Optimal. Leaf size=34 \[ -\frac {x^{13} f^a \Gamma \left (\frac {13}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{13/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2218} \[ -\frac {x^{13} f^a \text {Gamma}\left (\frac {13}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{13/2}} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{a+b x^2} x^{12} \, dx &=-\frac {f^a x^{13} \Gamma \left (\frac {13}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \[ -\frac {x^{13} f^a \Gamma \left (\frac {13}{2},-b x^2 \log (f)\right )}{2 \left (-b x^2 \log (f)\right )^{13/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 113, normalized size = 3.32 \[ -\frac {10395 \, \sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right ) - 2 \, {\left (32 \, b^{6} x^{11} \log \relax (f)^{6} - 176 \, b^{5} x^{9} \log \relax (f)^{5} + 792 \, b^{4} x^{7} \log \relax (f)^{4} - 2772 \, b^{3} x^{5} \log \relax (f)^{3} + 6930 \, b^{2} x^{3} \log \relax (f)^{2} - 10395 \, b x \log \relax (f)\right )} f^{b x^{2} + a}}{128 \, b^{7} \log \relax (f)^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 116, normalized size = 3.41 \[ -\frac {10395 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (f)} x\right )}{128 \, \sqrt {-b \log \relax (f)} b^{6} \log \relax (f)^{6}} + \frac {{\left (32 \, b^{5} x^{11} \log \relax (f)^{5} - 176 \, b^{4} x^{9} \log \relax (f)^{4} + 792 \, b^{3} x^{7} \log \relax (f)^{3} - 2772 \, b^{2} x^{5} \log \relax (f)^{2} + 6930 \, b x^{3} \log \relax (f) - 10395 \, x\right )} e^{\left (b x^{2} \log \relax (f) + a \log \relax (f)\right )}}{64 \, b^{6} \log \relax (f)^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 164, normalized size = 4.82 \[ \frac {x^{11} f^{a} f^{b \,x^{2}}}{2 b \ln \relax (f )}-\frac {11 x^{9} f^{a} f^{b \,x^{2}}}{4 b^{2} \ln \relax (f )^{2}}+\frac {99 x^{7} f^{a} f^{b \,x^{2}}}{8 b^{3} \ln \relax (f )^{3}}-\frac {693 x^{5} f^{a} f^{b \,x^{2}}}{16 b^{4} \ln \relax (f )^{4}}+\frac {3465 x^{3} f^{a} f^{b \,x^{2}}}{32 b^{5} \ln \relax (f )^{5}}-\frac {10395 x \,f^{a} f^{b \,x^{2}}}{64 b^{6} \ln \relax (f )^{6}}+\frac {10395 \sqrt {\pi }\, f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x \right )}{128 \sqrt {-b \ln \relax (f )}\, b^{6} \ln \relax (f )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 127, normalized size = 3.74 \[ \frac {{\left (32 \, b^{5} f^{a} x^{11} \log \relax (f)^{5} - 176 \, b^{4} f^{a} x^{9} \log \relax (f)^{4} + 792 \, b^{3} f^{a} x^{7} \log \relax (f)^{3} - 2772 \, b^{2} f^{a} x^{5} \log \relax (f)^{2} + 6930 \, b f^{a} x^{3} \log \relax (f) - 10395 \, f^{a} x\right )} f^{b x^{2}}}{64 \, b^{6} \log \relax (f)^{6}} + \frac {10395 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right )}{128 \, \sqrt {-b \log \relax (f)} b^{6} \log \relax (f)^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 154, normalized size = 4.53 \[ \frac {\frac {f^a\,\left (\frac {10395\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}\right )}{128}-\frac {10395\,f^{b\,x^2}\,x\,\sqrt {b\,\ln \relax (f)}}{64}\right )}{\sqrt {b\,\ln \relax (f)}}-\frac {693\,b^2\,f^{b\,x^2+a}\,x^5\,{\ln \relax (f)}^2}{16}+\frac {99\,b^3\,f^{b\,x^2+a}\,x^7\,{\ln \relax (f)}^3}{8}-\frac {11\,b^4\,f^{b\,x^2+a}\,x^9\,{\ln \relax (f)}^4}{4}+\frac {b^5\,f^{b\,x^2+a}\,x^{11}\,{\ln \relax (f)}^5}{2}+\frac {3465\,b\,f^{b\,x^2+a}\,x^3\,\ln \relax (f)}{32}}{b^6\,{\ln \relax (f)}^6} \]
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 \[ \int f^{a + b x^{2}} x^{12}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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