Optimal. Leaf size=102 \[ \frac{\left (3-n^2\right ) (n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}-\frac{(n-3 a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.189252, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {6189, 6184} \[ \frac{\left (3-n^2\right ) (n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}-\frac{(n-3 a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6189
Rule 6184
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-3 a x)}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac{\left (3-n^2\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{a^2 c \left (9-n^2\right )}\\ &=-\frac{e^{n \coth ^{-1}(a x)} (n-3 a x)}{a^3 c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac{e^{n \coth ^{-1}(a x)} \left (3-n^2\right ) (n-a x)}{a^3 c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.688619, size = 109, normalized size = 1.07 \[ \frac{e^{n \coth ^{-1}(a x)} \left (3 a \left (n^2-1\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )+a n^2 x-2 \left (n^2-1\right ) n \cosh \left (2 \coth ^{-1}(a x)\right )-9 a x-2 n^3+10 n\right )}{4 a^3 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 96, normalized size = 0.9 \begin{align*}{\frac{ \left ({a}^{3}{n}^{2}{x}^{3}-{a}^{2}{n}^{3}{x}^{2}-3\,{x}^{3}{a}^{3}+3\,n{x}^{2}{a}^{2}+2\,{n}^{2}xa-2\,n \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{ \left ({n}^{4}-10\,{n}^{2}+9 \right ){a}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{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{x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3283, size = 356, normalized size = 3.49 \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c}{\left (2 \, a n^{2} x +{\left (a^{3} n^{2} - 3 \, a^{3}\right )} x^{3} +{\left (a^{2} n^{3} - 3 \, a^{2} n\right )} x^{2} + 2 \, n\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{3} c^{3} n^{4} - 10 \, a^{3} c^{3} n^{2} + 9 \, a^{3} c^{3} +{\left (a^{7} c^{3} n^{4} - 10 \, a^{7} c^{3} n^{2} + 9 \, a^{7} c^{3}\right )} x^{4} - 2 \,{\left (a^{5} c^{3} n^{4} - 10 \, a^{5} c^{3} n^{2} + 9 \, a^{5} c^{3}\right )} x^{2}} \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{x^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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