Optimal. Leaf size=62 \[ \frac {\sqrt {x} F^{a+b x}}{b \log (F)}-\frac {\sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{2 b^{3/2} \log ^{\frac {3}{2}}(F)} \]
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Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2176, 2180, 2204} \[ \frac {\sqrt {x} F^{a+b x}}{b \log (F)}-\frac {\sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{2 b^{3/2} \log ^{\frac {3}{2}}(F)} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2180
Rule 2204
Rubi steps
\begin {align*} \int F^{a+b x} \sqrt {x} \, dx &=\frac {F^{a+b x} \sqrt {x}}{b \log (F)}-\frac {\int \frac {F^{a+b x}}{\sqrt {x}} \, dx}{2 b \log (F)}\\ &=\frac {F^{a+b x} \sqrt {x}}{b \log (F)}-\frac {\operatorname {Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt {x}\right )}{b \log (F)}\\ &=-\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{2 b^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{a+b x} \sqrt {x}}{b \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.48 \[ -\frac {x^{3/2} F^a \Gamma \left (\frac {3}{2},-b x \log (F)\right )}{(-b x \log (F))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 51, normalized size = 0.82 \[ \frac {2 \, F^{b x + a} b \sqrt {x} \log \relax (F) + \sqrt {\pi } \sqrt {-b \log \relax (F)} F^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (F)} \sqrt {x}\right )}{2 \, b^{2} \log \relax (F)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 58, normalized size = 0.94 \[ \frac {\sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} \sqrt {x}\right )}{2 \, \sqrt {-b \log \relax (F)} b \log \relax (F)} + \frac {\sqrt {x} e^{\left (b x \log \relax (F) + a \log \relax (F)\right )}}{b \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 1.06 \[ -\frac {\left (\frac {\left (-b \right )^{\frac {3}{2}} \sqrt {x}\, {\mathrm e}^{b x \ln \relax (F )} \sqrt {\ln \relax (F )}}{b}-\frac {\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, \erfi \left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \relax (F )}\right )}{2 b^{\frac {3}{2}}}\right ) F^{a}}{\sqrt {-b}\, b \ln \relax (F )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 24, normalized size = 0.39 \[ -\frac {F^{a} x^{\frac {3}{2}} \Gamma \left (\frac {3}{2}, -b x \log \relax (F)\right )}{\left (-b x \log \relax (F)\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.41, size = 55, normalized size = 0.89 \[ \frac {F^a\,F^{b\,x}\,\sqrt {x}}{b\,\ln \relax (F)}+\frac {F^a\,\sqrt {x}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \relax (F)}\right )}{2\,b\,\ln \relax (F)\,\sqrt {-b\,x\,\ln \relax (F)}} \]
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 \[ \int F^{a + b x} \sqrt {x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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