Optimal. Leaf size=62 \[ \frac {f^{a+b x^2}}{b^3 \log ^3(f)}-\frac {x^2 f^{a+b x^2}}{b^2 \log ^2(f)}+\frac {x^4 f^{a+b x^2}}{2 b \log (f)} \]
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Rubi [A] time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ -\frac {x^2 f^{a+b x^2}}{b^2 \log ^2(f)}+\frac {f^{a+b x^2}}{b^3 \log ^3(f)}+\frac {x^4 f^{a+b x^2}}{2 b \log (f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int f^{a+b x^2} x^5 \, dx &=\frac {f^{a+b x^2} x^4}{2 b \log (f)}-\frac {2 \int f^{a+b x^2} x^3 \, dx}{b \log (f)}\\ &=-\frac {f^{a+b x^2} x^2}{b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^4}{2 b \log (f)}+\frac {2 \int f^{a+b x^2} x \, dx}{b^2 \log ^2(f)}\\ &=\frac {f^{a+b x^2}}{b^3 \log ^3(f)}-\frac {f^{a+b x^2} x^2}{b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^4}{2 b \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.66 \[ \frac {f^{a+b x^2} \left (b^2 x^4 \log ^2(f)-2 b x^2 \log (f)+2\right )}{2 b^3 \log ^3(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 39, normalized size = 0.63 \[ \frac {{\left (b^{2} x^{4} \log \relax (f)^{2} - 2 \, b x^{2} \log \relax (f) + 2\right )} f^{b x^{2} + a}}{2 \, b^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 43, normalized size = 0.69 \[ \frac {{\left (b^{2} x^{4} \log \relax (f)^{2} - 2 \, b x^{2} \log \relax (f) + 2\right )} e^{\left (b x^{2} \log \relax (f) + a \log \relax (f)\right )}}{2 \, b^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.65 \[ \frac {\left (b^{2} x^{4} \ln \relax (f )^{2}-2 b \,x^{2} \ln \relax (f )+2\right ) f^{b \,x^{2}+a}}{2 b^{3} \ln \relax (f )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 47, normalized size = 0.76 \[ \frac {{\left (b^{2} f^{a} x^{4} \log \relax (f)^{2} - 2 \, b f^{a} x^{2} \log \relax (f) + 2 \, f^{a}\right )} f^{b x^{2}}}{2 \, b^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 39, normalized size = 0.63 \[ \frac {f^{b\,x^2+a}\,\left (\frac {b^2\,x^4\,{\ln \relax (f)}^2}{2}-b\,x^2\,\ln \relax (f)+1\right )}{b^3\,{\ln \relax (f)}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 54, normalized size = 0.87 \[ \begin {cases} \frac {f^{a + b x^{2}} \left (b^{2} x^{4} \log {\relax (f )}^{2} - 2 b x^{2} \log {\relax (f )} + 2\right )}{2 b^{3} \log {\relax (f )}^{3}} & \text {for}\: 2 b^{3} \log {\relax (f )}^{3} \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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