Optimal. Leaf size=59 \[ \frac {x f^{a+b x^2}}{2 b \log (f)}-\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2212, 2204} \[ \frac {x f^{a+b x^2}}{2 b \log (f)}-\frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2212
Rubi steps
\begin {align*} \int f^{a+b x^2} x^2 \, dx &=\frac {f^{a+b x^2} x}{2 b \log (f)}-\frac {\int f^{a+b x^2} \, dx}{2 b \log (f)}\\ &=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {f^{a+b x^2} x}{2 b \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 1.00 \[ \frac {x f^{a+b x^2}}{2 b \log (f)}-\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{4 b^{3/2} \log ^{\frac {3}{2}}(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 49, normalized size = 0.83 \[ \frac {2 \, b f^{b x^{2} + a} x \log \relax (f) + \sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right )}{4 \, b^{2} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 57, normalized size = 0.97 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (f)} x\right )}{4 \, \sqrt {-b \log \relax (f)} b \log \relax (f)} + \frac {x e^{\left (b x^{2} \log \relax (f) + a \log \relax (f)\right )}}{2 \, b \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 54, normalized size = 0.92 \[ \frac {x \,f^{a} f^{b \,x^{2}}}{2 b \ln \relax (f )}-\frac {\sqrt {\pi }\, f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x \right )}{4 \sqrt {-b \ln \relax (f )}\, b \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 53, normalized size = 0.90 \[ \frac {f^{b x^{2}} f^{a} x}{2 \, b \log \relax (f)} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right )}{4 \, \sqrt {-b \log \relax (f)} b \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.61, size = 54, normalized size = 0.92 \[ \frac {f^a\,f^{b\,x^2}\,x}{2\,b\,\ln \relax (f)}-\frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}\right )}{4\,b\,\ln \relax (f)\,\sqrt {b\,\ln \relax (f)}} \]
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 \[ \int f^{a + b x^{2}} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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