Optimal. Leaf size=62 \[ -\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{2 b^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{a+b x} \sqrt {x}}{b \log (F)} \]
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Rubi [A]
time = 0.03, 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 = {2207, 2211,
2235} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2211
Rule 2235
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 {\text {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 = 0.05, size = 30, normalized size = 0.48 \begin {gather*} -\frac {F^a x^{3/2} \Gamma \left (\frac {3}{2},-b x \log (F)\right )}{(-b x \log (F))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 66, normalized size = 1.06
method | result | size |
meijerg | \(-\frac {F^{a} \left (\frac {\sqrt {x}\, \left (-b \right )^{\frac {3}{2}} \sqrt {\ln \left (F \right )}\, {\mathrm e}^{x b \ln \left (F \right )}}{b}-\frac {\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, \erfi \left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \left (F \right )}\right )}{2 b^{\frac {3}{2}}}\right )}{\sqrt {-b}\, \ln \left (F \right )^{\frac {3}{2}} b}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 24, normalized size = 0.39 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 51, normalized size = 0.82 \begin {gather*} \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 {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 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 = 3.13, size = 58, normalized size = 0.94 \begin {gather*} \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 {gather*}
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 \begin {gather*} \frac {F^a\,F^{b\,x}\,\sqrt {x}}{b\,\ln \left (F\right )}+\frac {F^a\,\sqrt {x}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \left (F\right )}\right )}{2\,b\,\ln \left (F\right )\,\sqrt {-b\,x\,\ln \left (F\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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