Optimal. Leaf size=203 \[ \frac{(2-m) \left (2 m^2-8 m+3\right ) x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,a x)}{48 c^3 (m+1)}+\frac{(2-m) x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,-a x)}{16 c^3 (m+1)}-\frac{(2-m) (4-m) x^{m+1}}{24 c^3 (a x+1)}+\frac{(7-2 m) (2-m) x^{m+1}}{24 c^3 (1-a x) (a x+1)}+\frac{(4-m) x^{m+1}}{12 c^3 (1-a x)^2 (a x+1)}+\frac{x^{m+1}}{6 c^3 (1-a x)^3 (a x+1)} \]
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Rubi [A] time = 0.348774, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6150, 103, 151, 156, 64} \[ \frac{(2-m) \left (2 m^2-8 m+3\right ) x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{48 c^3 (m+1)}+\frac{(2-m) x^{m+1} \, _2F_1(1,m+1;m+2;-a x)}{16 c^3 (m+1)}-\frac{(2-m) (4-m) x^{m+1}}{24 c^3 (a x+1)}+\frac{(7-2 m) (2-m) x^{m+1}}{24 c^3 (1-a x) (a x+1)}+\frac{(4-m) x^{m+1}}{12 c^3 (1-a x)^2 (a x+1)}+\frac{x^{m+1}}{6 c^3 (1-a x)^3 (a x+1)} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 103
Rule 151
Rule 156
Rule 64
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^m}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^m}{(1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac{x^{1+m}}{6 c^3 (1-a x)^3 (1+a x)}-\frac{\int \frac{x^m \left (-a (5-m)-a^2 (3-m) x\right )}{(1-a x)^3 (1+a x)^2} \, dx}{6 a c^3}\\ &=\frac{x^{1+m}}{6 c^3 (1-a x)^3 (1+a x)}+\frac{(4-m) x^{1+m}}{12 c^3 (1-a x)^2 (1+a x)}+\frac{\int \frac{x^m \left (2 a^2 (2-m) (3-m)+2 a^3 (2-m) (4-m) x\right )}{(1-a x)^2 (1+a x)^2} \, dx}{24 a^2 c^3}\\ &=\frac{x^{1+m}}{6 c^3 (1-a x)^3 (1+a x)}+\frac{(4-m) x^{1+m}}{12 c^3 (1-a x)^2 (1+a x)}+\frac{(7-2 m) (2-m) x^{1+m}}{24 c^3 (1-a x) (1+a x)}-\frac{\int \frac{x^m \left (2 a^3 (2-m) \left (1+7 m-2 m^2\right )-2 a^4 (7-2 m) (1-m) (2-m) x\right )}{(1-a x) (1+a x)^2} \, dx}{48 a^3 c^3}\\ &=-\frac{(2-m) (4-m) x^{1+m}}{24 c^3 (1+a x)}+\frac{x^{1+m}}{6 c^3 (1-a x)^3 (1+a x)}+\frac{(4-m) x^{1+m}}{12 c^3 (1-a x)^2 (1+a x)}+\frac{(7-2 m) (2-m) x^{1+m}}{24 c^3 (1-a x) (1+a x)}-\frac{\int \frac{x^m \left (-4 a^4 (1-m) (2-m) (3-m)+4 a^5 (2-m) (4-m) m x\right )}{(1-a x) (1+a x)} \, dx}{96 a^4 c^3}\\ &=-\frac{(2-m) (4-m) x^{1+m}}{24 c^3 (1+a x)}+\frac{x^{1+m}}{6 c^3 (1-a x)^3 (1+a x)}+\frac{(4-m) x^{1+m}}{12 c^3 (1-a x)^2 (1+a x)}+\frac{(7-2 m) (2-m) x^{1+m}}{24 c^3 (1-a x) (1+a x)}+\frac{(2-m) \int \frac{x^m}{1+a x} \, dx}{16 c^3}+\frac{\left ((2-m) \left (3-8 m+2 m^2\right )\right ) \int \frac{x^m}{1-a x} \, dx}{48 c^3}\\ &=-\frac{(2-m) (4-m) x^{1+m}}{24 c^3 (1+a x)}+\frac{x^{1+m}}{6 c^3 (1-a x)^3 (1+a x)}+\frac{(4-m) x^{1+m}}{12 c^3 (1-a x)^2 (1+a x)}+\frac{(7-2 m) (2-m) x^{1+m}}{24 c^3 (1-a x) (1+a x)}+\frac{(2-m) x^{1+m} \, _2F_1(1,1+m;2+m;-a x)}{16 c^3 (1+m)}+\frac{(2-m) \left (3-8 m+2 m^2\right ) x^{1+m} \, _2F_1(1,1+m;2+m;a x)}{48 c^3 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.149856, size = 194, normalized size = 0.96 \[ \frac{x^{m+1} \left (-\left (2 m^3-12 m^2+19 m-6\right ) (a x+1) (a x-1)^3 \text{Hypergeometric2F1}(1,m+1,m+2,a x)-3 (m-2) (a x+1) (a x-1)^3 \text{Hypergeometric2F1}(1,m+1,m+2,-a x)-2 \left (m^2 \left (-5 a^3 x^3+6 a^2 x^2+5 a x-6\right )+m \left (2 a^3 x^3-3 a^2 x^2-6 a x+11\right )+2 \left (4 a^3 x^3-5 a^2 x^2-6 a x+9\right )+m^3 (a x-1)^2 (a x+1)\right )\right )}{48 c^3 (m+1) (a x-1)^3 (a x+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.403, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ) ^{2}{x}^{m}}{ \left ( -{a}^{2}{x}^{2}+1 \right ) \left ( -{a}^{2}c{x}^{2}+c \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{2} x^{m}}{{\left (a^{2} c x^{2} - c\right )}^{3}{\left (a^{2} x^{2} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m}}{a^{6} c^{3} x^{6} - 2 \, a^{5} c^{3} x^{5} - a^{4} c^{3} x^{4} + 4 \, a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2} - 2 \, a c^{3} x + c^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{m}}{a^{6} x^{6} - 2 a^{5} x^{5} - a^{4} x^{4} + 4 a^{3} x^{3} - a^{2} x^{2} - 2 a x + 1}\, dx}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{2} x^{m}}{{\left (a^{2} c x^{2} - c\right )}^{3}{\left (a^{2} x^{2} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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