Optimal. Leaf size=34 \[ \frac {1}{2} x^{11} f^a \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \]
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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} \[ \frac {1}{2} x^{11} f^a \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \text {Gamma}\left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^2}} x^{10} \, dx &=\frac {1}{2} f^a x^{11} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{11/2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 34, normalized size = 1.00 \[ \frac {1}{2} x^{11} f^a \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 110, normalized size = 3.24 \[ \frac {32}{10395} \, \sqrt {\pi } \sqrt {-b \log \relax (f)} b^{5} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (f)}}{x}\right ) \log \relax (f)^{5} + \frac {1}{10395} \, {\left (945 \, x^{11} + 210 \, b x^{9} \log \relax (f) + 60 \, b^{2} x^{7} \log \relax (f)^{2} + 24 \, b^{3} x^{5} \log \relax (f)^{3} + 16 \, b^{4} x^{3} \log \relax (f)^{4} + 32 \, b^{5} x \log \relax (f)^{5}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{10}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 155, normalized size = 4.56 \[ -\frac {32 \sqrt {\pi }\, b^{6} f^{a} \erf \left (\frac {\sqrt {-b \ln \relax (f )}}{x}\right ) \ln \relax (f )^{6}}{10395 \sqrt {-b \ln \relax (f )}}+\frac {32 b^{5} x \,f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{5}}{10395}+\frac {16 b^{4} x^{3} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{4}}{10395}+\frac {8 b^{3} x^{5} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{3}}{3465}+\frac {4 b^{2} x^{7} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{2}}{693}+\frac {2 b \,x^{9} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )}{99}+\frac {x^{11} f^{a} f^{\frac {b}{x^{2}}}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 28, normalized size = 0.82 \[ \frac {1}{2} \, f^{a} x^{11} \left (-\frac {b \log \relax (f)}{x^{2}}\right )^{\frac {11}{2}} \Gamma \left (-\frac {11}{2}, -\frac {b \log \relax (f)}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 173, normalized size = 5.09 \[ \frac {f^a\,f^{\frac {b}{x^2}}\,x^{11}}{11}-\frac {32\,f^a\,x^{11}\,\sqrt {\pi }\,{\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}^{11/2}}{10395}+\frac {32\,f^a\,x^{11}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \relax (f)}{x^2}}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}^{11/2}}{10395}+\frac {32\,b^5\,f^a\,f^{\frac {b}{x^2}}\,x\,{\ln \relax (f)}^5}{10395}+\frac {4\,b^2\,f^a\,f^{\frac {b}{x^2}}\,x^7\,{\ln \relax (f)}^2}{693}+\frac {8\,b^3\,f^a\,f^{\frac {b}{x^2}}\,x^5\,{\ln \relax (f)}^3}{3465}+\frac {16\,b^4\,f^a\,f^{\frac {b}{x^2}}\,x^3\,{\ln \relax (f)}^4}{10395}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,x^9\,\ln \relax (f)}{99} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{10}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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