Optimal. Leaf size=359 \[ -\frac{2 n x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2},\frac{n+1}{2},\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right )}{a (1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (n^2+2 n+2\right ) x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{a (1-n) (n+1) \left (c-a^2 c x^2\right )^{3/2}}+\frac{x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{(n+2) x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{a (n+1) \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.332594, antiderivative size = 363, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6192, 6194, 129, 155, 12, 131} \[ \frac{2 n x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}} \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (3-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (n^2+2 n+2\right ) x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{a (1-n) (n+1) \left (c-a^2 c x^2\right )^{3/2}}+\frac{x^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{(n+2) x^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{a (n+1) \left (c-a^2 c x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Rule 6192
Rule 6194
Rule 129
Rule 155
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}}{x^2} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (\left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{n}{a}-\frac{2 x}{a^2}\right ) \left (1-\frac{x}{a}\right )^{-\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{(2+n) \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{\left (a \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{-\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}} \left (\frac{n (1+n)}{a^2}+\frac{(2+n) x}{a^3}\right )}{x} \, dx,x,\frac{1}{x}\right )}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{(2+n) \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (2+2 n+n^2\right ) \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{\left (a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \frac{n \left (1-n^2\right ) \left (1-\frac{x}{a}\right )^{\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}}{a^3 x} \, dx,x,\frac{1}{x}\right )}{(1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{(2+n) \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (2+2 n+n^2\right ) \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac{\left (n \left (1-n^2\right ) \left (1-\frac{1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{\frac{1}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac{(2+n) \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (2+2 n+n^2\right ) \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^3}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-1-n)} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}+\frac{2 n \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{1}{2} (-3+n)} x^3 \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{a (3-n) \left (c-a^2 c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.552972, size = 133, normalized size = 0.37 \[ \frac{\frac{c (a n x-1) e^{n \coth ^{-1}(a x)}}{n^2-1}-\frac{c \left (a^2 x^2-1\right ) \left (\frac{2 n e^{(n+1) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )}{a x \sqrt{1-\frac{1}{a^2 x^2}}}+(n+1) e^{n \coth ^{-1}(a x)}\right )}{n+1}}{a^4 c^2 \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.307, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}{x}^{3} \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{x^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{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^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, 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{x^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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