Optimal. Leaf size=45 \[ -4 x^{m+1} \, _2F_1(1,m+1;m+2;a x)+\frac {4 x^{m+1}}{1-a x}+\frac {x^{m+1}}{m+1} \]
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Rubi [A] time = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6167, 6126, 89, 80, 64} \[ -4 x^{m+1} \, _2F_1(1,m+1;m+2;a x)+\frac {4 x^{m+1}}{1-a x}+\frac {x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
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Rule 64
Rule 80
Rule 89
Rule 6126
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} x^m \, dx &=\int e^{4 \tanh ^{-1}(a x)} x^m \, dx\\ &=\int \frac {x^m (1+a x)^2}{(1-a x)^2} \, dx\\ &=\frac {4 x^{1+m}}{1-a x}-\frac {\int \frac {x^m \left (a^2 (3+4 m)+a^3 x\right )}{1-a x} \, dx}{a^2}\\ &=\frac {x^{1+m}}{1+m}+\frac {4 x^{1+m}}{1-a x}-(4 (1+m)) \int \frac {x^m}{1-a x} \, dx\\ &=\frac {x^{1+m}}{1+m}+\frac {4 x^{1+m}}{1-a x}-4 x^{1+m} \, _2F_1(1,1+m;2+m;a x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 1.04 \[ \frac {x^{m+1} (-4 (m+1) (a x-1) \, _2F_1(1,m+1;m+2;a x)+a x-4 m-5)}{(m+1) (a x-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 1\right )} x^{m}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )}^{2} x^{m}}{{\left (a x - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.51, size = 201, normalized size = 4.47 \[ -\frac {\left (-a \right )^{-m} \left (\frac {x^{m} \left (-a \right )^{m} \left (a^{2} m \,x^{2}+a m x +2 a x -m^{2}-3 m -2\right )}{\left (1+m \right ) m \left (-a x +1\right )}+x^{m} \left (-a \right )^{m} \left (2+m \right ) \Phi \left (a x , 1, m\right )\right )}{a}+\frac {2 \left (-a \right )^{-m} \left (-\frac {x^{m} \left (-a \right )^{m} \left (a x -m -1\right )}{m \left (-a x +1\right )}-x^{m} \left (-a \right )^{m} \left (1+m \right ) \Phi \left (a x , 1, m\right )\right )}{a}-\frac {\left (-a \right )^{-m} \left (\frac {x^{m} \left (-a \right )^{m} \left (-1-m \right )}{\left (1+m \right ) \left (-a x +1\right )}+x^{m} \left (-a \right )^{m} m \Phi \left (a x , 1, m\right )\right )}{a} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{2} x^{m}}{{\left (a x - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m\,{\left (a\,x+1\right )}^2}{{\left (a\,x-1\right )}^2} \,d x \]
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 \[ \int \frac {x^{m} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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