Optimal. Leaf size=100 \[ \frac {8}{15} \sqrt {\pi } b^{5/2} F^a \log ^{\frac {5}{2}}(F) \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )-\frac {8 b^2 \log ^2(F) F^{a+b x}}{15 \sqrt {x}}-\frac {2 F^{a+b x}}{5 x^{5/2}}-\frac {4 b \log (F) F^{a+b x}}{15 x^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2177, 2180, 2204} \[ \frac {8}{15} \sqrt {\pi } b^{5/2} F^a \log ^{\frac {5}{2}}(F) \text {Erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )-\frac {8 b^2 \log ^2(F) F^{a+b x}}{15 \sqrt {x}}-\frac {2 F^{a+b x}}{5 x^{5/2}}-\frac {4 b \log (F) F^{a+b x}}{15 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2180
Rule 2204
Rubi steps
\begin {align*} \int \frac {F^{a+b x}}{x^{7/2}} \, dx &=-\frac {2 F^{a+b x}}{5 x^{5/2}}+\frac {1}{5} (2 b \log (F)) \int \frac {F^{a+b x}}{x^{5/2}} \, dx\\ &=-\frac {2 F^{a+b x}}{5 x^{5/2}}-\frac {4 b F^{a+b x} \log (F)}{15 x^{3/2}}+\frac {1}{15} \left (4 b^2 \log ^2(F)\right ) \int \frac {F^{a+b x}}{x^{3/2}} \, dx\\ &=-\frac {2 F^{a+b x}}{5 x^{5/2}}-\frac {4 b F^{a+b x} \log (F)}{15 x^{3/2}}-\frac {8 b^2 F^{a+b x} \log ^2(F)}{15 \sqrt {x}}+\frac {1}{15} \left (8 b^3 \log ^3(F)\right ) \int \frac {F^{a+b x}}{\sqrt {x}} \, dx\\ &=-\frac {2 F^{a+b x}}{5 x^{5/2}}-\frac {4 b F^{a+b x} \log (F)}{15 x^{3/2}}-\frac {8 b^2 F^{a+b x} \log ^2(F)}{15 \sqrt {x}}+\frac {1}{15} \left (16 b^3 \log ^3(F)\right ) \operatorname {Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 F^{a+b x}}{5 x^{5/2}}-\frac {4 b F^{a+b x} \log (F)}{15 x^{3/2}}-\frac {8 b^2 F^{a+b x} \log ^2(F)}{15 \sqrt {x}}+\frac {8}{15} b^{5/2} F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right ) \log ^{\frac {5}{2}}(F)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 0.61 \[ -\frac {2 F^a \left (F^{b x} \left (4 b^2 x^2 \log ^2(F)+2 b x \log (F)+3\right )-4 (-b x \log (F))^{5/2} \Gamma \left (\frac {1}{2},-b x \log (F)\right )\right )}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 74, normalized size = 0.74 \[ -\frac {2 \, {\left (4 \, \sqrt {\pi } \sqrt {-b \log \relax (F)} F^{a} b^{2} x^{3} \operatorname {erf}\left (\sqrt {-b \log \relax (F)} \sqrt {x}\right ) \log \relax (F)^{2} + {\left (4 \, b^{2} x^{2} \log \relax (F)^{2} + 2 \, b x \log \relax (F) + 3\right )} F^{b x + a} \sqrt {x}\right )}}{15 \, x^{3}} \]
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 \[ \int \frac {F^{b x + a}}{x^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 0.84 \[ -\frac {\left (-b \right )^{\frac {7}{2}} \left (\frac {8 \sqrt {\pi }\, b^{\frac {5}{2}} \erfi \left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \relax (F )}\right )}{15 \left (-b \right )^{\frac {5}{2}}}-\frac {2 \left (\frac {4 b^{2} x^{2} \ln \relax (F )^{2}}{3}+\frac {2 b x \ln \relax (F )}{3}+1\right ) {\mathrm e}^{b x \ln \relax (F )}}{5 \left (-b \right )^{\frac {5}{2}} x^{\frac {5}{2}} \ln \relax (F )^{\frac {5}{2}}}\right ) F^{a} \ln \relax (F )^{\frac {5}{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 24, normalized size = 0.24 \[ -\frac {\left (-b x \log \relax (F)\right )^{\frac {5}{2}} F^{a} \Gamma \left (-\frac {5}{2}, -b x \log \relax (F)\right )}{x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.45, size = 80, normalized size = 0.80 \[ -\frac {\frac {2\,F^{a+b\,x}}{5}+\frac {4\,F^{a+b\,x}\,b\,x\,\ln \relax (F)}{15}+\frac {8\,F^{a+b\,x}\,b^2\,x^2\,{\ln \relax (F)}^2}{15}-\frac {8\,F^a\,b^2\,x^2\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \relax (F)}\right )\,{\ln \relax (F)}^2\,\sqrt {-\pi \,b\,x\,\ln \relax (F)}}{15}}{x^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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