Optimal. Leaf size=330 \[ -\frac{3 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.366695, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {6192, 6195, 45, 37} \[ -\frac{3 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (n^2+4 n+3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(n+3) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{a x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6195
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^2} \, dx}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-\frac{5}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{\frac{1-n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}} \, dx,x,\frac{1}{x}\right )}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-3-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{3 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac{6 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (3+n-3 n^2-n^3\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{6 a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.654351, size = 110, normalized size = 0.33 \[ -\frac{e^{n \coth ^{-1}(a x)} \left (a n \left (n^2-1\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )+3 \left (a n^3 x-9 a n x-2 n^2+10\right )-6 \left (n^2-1\right ) \cosh \left (2 \coth ^{-1}(a x)\right )\right )}{4 a^4 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 93, normalized size = 0.3 \begin{align*} -{\frac{ \left ({a}^{3}{n}^{3}{x}^{3}-7\,n{x}^{3}{a}^{3}-3\,{a}^{2}{n}^{2}{x}^{2}+9\,{a}^{2}{x}^{2}+6\,nax-6 \right ) \left ( ax-1 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{{a}^{4} \left ({n}^{4}-10\,{n}^{2}+9 \right ) } \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^{3} \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 = 1.63409, size = 354, normalized size = 1.07 \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c}{\left ({\left (a^{3} n^{3} - 7 \, a^{3} n\right )} x^{3} + 6 \, a n x + 3 \,{\left (a^{2} n^{2} - 3 \, a^{2}\right )} x^{2} + 6\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{3} n^{4} - 10 \, a^{4} c^{3} n^{2} + 9 \, a^{4} c^{3} +{\left (a^{8} c^{3} n^{4} - 10 \, a^{8} c^{3} n^{2} + 9 \, a^{8} c^{3}\right )} x^{4} - 2 \,{\left (a^{6} c^{3} n^{4} - 10 \, a^{6} c^{3} n^{2} + 9 \, a^{6} 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^{3} \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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