Optimal. Leaf size=34 \[ \frac {1}{2} x^9 f^a \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2218} \[ \frac {1}{2} x^9 f^a \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \text {Gamma}\left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^2}} x^8 \, dx &=\frac {1}{2} f^a x^9 \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 34, normalized size = 1.00 \[ \frac {1}{2} x^9 f^a \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 98, normalized size = 2.88 \[ \frac {16}{945} \, \sqrt {\pi } \sqrt {-b \log \relax (f)} b^{4} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (f)}}{x}\right ) \log \relax (f)^{4} + \frac {1}{945} \, {\left (105 \, x^{9} + 30 \, b x^{7} \log \relax (f) + 12 \, b^{2} x^{5} \log \relax (f)^{2} + 8 \, b^{3} x^{3} \log \relax (f)^{3} + 16 \, b^{4} x \log \relax (f)^{4}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 133, normalized size = 3.91 \[ -\frac {16 \sqrt {\pi }\, b^{5} f^{a} \erf \left (\frac {\sqrt {-b \ln \relax (f )}}{x}\right ) \ln \relax (f )^{5}}{945 \sqrt {-b \ln \relax (f )}}+\frac {16 b^{4} x \,f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{4}}{945}+\frac {8 b^{3} x^{3} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{3}}{945}+\frac {4 b^{2} x^{5} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{2}}{315}+\frac {2 b \,x^{7} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )}{63}+\frac {x^{9} f^{a} f^{\frac {b}{x^{2}}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 28, normalized size = 0.82 \[ \frac {1}{2} \, f^{a} x^{9} \left (-\frac {b \log \relax (f)}{x^{2}}\right )^{\frac {9}{2}} \Gamma \left (-\frac {9}{2}, -\frac {b \log \relax (f)}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 151, normalized size = 4.44 \[ \frac {f^a\,f^{\frac {b}{x^2}}\,x^9}{9}+\frac {16\,f^a\,x^9\,\sqrt {\pi }\,{\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}^{9/2}}{945}-\frac {16\,f^a\,x^9\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \relax (f)}{x^2}}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}^{9/2}}{945}+\frac {16\,b^4\,f^a\,f^{\frac {b}{x^2}}\,x\,{\ln \relax (f)}^4}{945}+\frac {4\,b^2\,f^a\,f^{\frac {b}{x^2}}\,x^5\,{\ln \relax (f)}^2}{315}+\frac {8\,b^3\,f^a\,f^{\frac {b}{x^2}}\,x^3\,{\ln \relax (f)}^3}{945}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,x^7\,\ln \relax (f)}{63} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{8}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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