Optimal. Leaf size=31 \[ \frac {x^{1+m} F_1\left (1+m;\frac {1}{4},-\frac {1}{4};2+m;a x,-a x\right )}{1+m} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6261, 138}
\begin {gather*} \frac {x^{m+1} F_1\left (m+1;\frac {1}{4},-\frac {1}{4};m+2;a x,-a x\right )}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 6261
Rubi steps
\begin {align*} \int e^{\frac {1}{2} \tanh ^{-1}(a x)} x^m \, dx &=\int \frac {x^m \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}} \, dx\\ &=\frac {x^{1+m} F_1\left (1+m;\frac {1}{4},-\frac {1}{4};2+m;a x,-a x\right )}{1+m}\\ \end {align*}
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Mathematica [F]
time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int e^{\frac {1}{2} \tanh ^{-1}(a x)} x^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, x^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{m} \sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x^m\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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