Optimal. Leaf size=119 \[ -\frac {8}{105} \sqrt {\pi } b^{7/2} f^a \log ^{\frac {7}{2}}(f) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )+\frac {8}{105} b^3 x \log ^3(f) f^{a+\frac {b}{x^2}}+\frac {4}{105} b^2 x^3 \log ^2(f) f^{a+\frac {b}{x^2}}+\frac {1}{7} x^7 f^{a+\frac {b}{x^2}}+\frac {2}{35} b x^5 \log (f) f^{a+\frac {b}{x^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2214, 2206, 2211, 2204} \[ -\frac {8}{105} \sqrt {\pi } b^{7/2} f^a \log ^{\frac {7}{2}}(f) \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )+\frac {8}{105} b^3 x \log ^3(f) f^{a+\frac {b}{x^2}}+\frac {4}{105} b^2 x^3 \log ^2(f) f^{a+\frac {b}{x^2}}+\frac {1}{7} x^7 f^{a+\frac {b}{x^2}}+\frac {2}{35} b x^5 \log (f) f^{a+\frac {b}{x^2}} \]
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^6 \, dx &=\frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {1}{7} (2 b \log (f)) \int f^{a+\frac {b}{x^2}} x^4 \, dx\\ &=\frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {1}{35} \left (4 b^2 \log ^2(f)\right ) \int f^{a+\frac {b}{x^2}} x^2 \, dx\\ &=\frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {4}{105} b^2 f^{a+\frac {b}{x^2}} x^3 \log ^2(f)+\frac {1}{105} \left (8 b^3 \log ^3(f)\right ) \int f^{a+\frac {b}{x^2}} \, dx\\ &=\frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {4}{105} b^2 f^{a+\frac {b}{x^2}} x^3 \log ^2(f)+\frac {8}{105} b^3 f^{a+\frac {b}{x^2}} x \log ^3(f)+\frac {1}{105} \left (16 b^4 \log ^4(f)\right ) \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx\\ &=\frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {4}{105} b^2 f^{a+\frac {b}{x^2}} x^3 \log ^2(f)+\frac {8}{105} b^3 f^{a+\frac {b}{x^2}} x \log ^3(f)-\frac {1}{105} \left (16 b^4 \log ^4(f)\right ) \operatorname {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{7} f^{a+\frac {b}{x^2}} x^7+\frac {2}{35} b f^{a+\frac {b}{x^2}} x^5 \log (f)+\frac {4}{105} b^2 f^{a+\frac {b}{x^2}} x^3 \log ^2(f)+\frac {8}{105} b^3 f^{a+\frac {b}{x^2}} x \log ^3(f)-\frac {8}{105} b^{7/2} f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \log ^{\frac {7}{2}}(f)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 86, normalized size = 0.72 \[ \frac {1}{105} f^a \left (x f^{\frac {b}{x^2}} \left (8 b^3 \log ^3(f)+4 b^2 x^2 \log ^2(f)+6 b x^4 \log (f)+15 x^6\right )-8 \sqrt {\pi } b^{7/2} \log ^{\frac {7}{2}}(f) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 86, normalized size = 0.72 \[ \frac {8}{105} \, \sqrt {\pi } \sqrt {-b \log \relax (f)} b^{3} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (f)}}{x}\right ) \log \relax (f)^{3} + \frac {1}{105} \, {\left (15 \, x^{7} + 6 \, b x^{5} \log \relax (f) + 4 \, b^{2} x^{3} \log \relax (f)^{2} + 8 \, b^{3} x \log \relax (f)^{3}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 111, normalized size = 0.93 \[ -\frac {8 \sqrt {\pi }\, b^{4} f^{a} \erf \left (\frac {\sqrt {-b \ln \relax (f )}}{x}\right ) \ln \relax (f )^{4}}{105 \sqrt {-b \ln \relax (f )}}+\frac {8 b^{3} x \,f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{3}}{105}+\frac {4 b^{2} x^{3} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )^{2}}{105}+\frac {2 b \,x^{5} f^{a} f^{\frac {b}{x^{2}}} \ln \relax (f )}{35}+\frac {x^{7} f^{a} f^{\frac {b}{x^{2}}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 28, normalized size = 0.24 \[ \frac {1}{2} \, f^{a} x^{7} \left (-\frac {b \log \relax (f)}{x^{2}}\right )^{\frac {7}{2}} \Gamma \left (-\frac {7}{2}, -\frac {b \log \relax (f)}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 129, normalized size = 1.08 \[ \frac {f^a\,f^{\frac {b}{x^2}}\,x^7}{7}-\frac {8\,f^a\,x^7\,\sqrt {\pi }\,{\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}^{7/2}}{105}+\frac {8\,f^a\,x^7\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \relax (f)}{x^2}}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^2}\right )}^{7/2}}{105}+\frac {8\,b^3\,f^a\,f^{\frac {b}{x^2}}\,x\,{\ln \relax (f)}^3}{105}+\frac {4\,b^2\,f^a\,f^{\frac {b}{x^2}}\,x^3\,{\ln \relax (f)}^2}{105}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,x^5\,\ln \relax (f)}{35} \]
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 \[ \int f^{a + \frac {b}{x^{2}}} x^{6}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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