Integrand size = 13, antiderivative size = 85 \[ \int F^{a+b x} x^{3/2} \, dx=\frac {3 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{4 b^{5/2} \log ^{\frac {5}{2}}(F)}-\frac {3 F^{a+b x} \sqrt {x}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{3/2}}{b \log (F)} \]
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Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2207, 2211, 2235} \[ \int F^{a+b x} x^{3/2} \, dx=\frac {3 \sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{4 b^{5/2} \log ^{\frac {5}{2}}(F)}-\frac {3 \sqrt {x} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac {x^{3/2} F^{a+b x}}{b \log (F)} \]
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Rule 2207
Rule 2211
Rule 2235
Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+b x} x^{3/2}}{b \log (F)}-\frac {3 \int F^{a+b x} \sqrt {x} \, dx}{2 b \log (F)} \\ & = -\frac {3 F^{a+b x} \sqrt {x}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{3/2}}{b \log (F)}+\frac {3 \int \frac {F^{a+b x}}{\sqrt {x}} \, dx}{4 b^2 \log ^2(F)} \\ & = -\frac {3 F^{a+b x} \sqrt {x}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{3/2}}{b \log (F)}+\frac {3 \text {Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt {x}\right )}{2 b^2 \log ^2(F)} \\ & = \frac {3 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{4 b^{5/2} \log ^{\frac {5}{2}}(F)}-\frac {3 F^{a+b x} \sqrt {x}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{3/2}}{b \log (F)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.42 \[ \int F^{a+b x} x^{3/2} \, dx=\frac {F^a \Gamma \left (\frac {5}{2},-b x \log (F)\right ) \sqrt {-b x \log (F)}}{b^3 \sqrt {x} \log ^3(F)} \]
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Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.88
method | result | size |
meijerg | \(-\frac {F^{a} \left (-\frac {\sqrt {x}\, \left (-b \right )^{\frac {5}{2}} \sqrt {\ln \left (F \right )}\, \left (-10 x b \ln \left (F \right )+15\right ) {\mathrm e}^{x b \ln \left (F \right )}}{10 b^{2}}+\frac {3 \left (-b \right )^{\frac {5}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \left (F \right )}\right )}{4 b^{\frac {5}{2}}}\right )}{\left (-b \right )^{\frac {3}{2}} \ln \left (F \right )^{\frac {5}{2}} b}\) | \(75\) |
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Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int F^{a+b x} x^{3/2} \, dx=-\frac {3 \, \sqrt {\pi } \sqrt {-b \log \left (F\right )} F^{a} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} \sqrt {x}\right ) - 2 \, {\left (2 \, b^{2} x \log \left (F\right )^{2} - 3 \, b \log \left (F\right )\right )} F^{b x + a} \sqrt {x}}{4 \, b^{3} \log \left (F\right )^{3}} \]
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\[ \int F^{a+b x} x^{3/2} \, dx=\int F^{a + b x} x^{\frac {3}{2}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.28 \[ \int F^{a+b x} x^{3/2} \, dx=-\frac {F^{a} x^{\frac {5}{2}} \Gamma \left (\frac {5}{2}, -b x \log \left (F\right )\right )}{\left (-b x \log \left (F\right )\right )^{\frac {5}{2}}} \]
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Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.82 \[ \int F^{a+b x} x^{3/2} \, dx=-\frac {3 \, \sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} \sqrt {x}\right )}{4 \, \sqrt {-b \log \left (F\right )} b^{2} \log \left (F\right )^{2}} + \frac {{\left (2 \, b x^{\frac {3}{2}} \log \left (F\right ) - 3 \, \sqrt {x}\right )} e^{\left (b x \log \left (F\right ) + a \log \left (F\right )\right )}}{2 \, b^{2} \log \left (F\right )^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.88 \[ \int F^{a+b x} x^{3/2} \, dx=\frac {F^a\,F^{b\,x}\,x^{3/2}}{b\,\ln \left (F\right )}-\frac {3\,F^a\,F^{b\,x}\,\sqrt {x}}{2\,b^2\,{\ln \left (F\right )}^2}+\frac {3\,F^a\,x^{3/2}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \left (F\right )}\right )}{4\,b\,\ln \left (F\right )\,{\left (-b\,x\,\ln \left (F\right )\right )}^{3/2}} \]
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