Optimal. Leaf size=80 \[ \frac{c^2 x^{m+1} \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{m+1}+\frac{a c^2 x^{m+2} \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{m+2} \]
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Rubi [A] time = 0.095878, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6148, 808, 364} \[ \frac{c^2 x^{m+1} \, _2F_1\left (-\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{m+1}+\frac{a c^2 x^{m+2} \, _2F_1\left (-\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{m+2} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 808
Rule 364
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int x^m (1+a x) \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=c^2 \int x^m \left (1-a^2 x^2\right )^{3/2} \, dx+\left (a c^2\right ) \int x^{1+m} \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{c^2 x^{1+m} \, _2F_1\left (-\frac{3}{2},\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{1+m}+\frac{a c^2 x^{2+m} \, _2F_1\left (-\frac{3}{2},\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{2+m}\\ \end{align*}
Mathematica [A] time = 0.0358562, size = 82, normalized size = 1.02 \[ c^2 \left (\frac{x^{m+1} \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{m+1}{2},\frac{m+1}{2}+1,a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{m+2}{2},\frac{m+2}{2}+1,a^2 x^2\right )}{m+2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.323, size = 227, normalized size = 2.8 \begin{align*}{\frac{{a}^{5}{c}^{2}{x}^{6+m}}{6+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},3+{\frac{m}{2}};\,4+{\frac{m}{2}};\,{a}^{2}{x}^{2})}}-2\,{\frac{{a}^{3}{c}^{2}{x}^{4+m}{\mbox{$_2$F$_1$}(1/2,2+m/2;\,3+m/2;\,{a}^{2}{x}^{2})}}{4+m}}+{\frac{a{c}^{2}{x}^{2+m}}{2+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},1+{\frac{m}{2}};\,2+{\frac{m}{2}};\,{a}^{2}{x}^{2})}}+{\frac{{a}^{4}{c}^{2}{x}^{5+m}}{5+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{5}{2}}+{\frac{m}{2}};\,{\frac{7}{2}}+{\frac{m}{2}};\,{a}^{2}{x}^{2})}}-2\,{\frac{{a}^{2}{c}^{2}{x}^{3+m}{\mbox{$_2$F$_1$}(1/2,3/2+m/2;\,5/2+m/2;\,{a}^{2}{x}^{2})}}{3+m}}+{\frac{{c}^{2}{x}^{1+m}}{1+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{1}{2}}+{\frac{m}{2}};\,{\frac{3}{2}}+{\frac{m}{2}};\,{a}^{2}{x}^{2})}} \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^{2} c x^{2} - c\right )}^{2}{\left (a x + 1\right )} x^{m}}{\sqrt{-a^{2} x^{2} + 1}}\,{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 (-{\left (a^{3} c^{2} x^{3} + a^{2} c^{2} x^{2} - a c^{2} x - c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 11.6798, size = 223, normalized size = 2.79 \begin{align*} - \frac{a^{3} c^{2} x^{4} x^{m} \Gamma \left (\frac{m}{2} + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{m}{2} + 3\right )} - \frac{a^{2} c^{2} x^{3} x^{m} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{a c^{2} x^{2} x^{m} \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{m}{2} + 2\right )} + \frac{c^{2} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} \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^{2} c x^{2} - c\right )}^{2}{\left (a x + 1\right )} x^{m}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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