Optimal. Leaf size=507 \[ \frac{2 a n \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2},\frac{n+1}{2},\frac{a x+1}{1-a x}\right )}{c^2 (1-n) \sqrt{c-a^2 c x^2}}+\frac{a \left (n^2+6 n+12\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{c^2 (n+1) (n+3) \sqrt{c-a^2 c x^2}}-\frac{a \left (n^3+6 n^2+15 n+24\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1-n}{2}}}{c^2 (n+3) \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}+\frac{a \left (-n^4-2 n^3+7 n^2+18 n+24\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{3-n}{2}}}{c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}+\frac{a (n+4) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{c^2 (n+3) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{c^2 x \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.535474, antiderivative size = 511, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6153, 6150, 129, 155, 12, 131} \[ -\frac{2 a n \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{3-n}{2}} \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{1-a x}{a x+1}\right )}{c^2 (3-n) \sqrt{c-a^2 c x^2}}+\frac{a \left (n^2+6 n+12\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{c^2 (n+1) (n+3) \sqrt{c-a^2 c x^2}}-\frac{a \left (n^3+6 n^2+15 n+24\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1-n}{2}}}{c^2 (n+3) \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}+\frac{a \left (-n^4-2 n^3+7 n^2+18 n+24\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{3-n}{2}}}{c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}+\frac{a (n+4) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{c^2 (n+3) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{c^2 x \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Rule 6153
Rule 6150
Rule 129
Rule 155
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{n \tanh ^{-1}(a x)}}{x^2 \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{(1-a x)^{-\frac{5}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}}}{x^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 x \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \int \frac{(1-a x)^{-\frac{5}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \left (-a n-4 a^2 x\right )}{x} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{a (4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 x \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \int \frac{(1-a x)^{-\frac{3}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \left (a^2 n (3+n)+3 a^3 (4+n) x\right )}{x} \, dx}{a c^2 (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{a (4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 x \sqrt{c-a^2 c x^2}}+\frac{a \left (12+6 n+n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \int \frac{(1-a x)^{-\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \left (-a^3 n (1+n) (3+n)-2 a^4 \left (12+6 n+n^2\right ) x\right )}{x} \, dx}{a^2 c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{a (4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 x \sqrt{c-a^2 c x^2}}+\frac{a \left (12+6 n+n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{a \left (24+15 n+6 n^2+n^3\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \int \frac{(1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{5}{2}+\frac{n}{2}} \left (-a^4 (1-n) n (1+n) (3+n)+a^5 \left (24+15 n+6 n^2+n^3\right ) x\right )}{x} \, dx}{a^3 c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{a (4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 x \sqrt{c-a^2 c x^2}}+\frac{a \left (12+6 n+n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{a \left (24+15 n+6 n^2+n^3\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}+\frac{a \left (24+18 n+7 n^2-2 n^3-n^4\right ) (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \int \frac{a^5 (1-n) (3-n) n (1+n) (3+n) (1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx}{a^4 c^2 (1-n) (3-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{a (4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 x \sqrt{c-a^2 c x^2}}+\frac{a \left (12+6 n+n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{a \left (24+15 n+6 n^2+n^3\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}+\frac{a \left (24+18 n+7 n^2-2 n^3-n^4\right ) (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}+\frac{\left (a n \sqrt{1-a^2 x^2}\right ) \int \frac{(1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{a (4+n) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (3+n) \sqrt{c-a^2 c x^2}}-\frac{(1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 x \sqrt{c-a^2 c x^2}}+\frac{a \left (12+6 n+n^2\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{a \left (24+15 n+6 n^2+n^3\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}+\frac{a \left (24+18 n+7 n^2-2 n^3-n^4\right ) (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}-\frac{2 a n (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2} \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{1-a x}{1+a x}\right )}{c^2 (3-n) \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.303994, size = 268, normalized size = 0.53 \[ -\frac{\sqrt{1-a^2 x^2} (1-a x)^{\frac{1}{2} (-n-3)} (a x+1)^{\frac{n-3}{2}} \left (2 a \left (n^3+3 n^2-n-3\right ) n x (a x-1)^3 \text{Hypergeometric2F1}\left (1,\frac{3}{2}-\frac{n}{2},\frac{5}{2}-\frac{n}{2},\frac{1-a x}{a x+1}\right )+a n^3 x \left (-2 a^3 x^3+3 a^2 x^2+2 a x-4\right )+n^2 \left (7 a^4 x^4-18 a^3 x^3+6 a^2 x^2+18 a x-10\right )+a n x \left (18 a^3 x^3-33 a^2 x^2-18 a x+34\right )+24 a^4 x^4-36 a^2 x^2+n^4 \left (-(a x-1)^3\right ) (a x+1)+9\right )}{c^2 (n-3) (n-1) (n+1) (n+3) x \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.211, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{x}^{2}} \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 (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x^{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} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{6} c^{3} x^{8} - 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} - c^{3} x^{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{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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