Optimal. Leaf size=20 \[ \frac {f^{a+b x^2}}{2 b \log (f)} \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2240}
\begin {gather*} \frac {f^{a+b x^2}}{2 b \log (f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rubi steps
\begin {align*} \int f^{a+b x^2} x \, dx &=\frac {f^{a+b x^2}}{2 b \log (f)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} \frac {f^{a+b x^2}}{2 b \log (f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 19, normalized size = 0.95
method | result | size |
gosper | \(\frac {f^{b \,x^{2}+a}}{2 b \ln \left (f \right )}\) | \(19\) |
derivativedivides | \(\frac {f^{b \,x^{2}+a}}{2 b \ln \left (f \right )}\) | \(19\) |
default | \(\frac {f^{b \,x^{2}+a}}{2 b \ln \left (f \right )}\) | \(19\) |
risch | \(\frac {f^{b \,x^{2}+a}}{2 b \ln \left (f \right )}\) | \(19\) |
norman | \(\frac {{\mathrm e}^{\left (b \,x^{2}+a \right ) \ln \left (f \right )}}{2 b \ln \left (f \right )}\) | \(21\) |
meijerg | \(-\frac {f^{a} \left (1-{\mathrm e}^{b \,x^{2} \ln \left (f \right )}\right )}{2 b \ln \left (f \right )}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 18, normalized size = 0.90 \begin {gather*} \frac {f^{b x^{2} + a}}{2 \, b \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 18, normalized size = 0.90 \begin {gather*} \frac {f^{b x^{2} + a}}{2 \, b \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 22, normalized size = 1.10 \begin {gather*} \begin {cases} \frac {f^{a + b x^{2}}}{2 b \log {\left (f \right )}} & \text {for}\: b \log {\left (f \right )} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.58, size = 18, normalized size = 0.90 \begin {gather*} \frac {f^{b x^{2} + a}}{2 \, b \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.30, size = 18, normalized size = 0.90 \begin {gather*} \frac {f^{b\,x^2+a}}{2\,b\,\ln \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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