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 = 0.125317, antiderivative size = 119, normalized size of antiderivative = 1., 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 2214
Rule 2206
Rule 2211
Rule 2204
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*}
Mathematica [A] time = 0.0364954, size = 86, normalized size = 0.72 \[ \frac{1}{105} f^a \left (x f^{\frac{b}{x^2}} \left (4 b^2 x^2 \log ^2(f)+8 b^3 \log ^3(f)+6 b x^4 \log (f)+15 x^6\right )-8 \sqrt{\pi } b^{7/2} \log ^{\frac{7}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 111, normalized size = 0.9 \begin{align*}{\frac{{f}^{a}{x}^{7}}{7}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) b{x}^{5}}{35}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{4\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}{x}^{3}}{105}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}x}{105}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{8\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{4}{b}^{4}\sqrt{\pi }}{105}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22611, size = 38, normalized size = 0.32 \begin{align*} \frac{1}{2} \, f^{a} x^{7} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{7}{2}} \Gamma \left (-\frac{7}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78589, size = 224, normalized size = 1.88 \begin{align*} \frac{8}{105} \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} b^{3} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{3} + \frac{1}{105} \,{\left (15 \, x^{7} + 6 \, b x^{5} \log \left (f\right ) + 4 \, b^{2} x^{3} \log \left (f\right )^{2} + 8 \, b^{3} x \log \left (f\right )^{3}\right )} f^{\frac{a x^{2} + b}{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{2}}} x^{6}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{2}}} x^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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