Optimal. Leaf size=96 \[ -\frac {4}{15} \sqrt {\pi } b^{5/2} f^a \log ^{\frac {5}{2}}(f) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )+\frac {4}{15} b^2 x \log ^2(f) f^{a+\frac {b}{x^2}}+\frac {1}{5} x^5 f^{a+\frac {b}{x^2}}+\frac {2}{15} b x^3 \log (f) f^{a+\frac {b}{x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2214, 2206, 2211, 2204} \[ -\frac {4}{15} \sqrt {\pi } b^{5/2} f^a \log ^{\frac {5}{2}}(f) \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )+\frac {4}{15} b^2 x \log ^2(f) f^{a+\frac {b}{x^2}}+\frac {1}{5} x^5 f^{a+\frac {b}{x^2}}+\frac {2}{15} b x^3 \log (f) f^{a+\frac {b}{x^2}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2206
Rule 2211
Rule 2214
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^2}} x^4 \, dx &=\frac {1}{5} f^{a+\frac {b}{x^2}} x^5+\frac {1}{5} (2 b \log (f)) \int f^{a+\frac {b}{x^2}} x^2 \, dx\\ &=\frac {1}{5} f^{a+\frac {b}{x^2}} x^5+\frac {2}{15} b f^{a+\frac {b}{x^2}} x^3 \log (f)+\frac {1}{15} \left (4 b^2 \log ^2(f)\right ) \int f^{a+\frac {b}{x^2}} \, dx\\ &=\frac {1}{5} f^{a+\frac {b}{x^2}} x^5+\frac {2}{15} b f^{a+\frac {b}{x^2}} x^3 \log (f)+\frac {4}{15} b^2 f^{a+\frac {b}{x^2}} x \log ^2(f)+\frac {1}{15} \left (8 b^3 \log ^3(f)\right ) \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx\\ &=\frac {1}{5} f^{a+\frac {b}{x^2}} x^5+\frac {2}{15} b f^{a+\frac {b}{x^2}} x^3 \log (f)+\frac {4}{15} b^2 f^{a+\frac {b}{x^2}} x \log ^2(f)-\frac {1}{15} \left (8 b^3 \log ^3(f)\right ) \operatorname {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{5} f^{a+\frac {b}{x^2}} x^5+\frac {2}{15} b f^{a+\frac {b}{x^2}} x^3 \log (f)+\frac {4}{15} b^2 f^{a+\frac {b}{x^2}} x \log ^2(f)-\frac {4}{15} b^{5/2} f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \log ^{\frac {5}{2}}(f)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 74, normalized size = 0.77 \[ \frac {1}{15} f^a \left (x f^{\frac {b}{x^2}} \left (4 b^2 \log ^2(f)+2 b x^2 \log (f)+3 x^4\right )-4 \sqrt {\pi } b^{5/2} \log ^{\frac {5}{2}}(f) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 74, normalized size = 0.77 \[ \frac {4}{15} \, \sqrt {\pi } \sqrt {-b \log \relax (f)} b^{2} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (f)}}{x}\right ) \log \relax (f)^{2} + \frac {1}{15} \, {\left (3 \, x^{5} + 2 \, b x^{3} \log \relax (f) + 4 \, b^{2} x \log \relax (f)^{2}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 89, normalized size = 0.93 \[ -\frac {4 \sqrt {\pi }\, b^{3} f^{a} \erf \left (\frac {\sqrt {-b \ln \relax (f )}}{x}\right ) \ln \relax (f )^{3}}{15 \sqrt {-b \ln \relax (f )}}+\frac {4 b^{2} x \,f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{2}}{15}+\frac {2 b \,x^{3} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )}{15}+\frac {x^{5} f^{a} f^{\frac {b}{x^{2}}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 28, normalized size = 0.29 \[ \frac {1}{2} \, f^{a} x^{5} \left (-\frac {b \log \relax (f)}{x^{2}}\right )^{\frac {5}{2}} \Gamma \left (-\frac {5}{2}, -\frac {b \log \relax (f)}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.61, size = 107, normalized size = 1.11 \[ \frac {f^a\,f^{\frac {b}{x^2}}\,x^5}{5}+\frac {4\,f^a\,x^5\,\sqrt {\pi }\,{\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}^{5/2}}{15}-\frac {4\,f^a\,x^5\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \relax (f)}{x^2}}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}^{5/2}}{15}+\frac {4\,b^2\,f^a\,f^{\frac {b}{x^2}}\,x\,{\ln \relax (f)}^2}{15}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,x^3\,\ln \relax (f)}{15} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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