Optimal. Leaf size=24 \[ \frac {a x^{m+2}}{m+2}+\frac {x^{m+1}}{m+1} \]
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Rubi [A] time = 0.08, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6150, 43} \[ \frac {a x^{m+2}}{m+2}+\frac {x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^m \sqrt {1-a^2 x^2} \, dx &=\int x^m (1+a x) \, dx\\ &=\int \left (x^m+a x^{1+m}\right ) \, dx\\ &=\frac {x^{1+m}}{1+m}+\frac {a x^{2+m}}{2+m}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.83 \[ x^{m+1} \left (\frac {a x}{m+2}+\frac {1}{m+1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 29, normalized size = 1.21 \[ \frac {{\left ({\left (a m + a\right )} x^{2} + {\left (m + 2\right )} x\right )} x^{m}}{m^{2} + 3 \, m + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 41, normalized size = 1.71 \[ \frac {a m x^{2} x^{m} + a x^{2} x^{m} + m x x^{m} + 2 \, x x^{m}}{m^{2} + 3 \, m + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 27, normalized size = 1.12 \[ \frac {x^{1+m} \left (a m x +a x +m +2\right )}{\left (2+m \right ) \left (1+m \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 24, normalized size = 1.00 \[ \frac {a x^{m + 2}}{m + 2} + \frac {x^{m + 1}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 26, normalized size = 1.08 \[ \frac {x^{m+1}\,\left (m+a\,x+a\,m\,x+2\right )}{m^2+3\,m+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 82, normalized size = 3.42 \[ \begin {cases} a \log {\relax (x )} - \frac {1}{x} & \text {for}\: m = -2 \\a x + \log {\relax (x )} & \text {for}\: m = -1 \\\frac {a m x^{2} x^{m}}{m^{2} + 3 m + 2} + \frac {a x^{2} x^{m}}{m^{2} + 3 m + 2} + \frac {m x x^{m}}{m^{2} + 3 m + 2} + \frac {2 x x^{m}}{m^{2} + 3 m + 2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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