Optimal. Leaf size=44 \[ \frac {x^2 f^{a+b x^2}}{2 b \log (f)}-\frac {f^{a+b x^2}}{2 b^2 \log ^2(f)} \]
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Rubi [A] time = 0.04, antiderivative size = 44, 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, 2209} \[ \frac {x^2 f^{a+b x^2}}{2 b \log (f)}-\frac {f^{a+b x^2}}{2 b^2 \log ^2(f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int f^{a+b x^2} x^3 \, dx &=\frac {f^{a+b x^2} x^2}{2 b \log (f)}-\frac {\int f^{a+b x^2} x \, dx}{b \log (f)}\\ &=-\frac {f^{a+b x^2}}{2 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^2}{2 b \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.66 \[ \frac {f^{a+b x^2} \left (b x^2 \log (f)-1\right )}{2 b^2 \log ^2(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 27, normalized size = 0.61 \[ \frac {{\left (b x^{2} \log \relax (f) - 1\right )} f^{b x^{2} + a}}{2 \, b^{2} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 690, normalized size = 15.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 0.64 \[ \frac {\left (b \,x^{2} \ln \relax (f )-1\right ) f^{b \,x^{2}+a}}{2 b^{2} \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 32, normalized size = 0.73 \[ \frac {{\left (b f^{a} x^{2} \log \relax (f) - f^{a}\right )} f^{b x^{2}}}{2 \, b^{2} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.46, size = 27, normalized size = 0.61 \[ \frac {f^{b\,x^2+a}\,\left (\frac {b\,x^2\,\ln \relax (f)}{2}-\frac {1}{2}\right )}{b^2\,{\ln \relax (f)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 41, normalized size = 0.93 \[ \begin {cases} \frac {f^{a + b x^{2}} \left (b x^{2} \log {\relax (f )} - 1\right )}{2 b^{2} \log {\relax (f )}^{2}} & \text {for}\: 2 b^{2} \log {\relax (f )}^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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