Optimal. Leaf size=25 \[ \frac {x^{m+1} \, _2F_1(2,m+1;m+2;a x)}{c (m+1)} \]
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Rubi [A] time = 0.08, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 64} \[ \frac {x^{m+1} \, _2F_1(2,m+1;m+2;a x)}{c (m+1)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x^m}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x^m}{(1-a x)^2} \, dx}{c}\\ &=\frac {x^{1+m} \, _2F_1(2,1+m;2+m;a x)}{c (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ \frac {x^{m+1} \, _2F_1(2,m+1;m+2;a x)}{c (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{a^{2} c x^{2} - 2 \, a c x + c}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )}^{2} x^{m}}{{\left (a^{2} c x^{2} - c\right )} {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right )^{2} x^{m}}{\left (-a^{2} x^{2}+1\right ) \left (-a^{2} c \,x^{2}+c \right )}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )}^{2} x^{m}}{{\left (a^{2} c x^{2} - c\right )} {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ -\int \frac {x^m\,{\left (a\,x+1\right )}^2}{\left (c-a^2\,c\,x^2\right )\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ \frac {\int \frac {x^{m}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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