Optimal. Leaf size=134 \[ \frac{\sqrt{1-a^2 x^2} x^{m+1} \text{Hypergeometric2F1}\left (3,\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{c^2 (m+1) \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} x^{m+2} \text{Hypergeometric2F1}\left (3,\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{c^2 (m+2) \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.211823, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6153, 6150, 82, 73, 364} \[ \frac{\sqrt{1-a^2 x^2} x^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{c^2 (m+1) \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} x^{m+2} \, _2F_1\left (3,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{c^2 (m+2) \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 82
Rule 73
Rule 364
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^m}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{\tanh ^{-1}(a x)} x^m}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{x^m}{(1-a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{x^m}{(1-a x)^3 (1+a x)^3} \, dx}{c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (a \sqrt{1-a^2 x^2}\right ) \int \frac{x^{1+m}}{(1-a x)^3 (1+a x)^3} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{x^m}{\left (1-a^2 x^2\right )^3} \, dx}{c^2 \sqrt{c-a^2 c x^2}}+\frac{\left (a \sqrt{1-a^2 x^2}\right ) \int \frac{x^{1+m}}{\left (1-a^2 x^2\right )^3} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{x^{1+m} \sqrt{1-a^2 x^2} \, _2F_1\left (3,\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{c^2 (1+m) \sqrt{c-a^2 c x^2}}+\frac{a x^{2+m} \sqrt{1-a^2 x^2} \, _2F_1\left (3,\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{c^2 (2+m) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.040429, size = 107, normalized size = 0.8 \[ \frac{\sqrt{1-a^2 x^2} \left (\frac{x^{m+1} \text{Hypergeometric2F1}\left (3,\frac{m+1}{2},\frac{m+1}{2}+1,a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \text{Hypergeometric2F1}\left (3,\frac{m+2}{2},\frac{m+2}{2}+1,a^2 x^2\right )}{m+2}\right )}{c^2 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.344, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+1 \right ){x}^{m}{\frac{1}{\sqrt{-{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 x + 1\right )} x^{m}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \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 (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{m}}{a^{7} c^{3} x^{7} - a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )} x^{m}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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