Optimal. Leaf size=109 \[ \frac {15 \sqrt {\pi } f^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)}-\frac {15 f^{a+\frac {b}{x^2}}}{8 b^3 x \log ^3(f)}+\frac {5 f^{a+\frac {b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^5 \log (f)} \]
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Rubi [A] time = 0.11, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2212, 2211, 2204} \[ \frac {15 \sqrt {\pi } f^a \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)}+\frac {5 f^{a+\frac {b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac {15 f^{a+\frac {b}{x^2}}}{8 b^3 x \log ^3(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^5 \log (f)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2211
Rule 2212
Rubi steps
\begin {align*} \int \frac {f^{a+\frac {b}{x^2}}}{x^8} \, dx &=-\frac {f^{a+\frac {b}{x^2}}}{2 b x^5 \log (f)}-\frac {5 \int \frac {f^{a+\frac {b}{x^2}}}{x^6} \, dx}{2 b \log (f)}\\ &=\frac {5 f^{a+\frac {b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^5 \log (f)}+\frac {15 \int \frac {f^{a+\frac {b}{x^2}}}{x^4} \, dx}{4 b^2 \log ^2(f)}\\ &=-\frac {15 f^{a+\frac {b}{x^2}}}{8 b^3 x \log ^3(f)}+\frac {5 f^{a+\frac {b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^5 \log (f)}-\frac {15 \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx}{8 b^3 \log ^3(f)}\\ &=-\frac {15 f^{a+\frac {b}{x^2}}}{8 b^3 x \log ^3(f)}+\frac {5 f^{a+\frac {b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^5 \log (f)}+\frac {15 \operatorname {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right )}{8 b^3 \log ^3(f)}\\ &=\frac {15 f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)}-\frac {15 f^{a+\frac {b}{x^2}}}{8 b^3 x \log ^3(f)}+\frac {5 f^{a+\frac {b}{x^2}}}{4 b^2 x^3 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^5 \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 86, normalized size = 0.79 \[ \frac {15 \sqrt {\pi } f^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)}-\frac {f^{a+\frac {b}{x^2}} \left (4 b^2 \log ^2(f)-10 b x^2 \log (f)+15 x^4\right )}{8 b^3 x^5 \log ^3(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 88, normalized size = 0.81 \[ -\frac {15 \, \sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} x^{5} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (f)}}{x}\right ) + 2 \, {\left (15 \, b x^{4} \log \relax (f) - 10 \, b^{2} x^{2} \log \relax (f)^{2} + 4 \, b^{3} \log \relax (f)^{3}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{16 \, b^{4} x^{5} \log \relax (f)^{4}} \]
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 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 102, normalized size = 0.94 \[ \frac {15 \sqrt {\pi }\, f^{a} \erf \left (\frac {\sqrt {-b \ln \relax (f )}}{x}\right )}{16 \sqrt {-b \ln \relax (f )}\, b^{3} \ln \relax (f )^{3}}-\frac {f^{a} f^{\frac {b}{x^{2}}}}{2 b \,x^{5} \ln \relax (f )}+\frac {5 f^{a} f^{\frac {b}{x^{2}}}}{4 b^{2} x^{3} \ln \relax (f )^{2}}-\frac {15 f^{a} f^{\frac {b}{x^{2}}}}{8 b^{3} x \ln \relax (f )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 28, normalized size = 0.26 \[ \frac {f^{a} \Gamma \left (\frac {7}{2}, -\frac {b \log \relax (f)}{x^{2}}\right )}{2 \, x^{7} \left (-\frac {b \log \relax (f)}{x^{2}}\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 102, normalized size = 0.94 \[ \frac {5\,f^a\,f^{\frac {b}{x^2}}}{4\,b^2\,x^3\,{\ln \relax (f)}^2}-\frac {f^a\,f^{\frac {b}{x^2}}}{2\,b\,x^5\,\ln \relax (f)}-\frac {15\,f^a\,f^{\frac {b}{x^2}}}{8\,b^3\,x\,{\ln \relax (f)}^3}+\frac {15\,f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \relax (f)}{x\,\sqrt {b\,\ln \relax (f)}}\right )}{16\,b^3\,{\ln \relax (f)}^3\,\sqrt {b\,\ln \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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