Optimal. Leaf size=128 \[ \frac {105 \sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)}-\frac {105 x f^{a+b x^2}}{16 b^4 \log ^4(f)}+\frac {35 x^3 f^{a+b x^2}}{8 b^3 \log ^3(f)}-\frac {7 x^5 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac {x^7 f^{a+b x^2}}{2 b \log (f)} \]
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Rubi [A] time = 0.14, antiderivative size = 128, 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 = {2212, 2204} \[ \frac {105 \sqrt {\pi } f^a \text {Erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)}-\frac {7 x^5 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac {35 x^3 f^{a+b x^2}}{8 b^3 \log ^3(f)}-\frac {105 x f^{a+b x^2}}{16 b^4 \log ^4(f)}+\frac {x^7 f^{a+b x^2}}{2 b \log (f)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2212
Rubi steps
\begin {align*} \int f^{a+b x^2} x^8 \, dx &=\frac {f^{a+b x^2} x^7}{2 b \log (f)}-\frac {7 \int f^{a+b x^2} x^6 \, dx}{2 b \log (f)}\\ &=-\frac {7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^7}{2 b \log (f)}+\frac {35 \int f^{a+b x^2} x^4 \, dx}{4 b^2 \log ^2(f)}\\ &=\frac {35 f^{a+b x^2} x^3}{8 b^3 \log ^3(f)}-\frac {7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^7}{2 b \log (f)}-\frac {105 \int f^{a+b x^2} x^2 \, dx}{8 b^3 \log ^3(f)}\\ &=-\frac {105 f^{a+b x^2} x}{16 b^4 \log ^4(f)}+\frac {35 f^{a+b x^2} x^3}{8 b^3 \log ^3(f)}-\frac {7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^7}{2 b \log (f)}+\frac {105 \int f^{a+b x^2} \, dx}{16 b^4 \log ^4(f)}\\ &=\frac {105 f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)}-\frac {105 f^{a+b x^2} x}{16 b^4 \log ^4(f)}+\frac {35 f^{a+b x^2} x^3}{8 b^3 \log ^3(f)}-\frac {7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^7}{2 b \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 95, normalized size = 0.74 \[ \frac {f^a \left (2 \sqrt {b} x \sqrt {\log (f)} f^{b x^2} \left (8 b^3 x^6 \log ^3(f)-28 b^2 x^4 \log ^2(f)+70 b x^2 \log (f)-105\right )+105 \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 89, normalized size = 0.70 \[ -\frac {105 \, \sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right ) - 2 \, {\left (8 \, b^{4} x^{7} \log \relax (f)^{4} - 28 \, b^{3} x^{5} \log \relax (f)^{3} + 70 \, b^{2} x^{3} \log \relax (f)^{2} - 105 \, b x \log \relax (f)\right )} f^{b x^{2} + a}}{32 \, b^{5} \log \relax (f)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 92, normalized size = 0.72 \[ -\frac {105 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (f)} x\right )}{32 \, \sqrt {-b \log \relax (f)} b^{4} \log \relax (f)^{4}} + \frac {{\left (8 \, b^{3} x^{7} \log \relax (f)^{3} - 28 \, b^{2} x^{5} \log \relax (f)^{2} + 70 \, b x^{3} \log \relax (f) - 105 \, x\right )} e^{\left (b x^{2} \log \relax (f) + a \log \relax (f)\right )}}{16 \, b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 120, normalized size = 0.94 \[ \frac {x^{7} f^{a} f^{b \,x^{2}}}{2 b \ln \relax (f )}-\frac {7 x^{5} f^{a} f^{b \,x^{2}}}{4 b^{2} \ln \relax (f )^{2}}+\frac {35 x^{3} f^{a} f^{b \,x^{2}}}{8 b^{3} \ln \relax (f )^{3}}-\frac {105 x \,f^{a} f^{b \,x^{2}}}{16 b^{4} \ln \relax (f )^{4}}+\frac {105 \sqrt {\pi }\, f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x \right )}{32 \sqrt {-b \ln \relax (f )}\, b^{4} \ln \relax (f )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 97, normalized size = 0.76 \[ \frac {{\left (8 \, b^{3} f^{a} x^{7} \log \relax (f)^{3} - 28 \, b^{2} f^{a} x^{5} \log \relax (f)^{2} + 70 \, b f^{a} x^{3} \log \relax (f) - 105 \, f^{a} x\right )} f^{b x^{2}}}{16 \, b^{4} \log \relax (f)^{4}} + \frac {105 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right )}{32 \, \sqrt {-b \log \relax (f)} b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 116, normalized size = 0.91 \[ \frac {\frac {f^a\,\left (105\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}\right )-210\,f^{b\,x^2}\,x\,\sqrt {b\,\ln \relax (f)}\right )}{32\,\sqrt {b\,\ln \relax (f)}}-\frac {7\,b^2\,f^a\,f^{b\,x^2}\,x^5\,{\ln \relax (f)}^2}{4}+\frac {b^3\,f^a\,f^{b\,x^2}\,x^7\,{\ln \relax (f)}^3}{2}+\frac {35\,b\,f^a\,f^{b\,x^2}\,x^3\,\ln \relax (f)}{8}}{b^4\,{\ln \relax (f)}^4} \]
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 \[ \int f^{a + b x^{2}} x^{8}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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