Optimal. Leaf size=119 \[ \frac {8}{105} \sqrt {\pi } b^{7/2} f^a \log ^{\frac {7}{2}}(f) \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {8 b^3 \log ^3(f) f^{a+b x^2}}{105 x}-\frac {4 b^2 \log ^2(f) f^{a+b x^2}}{105 x^3}-\frac {f^{a+b x^2}}{7 x^7}-\frac {2 b \log (f) f^{a+b x^2}}{35 x^5} \]
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Rubi [A] time = 0.11, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2214, 2204} \[ \frac {8}{105} \sqrt {\pi } b^{7/2} f^a \log ^{\frac {7}{2}}(f) \text {Erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {8 b^3 \log ^3(f) f^{a+b x^2}}{105 x}-\frac {4 b^2 \log ^2(f) f^{a+b x^2}}{105 x^3}-\frac {f^{a+b x^2}}{7 x^7}-\frac {2 b \log (f) f^{a+b x^2}}{35 x^5} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2214
Rubi steps
\begin {align*} \int \frac {f^{a+b x^2}}{x^8} \, dx &=-\frac {f^{a+b x^2}}{7 x^7}+\frac {1}{7} (2 b \log (f)) \int \frac {f^{a+b x^2}}{x^6} \, dx\\ &=-\frac {f^{a+b x^2}}{7 x^7}-\frac {2 b f^{a+b x^2} \log (f)}{35 x^5}+\frac {1}{35} \left (4 b^2 \log ^2(f)\right ) \int \frac {f^{a+b x^2}}{x^4} \, dx\\ &=-\frac {f^{a+b x^2}}{7 x^7}-\frac {2 b f^{a+b x^2} \log (f)}{35 x^5}-\frac {4 b^2 f^{a+b x^2} \log ^2(f)}{105 x^3}+\frac {1}{105} \left (8 b^3 \log ^3(f)\right ) \int \frac {f^{a+b x^2}}{x^2} \, dx\\ &=-\frac {f^{a+b x^2}}{7 x^7}-\frac {2 b f^{a+b x^2} \log (f)}{35 x^5}-\frac {4 b^2 f^{a+b x^2} \log ^2(f)}{105 x^3}-\frac {8 b^3 f^{a+b x^2} \log ^3(f)}{105 x}+\frac {1}{105} \left (16 b^4 \log ^4(f)\right ) \int f^{a+b x^2} \, dx\\ &=-\frac {f^{a+b x^2}}{7 x^7}-\frac {2 b f^{a+b x^2} \log (f)}{35 x^5}-\frac {4 b^2 f^{a+b x^2} \log ^2(f)}{105 x^3}-\frac {8 b^3 f^{a+b x^2} \log ^3(f)}{105 x}+\frac {8}{105} b^{7/2} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right ) \log ^{\frac {7}{2}}(f)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 89, normalized size = 0.75 \[ \frac {f^a \left (8 \sqrt {\pi } b^{7/2} x^7 \log ^{\frac {7}{2}}(f) \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-f^{b x^2} \left (8 b^3 x^6 \log ^3(f)+4 b^2 x^4 \log ^2(f)+6 b x^2 \log (f)+15\right )\right )}{105 x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 85, normalized size = 0.71 \[ -\frac {8 \, \sqrt {\pi } \sqrt {-b \log \relax (f)} b^{3} f^{a} x^{7} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right ) \log \relax (f)^{3} + {\left (8 \, b^{3} x^{6} \log \relax (f)^{3} + 4 \, b^{2} x^{4} \log \relax (f)^{2} + 6 \, b x^{2} \log \relax (f) + 15\right )} f^{b x^{2} + a}}{105 \, x^{7}} \]
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 \[ \int \frac {f^{b x^{2} + a}}{x^{8}}\,{d x} \]
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 \[ \frac {8 \sqrt {\pi }\, b^{4} f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x \right ) \ln \relax (f )^{4}}{105 \sqrt {-b \ln \relax (f )}}-\frac {8 b^{3} f^{a} f^{b \,x^{2}} \ln \relax (f )^{3}}{105 x}-\frac {4 b^{2} f^{a} f^{b \,x^{2}} \ln \relax (f )^{2}}{105 x^{3}}-\frac {2 b \,f^{a} f^{b \,x^{2}} \ln \relax (f )}{35 x^{5}}-\frac {f^{a} f^{b \,x^{2}}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 28, normalized size = 0.24 \[ -\frac {\left (-b x^{2} \log \relax (f)\right )^{\frac {7}{2}} f^{a} \Gamma \left (-\frac {7}{2}, -b x^{2} \log \relax (f)\right )}{2 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.49, size = 131, normalized size = 1.10 \[ \frac {8\,f^a\,\sqrt {\pi }\,{\left (-b\,x^2\,\ln \relax (f)\right )}^{7/2}}{105\,x^7}-\frac {f^a\,f^{b\,x^2}}{7\,x^7}-\frac {8\,f^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x^2\,\ln \relax (f)}\right )\,{\left (-b\,x^2\,\ln \relax (f)\right )}^{7/2}}{105\,x^7}-\frac {4\,b^2\,f^a\,f^{b\,x^2}\,{\ln \relax (f)}^2}{105\,x^3}-\frac {8\,b^3\,f^a\,f^{b\,x^2}\,{\ln \relax (f)}^3}{105\,x}-\frac {2\,b\,f^a\,f^{b\,x^2}\,\ln \relax (f)}{35\,x^5} \]
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 \[ \int \frac {f^{a + b x^{2}}}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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