Optimal. Leaf size=463 \[ -\frac{2 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-1}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2},\frac{n+1}{2},\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right )}{(1-n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (n^2+6 n+15\right ) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(1-n) (n+1) (n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (-n^3-2 n^2+7 n+18\right ) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{(n+6) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{(n+1) (n+3) \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.534858, antiderivative size = 467, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6192, 6195, 129, 155, 12, 131} \[ \frac{2 x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}} \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{(3-n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (n^2+6 n+15\right ) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1-n}{2}}}{(1-n) (n+1) (n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (-n^3-2 n^2+7 n+18\right ) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{3-n}{2}}}{\left (n^4-10 n^2+9\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac{x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-3)}}{(n+3) \left (c-a^2 c x^2\right )^{5/2}}-\frac{(n+6) x^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{\frac{n-3}{2}} \left (1-\frac{1}{a x}\right )^{\frac{1}{2} (-n-1)}}{(n+1) (n+3) \left (c-a^2 c x^2\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Rule 6192
Rule 6195
Rule 129
Rule 155
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)} x^4}{\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} \, 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 \frac{\left (1-\frac{x}{a}\right )^{-\frac{5}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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 (a \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{3+n}{a}-\frac{3 x}{a^2}\right ) \left (1-\frac{x}{a}\right )^{-\frac{3}{2}-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{-\frac{5}{2}+\frac{n}{2}}}{x} \, dx,x,\frac{1}{x}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (a^2 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\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{5}{2}+\frac{n}{2}} \left (\frac{(1+n) (3+n)}{a^2}+\frac{2 (6+n) x}{a^3}\right )}{x} \, dx,x,\frac{1}{x}\right )}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (a^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\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{5}{2}+\frac{n}{2}} \left (\frac{(1-n) (1+n) (3+n)}{a^3}-\frac{\left (15+6 n+n^2\right ) x}{a^4}\right )}{x} \, dx,x,\frac{1}{x}\right )}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (18+7 n-2 n^2-n^3\right ) \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}}-\frac{\left (a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname{Subst}\left (\int \frac{\left (9-10 n^2+n^4\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^4 x} \, dx,x,\frac{1}{x}\right )}{(1-n) (3-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (18+7 n-2 n^2-n^3\right ) \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}}-\frac{\left (\left (9-10 n^2+n^4\right ) \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x^5\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 )}{(1-n) (3-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}\\ &=-\frac{\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{(6+n) \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}{(1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac{\left (15+6 n+n^2\right ) \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}{(1-n) (1+n) (3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac{\left (18+7 n-2 n^2-n^3\right ) \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}}+\frac{2 \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 \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )}{(3-n) \left (c-a^2 c x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.9755, size = 201, normalized size = 0.43 \[ \frac{\left (a^2 x^2-1\right ) \left (-\frac{8 a x \sqrt{1-\frac{1}{a^2 x^2}} e^{(n+1) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},e^{2 \coth ^{-1}(a x)}\right )}{n+1}+\frac{e^{n \coth ^{-1}(a x)} \left (-3 a \left (n^2-1\right ) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left (3 \coth ^{-1}(a x)\right )+3 a n^2 x+2 \left (n^2-1\right ) n \cosh \left (2 \coth ^{-1}(a x)\right )-27 a x-2 n^3+26 n\right )}{n^4-10 n^2+9}+\frac{8 (n-a x) e^{n \coth ^{-1}(a x)}}{n^2-1}\right )}{4 a^5 c \left (c-a^2 c x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.303, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}{x}^{4} \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{x^{4} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} c x^{2} + c} x^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - 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{x^{4} \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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