Optimal. Leaf size=54 \[ -\frac {2 F^{a+b x}}{\sqrt {x}}+2 \sqrt {b} F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right ) \sqrt {\log (F)} \]
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Rubi [A]
time = 0.03, antiderivative size = 54, 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 = {2208, 2211,
2235} \begin {gather*} 2 \sqrt {\pi } \sqrt {b} F^a \sqrt {\log (F)} \text {Erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )-\frac {2 F^{a+b x}}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2208
Rule 2211
Rule 2235
Rubi steps
\begin {align*} \int \frac {F^{a+b x}}{x^{3/2}} \, dx &=-\frac {2 F^{a+b x}}{\sqrt {x}}+(2 b \log (F)) \int \frac {F^{a+b x}}{\sqrt {x}} \, dx\\ &=-\frac {2 F^{a+b x}}{\sqrt {x}}+(4 b \log (F)) \text {Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 F^{a+b x}}{\sqrt {x}}+2 \sqrt {b} F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right ) \sqrt {\log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 38, normalized size = 0.70 \begin {gather*} -\frac {2 F^a \left (F^{b x}-\Gamma \left (\frac {1}{2},-b x \log (F)\right ) \sqrt {-b x \log (F)}\right )}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 64, normalized size = 1.19
method | result | size |
meijerg | \(-\frac {F^{a} \left (-b \right )^{\frac {3}{2}} \sqrt {\ln \left (F \right )}\, \left (-\frac {2 \,{\mathrm e}^{x b \ln \left (F \right )}}{\sqrt {x}\, \sqrt {-b}\, \sqrt {\ln \left (F \right )}}+\frac {2 \sqrt {b}\, \sqrt {\pi }\, \erfi \left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \left (F \right )}\right )}{\sqrt {-b}}\right )}{b}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 24, normalized size = 0.44 \begin {gather*} -\frac {\sqrt {-b x \log \left (F\right )} F^{a} \Gamma \left (-\frac {1}{2}, -b x \log \left (F\right )\right )}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 44, normalized size = 0.81 \begin {gather*} -\frac {2 \, {\left (\sqrt {\pi } \sqrt {-b \log \left (F\right )} F^{a} x \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} \sqrt {x}\right ) + F^{b x + a} \sqrt {x}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.93, size = 34, normalized size = 0.63 \begin {gather*} 4 F^{a} F^{b x} b \sqrt {x} \log {\left (F \right )} - \frac {2 F^{a} F^{b x}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.49, size = 42, normalized size = 0.78 \begin {gather*} \frac {2\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \left (F\right )}\right )\,\sqrt {-b\,x\,\ln \left (F\right )}}{\sqrt {x}}-\frac {2\,F^a\,F^{b\,x}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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