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 = 0.0443172, antiderivative size = 62, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0076998, size = 30, normalized size = 0.48 \[ -\frac{x^{3/2} F^a \text{Gamma}\left (\frac{3}{2},-b x \log (F)\right )}{(-b x \log (F))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 66, normalized size = 1.1 \begin{align*} -{\frac{{F}^{a}}{b} \left ({\frac{{{\rm e}^{b\ln \left ( F \right ) x}}}{b}\sqrt{x} \left ( -b \right ) ^{{\frac{3}{2}}}\sqrt{\ln \left ( F \right ) }}-{\frac{\sqrt{\pi }}{2} \left ( -b \right ) ^{{\frac{3}{2}}}{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ){b}^{-{\frac{3}{2}}}} \right ){\frac{1}{\sqrt{-b}}} \left ( \ln \left ( F \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24986, size = 32, normalized size = 0.52 \begin{align*} -\frac{F^{a} x^{\frac{3}{2}} \Gamma \left (\frac{3}{2}, -b x \log \left (F\right )\right )}{\left (-b x \log \left (F\right )\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5187, size = 153, normalized size = 2.47 \begin{align*} \frac{2 \, F^{b x + a} b \sqrt{x} \log \left (F\right ) + \sqrt{\pi } \sqrt{-b \log \left (F\right )} F^{a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{2 \, b^{2} \log \left (F\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{a + b x} \sqrt{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22632, size = 78, normalized size = 1.26 \begin{align*} \frac{\sqrt{\pi } F^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{2 \, \sqrt{-b \log \left (F\right )} b \log \left (F\right )} + \frac{\sqrt{x} e^{\left (b x \log \left (F\right ) + a \log \left (F\right )\right )}}{b \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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