Optimal. Leaf size=132 \[ -\frac {105 \sqrt {\pi } f^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)}+\frac {105 f^{a+\frac {b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac {35 f^{a+\frac {b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac {7 f^{a+\frac {b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^7 \log (f)} \]
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Rubi [A] time = 0.16, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2212, 2211, 2204} \[ -\frac {105 \sqrt {\pi } f^a \text {Erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)}+\frac {7 f^{a+\frac {b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac {35 f^{a+\frac {b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac {105 f^{a+\frac {b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^7 \log (f)} \]
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^{10}} \, dx &=-\frac {f^{a+\frac {b}{x^2}}}{2 b x^7 \log (f)}-\frac {7 \int \frac {f^{a+\frac {b}{x^2}}}{x^8} \, dx}{2 b \log (f)}\\ &=\frac {7 f^{a+\frac {b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^7 \log (f)}+\frac {35 \int \frac {f^{a+\frac {b}{x^2}}}{x^6} \, dx}{4 b^2 \log ^2(f)}\\ &=-\frac {35 f^{a+\frac {b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac {7 f^{a+\frac {b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^7 \log (f)}-\frac {105 \int \frac {f^{a+\frac {b}{x^2}}}{x^4} \, dx}{8 b^3 \log ^3(f)}\\ &=\frac {105 f^{a+\frac {b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac {35 f^{a+\frac {b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac {7 f^{a+\frac {b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^7 \log (f)}+\frac {105 \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx}{16 b^4 \log ^4(f)}\\ &=\frac {105 f^{a+\frac {b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac {35 f^{a+\frac {b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac {7 f^{a+\frac {b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^7 \log (f)}-\frac {105 \operatorname {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right )}{16 b^4 \log ^4(f)}\\ &=-\frac {105 f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)}+\frac {105 f^{a+\frac {b}{x^2}}}{16 b^4 x \log ^4(f)}-\frac {35 f^{a+\frac {b}{x^2}}}{8 b^3 x^3 \log ^3(f)}+\frac {7 f^{a+\frac {b}{x^2}}}{4 b^2 x^5 \log ^2(f)}-\frac {f^{a+\frac {b}{x^2}}}{2 b x^7 \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 100, normalized size = 0.76 \[ \frac {f^a \left (\frac {2 \sqrt {b} \sqrt {\log (f)} f^{\frac {b}{x^2}} \left (-8 b^3 \log ^3(f)+28 b^2 x^2 \log ^2(f)-70 b x^4 \log (f)+105 x^6\right )}{x^7}-105 \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 100, normalized size = 0.76 \[ \frac {105 \, \sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} x^{7} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (f)}}{x}\right ) + 2 \, {\left (105 \, b x^{6} \log \relax (f) - 70 \, b^{2} x^{4} \log \relax (f)^{2} + 28 \, b^{3} x^{2} \log \relax (f)^{3} - 8 \, b^{4} \log \relax (f)^{4}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{32 \, b^{5} x^{7} \log \relax (f)^{5}} \]
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 \[ \int \frac {f^{a + \frac {b}{x^{2}}}}{x^{10}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 124, normalized size = 0.94 \[ -\frac {105 \sqrt {\pi }\, f^{a} \erf \left (\frac {\sqrt {-b \ln \relax (f )}}{x}\right )}{32 \sqrt {-b \ln \relax (f )}\, b^{4} \ln \relax (f )^{4}}-\frac {f^{a} f^{\frac {b}{x^{2}}}}{2 b \,x^{7} \ln \relax (f )}+\frac {7 f^{a} f^{\frac {b}{x^{2}}}}{4 b^{2} x^{5} \ln \relax (f )^{2}}-\frac {35 f^{a} f^{\frac {b}{x^{2}}}}{8 b^{3} x^{3} \ln \relax (f )^{3}}+\frac {105 f^{a} f^{\frac {b}{x^{2}}}}{16 b^{4} x \ln \relax (f )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 28, normalized size = 0.21 \[ \frac {f^{a} \Gamma \left (\frac {9}{2}, -\frac {b \log \relax (f)}{x^{2}}\right )}{2 \, x^{9} \left (-\frac {b \log \relax (f)}{x^{2}}\right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 121, normalized size = 0.92 \[ -\frac {\frac {f^a\,\left (105\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \relax (f)}{x\,\sqrt {b\,\ln \relax (f)}}\right )-\frac {210\,f^{\frac {b}{x^2}}\,\sqrt {b\,\ln \relax (f)}}{x}\right )}{32\,\sqrt {b\,\ln \relax (f)}}-\frac {7\,b^2\,f^a\,f^{\frac {b}{x^2}}\,{\ln \relax (f)}^2}{4\,x^5}+\frac {b^3\,f^a\,f^{\frac {b}{x^2}}\,{\ln \relax (f)}^3}{2\,x^7}+\frac {35\,b\,f^a\,f^{\frac {b}{x^2}}\,\ln \relax (f)}{8\,x^3}}{b^4\,{\ln \relax (f)}^4} \]
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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