Optimal. Leaf size=38 \[ \frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{\sqrt {b} \sqrt {\log (F)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2211, 2235}
\begin {gather*} \frac {\sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{\sqrt {b} \sqrt {\log (F)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rubi steps
\begin {align*} \int \frac {F^{a+b x}}{\sqrt {x}} \, dx &=2 \text {Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{\sqrt {b} \sqrt {\log (F)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 30, normalized size = 0.79 \begin {gather*} -\frac {F^a \sqrt {x} \Gamma \left (\frac {1}{2},-b x \log (F)\right )}{\sqrt {-b x \log (F)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 27, normalized size = 0.71
method | result | size |
meijerg | \(\frac {F^{a} \erfi \left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \left (F \right )}\right ) \sqrt {\pi }}{\sqrt {b}\, \sqrt {\ln \left (F \right )}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 29, normalized size = 0.76 \begin {gather*} \frac {\sqrt {\pi } F^{a} \sqrt {x} {\left (\operatorname {erf}\left (\sqrt {-b x \log \left (F\right )}\right ) - 1\right )}}{\sqrt {-b x \log \left (F\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 34, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-b \log \left (F\right )} F^{a} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} \sqrt {x}\right )}{b \log \left (F\right )} \end {gather*}
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 \begin {gather*} \int \frac {F^{a + b x}}{\sqrt {x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.09, size = 28, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} \sqrt {x}\right )}{\sqrt {-b \log \left (F\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.50, size = 32, normalized size = 0.84 \begin {gather*} \frac {F^a\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \left (F\right )}\right )\,\sqrt {-\pi \,b\,x\,\ln \left (F\right )}}{b\,\sqrt {x}\,\ln \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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