Integrand size = 13, antiderivative size = 131 \[ \int F^{a+b x} x^{7/2} \, dx=\frac {105 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{16 b^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {105 F^{a+b x} \sqrt {x}}{8 b^4 \log ^4(F)}+\frac {35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)} \]
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Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2207, 2211, 2235} \[ \int F^{a+b x} x^{7/2} \, dx=\frac {105 \sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{16 b^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {105 \sqrt {x} F^{a+b x}}{8 b^4 \log ^4(F)}+\frac {35 x^{3/2} F^{a+b x}}{4 b^3 \log ^3(F)}-\frac {7 x^{5/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac {x^{7/2} F^{a+b x}}{b \log (F)} \]
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Rule 2207
Rule 2211
Rule 2235
Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+b x} x^{7/2}}{b \log (F)}-\frac {7 \int F^{a+b x} x^{5/2} \, dx}{2 b \log (F)} \\ & = -\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)}+\frac {35 \int F^{a+b x} x^{3/2} \, dx}{4 b^2 \log ^2(F)} \\ & = \frac {35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)}-\frac {105 \int F^{a+b x} \sqrt {x} \, dx}{8 b^3 \log ^3(F)} \\ & = -\frac {105 F^{a+b x} \sqrt {x}}{8 b^4 \log ^4(F)}+\frac {35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)}+\frac {105 \int \frac {F^{a+b x}}{\sqrt {x}} \, dx}{16 b^4 \log ^4(F)} \\ & = -\frac {105 F^{a+b x} \sqrt {x}}{8 b^4 \log ^4(F)}+\frac {35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)}+\frac {105 \text {Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt {x}\right )}{8 b^4 \log ^4(F)} \\ & = \frac {105 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{16 b^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {105 F^{a+b x} \sqrt {x}}{8 b^4 \log ^4(F)}+\frac {35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.27 \[ \int F^{a+b x} x^{7/2} \, dx=\frac {F^a \Gamma \left (\frac {9}{2},-b x \log (F)\right ) \sqrt {-b x \log (F)}}{b^5 \sqrt {x} \log ^5(F)} \]
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Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.76
method | result | size |
meijerg | \(-\frac {F^{a} \left (-\frac {\sqrt {x}\, \left (-b \right )^{\frac {9}{2}} \sqrt {\ln \left (F \right )}\, \left (-72 b^{3} x^{3} \ln \left (F \right )^{3}+252 b^{2} x^{2} \ln \left (F \right )^{2}-630 x b \ln \left (F \right )+945\right ) {\mathrm e}^{x b \ln \left (F \right )}}{72 b^{4}}+\frac {105 \left (-b \right )^{\frac {9}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \left (F \right )}\right )}{16 b^{\frac {9}{2}}}\right )}{\left (-b \right )^{\frac {7}{2}} \ln \left (F \right )^{\frac {9}{2}} b}\) | \(99\) |
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Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.68 \[ \int F^{a+b x} x^{7/2} \, dx=-\frac {105 \, \sqrt {\pi } \sqrt {-b \log \left (F\right )} F^{a} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} \sqrt {x}\right ) - 2 \, {\left (8 \, b^{4} x^{3} \log \left (F\right )^{4} - 28 \, b^{3} x^{2} \log \left (F\right )^{3} + 70 \, b^{2} x \log \left (F\right )^{2} - 105 \, b \log \left (F\right )\right )} F^{b x + a} \sqrt {x}}{16 \, b^{5} \log \left (F\right )^{5}} \]
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\[ \int F^{a+b x} x^{7/2} \, dx=\int F^{a + b x} x^{\frac {7}{2}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.18 \[ \int F^{a+b x} x^{7/2} \, dx=-\frac {F^{a} x^{\frac {9}{2}} \Gamma \left (\frac {9}{2}, -b x \log \left (F\right )\right )}{\left (-b x \log \left (F\right )\right )^{\frac {9}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.72 \[ \int F^{a+b x} x^{7/2} \, dx=-\frac {105 \, \sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} \sqrt {x}\right )}{16 \, \sqrt {-b \log \left (F\right )} b^{4} \log \left (F\right )^{4}} + \frac {{\left (8 \, b^{3} x^{\frac {7}{2}} \log \left (F\right )^{3} - 28 \, b^{2} x^{\frac {5}{2}} \log \left (F\right )^{2} + 70 \, b x^{\frac {3}{2}} \log \left (F\right ) - 105 \, \sqrt {x}\right )} e^{\left (b x \log \left (F\right ) + a \log \left (F\right )\right )}}{8 \, b^{4} \log \left (F\right )^{4}} \]
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Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.63 \[ \int F^{a+b x} x^{7/2} \, dx=\frac {F^a\,x^{7/2}\,\left (\frac {105\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \left (F\right )}\right )}{16}+F^{b\,x}\,\left (\frac {105\,\sqrt {-b\,x\,\ln \left (F\right )}}{8}+\frac {35\,{\left (-b\,x\,\ln \left (F\right )\right )}^{3/2}}{4}+\frac {7\,{\left (-b\,x\,\ln \left (F\right )\right )}^{5/2}}{2}+{\left (-b\,x\,\ln \left (F\right )\right )}^{7/2}\right )\right )}{b\,\ln \left (F\right )\,{\left (-b\,x\,\ln \left (F\right )\right )}^{7/2}} \]
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