Optimal. Leaf size=176 \[ \frac{c^2 (2 m+7) x^{m+1} \sqrt{c-a^2 c x^2} \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{(m+1) (m+6) \sqrt{1-a^2 x^2}}+\frac{2 a c^2 x^{m+2} \sqrt{c-a^2 c x^2} \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{(m+2) \sqrt{1-a^2 x^2}}-\frac{x^{m+1} \left (c-a^2 c x^2\right )^{5/2}}{m+6} \]
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Rubi [A] time = 0.314157, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6151, 1809, 808, 365, 364} \[ \frac{c^2 (2 m+7) x^{m+1} \sqrt{c-a^2 c x^2} \, _2F_1\left (-\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{(m+1) (m+6) \sqrt{1-a^2 x^2}}+\frac{2 a c^2 x^{m+2} \sqrt{c-a^2 c x^2} \, _2F_1\left (-\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{(m+2) \sqrt{1-a^2 x^2}}-\frac{x^{m+1} \left (c-a^2 c x^2\right )^{5/2}}{m+6} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1809
Rule 808
Rule 365
Rule 364
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^{5/2} \, dx &=c \int x^m (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac{x^{1+m} \left (c-a^2 c x^2\right )^{5/2}}{6+m}-\frac{\int x^m \left (-a^2 c (7+2 m)-2 a^3 c (6+m) x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{a^2 (6+m)}\\ &=-\frac{x^{1+m} \left (c-a^2 c x^2\right )^{5/2}}{6+m}+(2 a c) \int x^{1+m} \left (c-a^2 c x^2\right )^{3/2} \, dx+\frac{(c (7+2 m)) \int x^m \left (c-a^2 c x^2\right )^{3/2} \, dx}{6+m}\\ &=-\frac{x^{1+m} \left (c-a^2 c x^2\right )^{5/2}}{6+m}+\frac{\left (2 a c^2 \sqrt{c-a^2 c x^2}\right ) \int x^{1+m} \left (1-a^2 x^2\right )^{3/2} \, dx}{\sqrt{1-a^2 x^2}}+\frac{\left (c^2 (7+2 m) \sqrt{c-a^2 c x^2}\right ) \int x^m \left (1-a^2 x^2\right )^{3/2} \, dx}{(6+m) \sqrt{1-a^2 x^2}}\\ &=-\frac{x^{1+m} \left (c-a^2 c x^2\right )^{5/2}}{6+m}+\frac{c^2 (7+2 m) x^{1+m} \sqrt{c-a^2 c x^2} \, _2F_1\left (-\frac{3}{2},\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{(1+m) (6+m) \sqrt{1-a^2 x^2}}+\frac{2 a c^2 x^{2+m} \sqrt{c-a^2 c x^2} \, _2F_1\left (-\frac{3}{2},\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{(2+m) \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.220947, size = 180, normalized size = 1.02 \[ \frac{c^2 x^{m+1} \sqrt{c-a^2 c x^2} \left (-\frac{a^4 x^4 \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m+5}{2},\frac{m+7}{2},a^2 x^2\right )}{m+5}-\frac{2 a^3 x^3 \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m}{2}+2,\frac{m}{2}+3,a^2 x^2\right )}{m+4}+\frac{2 a x \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m}{2}+1,\frac{m}{2}+2,a^2 x^2\right )}{m+2}+\frac{\text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{m+1}\right )}{\sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.402, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ) ^{2}{x}^{m}}{-{a}^{2}{x}^{2}+1} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}\, 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^{2} c x^{2} + c\right )}^{\frac{5}{2}}{\left (a x + 1\right )}^{2} x^{m}}{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^{4} c^{2} x^{4} + 2 \, a^{3} c^{2} x^{3} - 2 \, a c^{2} x - c^{2}\right )} \sqrt{-a^{2} c x^{2} + c} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 88.0958, size = 226, normalized size = 1.28 \begin{align*} - \frac{a^{4} c^{\frac{5}{2}} x^{5} x^{m} \Gamma \left (\frac{m}{2} + \frac{5}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{5}{2} \\ \frac{m}{2} + \frac{7}{2} \end{matrix}\middle |{a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} - \frac{a^{3} c^{\frac{5}{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 )}}{\Gamma \left (\frac{m}{2} + 3\right )} + \frac{a c^{\frac{5}{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 )}}{\Gamma \left (\frac{m}{2} + 2\right )} + \frac{c^{\frac{5}{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 )}^{\frac{5}{2}}{\left (a x + 1\right )}^{2} x^{m}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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