Optimal. Leaf size=183 \[ \frac{(1-2 m) \sqrt{1-a^2 x^2} x^{m+1} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{3 c (m+1) \sqrt{c-a^2 c x^2}}+\frac{2 a (1-m) \sqrt{1-a^2 x^2} x^{m+2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{3 c (m+2) \sqrt{c-a^2 c x^2}}+\frac{2 (a x+1) x^{m+1}}{3 \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.319657, antiderivative size = 183, 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, 1806, 808, 365, 364} \[ \frac{(1-2 m) \sqrt{1-a^2 x^2} x^{m+1} \, _2F_1\left (\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{3 c (m+1) \sqrt{c-a^2 c x^2}}+\frac{2 a (1-m) \sqrt{1-a^2 x^2} x^{m+2} \, _2F_1\left (\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{3 c (m+2) \sqrt{c-a^2 c x^2}}+\frac{2 (a x+1) x^{m+1}}{3 \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1806
Rule 808
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^m}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac{x^m (1+a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac{2 x^{1+m} (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{1}{3} \int \frac{x^m (-1+2 m-2 a (1-m) x)}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{2 x^{1+m} (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{1}{3} (2 a (1-m)) \int \frac{x^{1+m}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx-\frac{1}{3} (-1+2 m) \int \frac{x^m}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{2 x^{1+m} (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (2 a (1-m) \sqrt{1-a^2 x^2}\right ) \int \frac{x^{1+m}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c \sqrt{c-a^2 c x^2}}-\frac{\left ((-1+2 m) \sqrt{1-a^2 x^2}\right ) \int \frac{x^m}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c \sqrt{c-a^2 c x^2}}\\ &=\frac{2 x^{1+m} (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{(1-2 m) x^{1+m} \sqrt{1-a^2 x^2} \, _2F_1\left (\frac{3}{2},\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{3 c (1+m) \sqrt{c-a^2 c x^2}}+\frac{2 a (1-m) x^{2+m} \sqrt{1-a^2 x^2} \, _2F_1\left (\frac{3}{2},\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{3 c (2+m) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 0.241863, size = 173, normalized size = 0.95 \[ \frac{x^{m+1} \left (\sqrt{a x-1} \sqrt{a x+1} \sqrt{c-a c x} F_1\left (m+1;\frac{1}{2},-\frac{1}{2};m+2;-a x,a x\right )+(a x-1) \sqrt{-c (a x+1)} \left (F_1\left (m+1;\frac{1}{2},-\frac{1}{2};m+2;a x,-a x\right )+2 F_1\left (m+1;\frac{3}{2},-\frac{1}{2};m+2;a x,-a x\right )+4 F_1\left (m+1;\frac{5}{2},-\frac{1}{2};m+2;a x,-a x\right )\right )\right )}{8 c^2 (m+1) \sqrt{-(a x-1)^2} \sqrt{a x+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.401, 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{3}{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 x + 1\right )}^{2} x^{m}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}{\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{\sqrt{-a^{2} c x^{2} + c} x^{m}}{a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a c^{2} x - c^{2}}, 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*} - \int \frac{x^{m}}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x x^{m}}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx \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 )}^{\frac{3}{2}}{\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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