sqrt {-b \log \relax (F)} d x - \frac {b c \log \relax (F)}{\sqrt {-b \log \relax (F)}}\right )}{2 \, \sqrt {-b \log \relax (F)} F^{b c^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 48, normalized size = 1.09 \[ -\frac {F^a\,\sqrt {\pi }\,\mathrm {erf}\left (\frac {1{}\mathrm {i}\,b\,x\,\ln \relax (F)\,d^2+1{}\mathrm {i}\,b\,c\,\ln \relax (F)\,d}{\sqrt {b\,d^2\,\ln \relax (F)}}\right )\,1{}\mathrm {i}}{2\,\sqrt {b\,d^2\,\ln \relax (F)}} \]
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 \[ \int F^{a + b \left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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