Optimal. Leaf size=105 \[ -\frac {15 \sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)}+\frac {15 x f^{a+b x^2}}{8 b^3 \log ^3(f)}-\frac {5 x^3 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac {x^5 f^{a+b x^2}}{2 b \log (f)} \]
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Rubi [A] time = 0.09, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2212, 2204} \[ -\frac {15 \sqrt {\pi } f^a \text {Erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)}-\frac {5 x^3 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac {15 x f^{a+b x^2}}{8 b^3 \log ^3(f)}+\frac {x^5 f^{a+b x^2}}{2 b \log (f)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2212
Rubi steps
\begin {align*} \int f^{a+b x^2} x^6 \, dx &=\frac {f^{a+b x^2} x^5}{2 b \log (f)}-\frac {5 \int f^{a+b x^2} x^4 \, dx}{2 b \log (f)}\\ &=-\frac {5 f^{a+b x^2} x^3}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^5}{2 b \log (f)}+\frac {15 \int f^{a+b x^2} x^2 \, dx}{4 b^2 \log ^2(f)}\\ &=\frac {15 f^{a+b x^2} x}{8 b^3 \log ^3(f)}-\frac {5 f^{a+b x^2} x^3}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^5}{2 b \log (f)}-\frac {15 \int f^{a+b x^2} \, dx}{8 b^3 \log ^3(f)}\\ &=-\frac {15 f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)}+\frac {15 f^{a+b x^2} x}{8 b^3 \log ^3(f)}-\frac {5 f^{a+b x^2} x^3}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^5}{2 b \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 83, normalized size = 0.79 \[ \frac {f^a \left (2 \sqrt {b} x \sqrt {\log (f)} f^{b x^2} \left (4 b^2 x^4 \log ^2(f)-10 b x^2 \log (f)+15\right )-15 \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )\right )}{16 b^{7/2} \log ^{\frac {7}{2}}(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 77, normalized size = 0.73 \[ \frac {15 \, \sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right ) + 2 \, {\left (4 \, b^{3} x^{5} \log \relax (f)^{3} - 10 \, b^{2} x^{3} \log \relax (f)^{2} + 15 \, b x \log \relax (f)\right )} f^{b x^{2} + a}}{16 \, b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 80, normalized size = 0.76 \[ \frac {15 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (f)} x\right )}{16 \, \sqrt {-b \log \relax (f)} b^{3} \log \relax (f)^{3}} + \frac {{\left (4 \, b^{2} x^{5} \log \relax (f)^{2} - 10 \, b x^{3} \log \relax (f) + 15 \, x\right )} e^{\left (b x^{2} \log \relax (f) + a \log \relax (f)\right )}}{8 \, b^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 98, normalized size = 0.93 \[ \frac {x^{5} f^{a} f^{b \,x^{2}}}{2 b \ln \relax (f )}-\frac {5 x^{3} f^{a} f^{b \,x^{2}}}{4 b^{2} \ln \relax (f )^{2}}+\frac {15 x \,f^{a} f^{b \,x^{2}}}{8 b^{3} \ln \relax (f )^{3}}-\frac {15 \sqrt {\pi }\, f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x \right )}{16 \sqrt {-b \ln \relax (f )}\, b^{3} \ln \relax (f )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 82, normalized size = 0.78 \[ \frac {{\left (4 \, b^{2} f^{a} x^{5} \log \relax (f)^{2} - 10 \, b f^{a} x^{3} \log \relax (f) + 15 \, f^{a} x\right )} f^{b x^{2}}}{8 \, b^{3} \log \relax (f)^{3}} - \frac {15 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right )}{16 \, \sqrt {-b \log \relax (f)} b^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.52, size = 98, normalized size = 0.93 \[ \frac {15\,f^a\,f^{b\,x^2}\,x}{8\,b^3\,{\ln \relax (f)}^3}+\frac {f^a\,f^{b\,x^2}\,x^5}{2\,b\,\ln \relax (f)}-\frac {5\,f^a\,f^{b\,x^2}\,x^3}{4\,b^2\,{\ln \relax (f)}^2}-\frac {15\,f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}\right )}{16\,b^3\,{\ln \relax (f)}^3\,\sqrt {b\,\ln \relax (f)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + b x^{2}} x^{6}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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