Optimal. Leaf size=31 \[ \frac {x^{m+1} F_1\left (m+1;\frac {1}{8},-\frac {1}{8};m+2;a x,-a x\right )}{m+1} \]
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Rubi [A] time = 0.03, 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 = {6126, 133} \[ \frac {x^{m+1} F_1\left (m+1;\frac {1}{8},-\frac {1}{8};m+2;a x,-a x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 133
Rule 6126
Rubi steps
\begin {align*} \int e^{\frac {1}{4} \tanh ^{-1}(a x)} x^m \, dx &=\int \frac {x^m \sqrt [8]{1+a x}}{\sqrt [8]{1-a x}} \, dx\\ &=\frac {x^{1+m} F_1\left (1+m;\frac {1}{8},-\frac {1}{8};2+m;a x,-a x\right )}{1+m}\\ \end {align*}
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Mathematica [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int e^{\frac {1}{4} \tanh ^{-1}(a x)} x^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac {1}{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {1}{4}} x^{m}\, dx \]
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 \[ \int x^{m} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {1}{4}}\,{d x} \]
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 \[ \int x^m\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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