Optimal. Leaf size=169 \[ -\frac{(2 m+1) \sqrt{1-a^2 x^2} x^{m+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{(m+1) \sqrt{c-a^2 c x^2}}-\frac{2 a (m+1) \sqrt{1-a^2 x^2} x^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{(m+2) \sqrt{c-a^2 c x^2}}+\frac{2 (a x+1) x^{m+1}}{\sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.293024, antiderivative size = 169, 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} \[ -\frac{(2 m+1) \sqrt{1-a^2 x^2} x^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{(m+1) \sqrt{c-a^2 c x^2}}-\frac{2 a (m+1) \sqrt{1-a^2 x^2} x^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{(m+2) \sqrt{c-a^2 c x^2}}+\frac{2 (a x+1) x^{m+1}}{\sqrt{c-a^2 c x^2}} \]
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}{\sqrt{c-a^2 c x^2}} \, dx &=c \int \frac{x^m (1+a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{2 x^{1+m} (1+a x)}{\sqrt{c-a^2 c x^2}}-\int \frac{x^m (1+2 m+2 a (1+m) x)}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{2 x^{1+m} (1+a x)}{\sqrt{c-a^2 c x^2}}-(2 a (1+m)) \int \frac{x^{1+m}}{\sqrt{c-a^2 c x^2}} \, dx-(1+2 m) \int \frac{x^m}{\sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{2 x^{1+m} (1+a x)}{\sqrt{c-a^2 c x^2}}-\frac{\left (2 a (1+m) \sqrt{1-a^2 x^2}\right ) \int \frac{x^{1+m}}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-a^2 c x^2}}-\frac{\left ((1+2 m) \sqrt{1-a^2 x^2}\right ) \int \frac{x^m}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{2 x^{1+m} (1+a x)}{\sqrt{c-a^2 c x^2}}-\frac{(1+2 m) x^{1+m} \sqrt{1-a^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{(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{1}{2},\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{(2+m) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 0.114016, size = 66, normalized size = 0.39 \[ \frac{\sqrt{1-a^2 x^2} x^{m+1} F_1\left (m+1;\frac{3}{2},-\frac{1}{2};m+2;a x,-a x\right )}{(m+1) \sqrt{a x-1} \sqrt{-c (a x+1)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.397, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ) ^{2}{x}^{m}}{-{a}^{2}{x}^{2}+1}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}}\, 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}}{\sqrt{-a^{2} c x^{2} + c}{\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^{2} c x^{2} - 2 \, a c x + c}, 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 x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x x^{m}}{a x \sqrt{- a^{2} c x^{2} + 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}}{\sqrt{-a^{2} c x^{2} + c}{\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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