Optimal. Leaf size=623 \[ \frac{a^2 \left (n^2+5\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-1}{2}} (1-a x)^{\frac{1-n}{2}} \text{Hypergeometric2F1}\left (1,\frac{n-1}{2},\frac{n+1}{2},\frac{a x+1}{1-a x}\right )}{c^2 (1-n) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (n^2+4 n+5\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{2 c^2 (n+3) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (n^3+6 n^2+17 n+30\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{2 c^2 (n+1) (n+3) \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (n^4+6 n^3+20 n^2+54 n+75\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1-n}{2}}}{2 c^2 (n+3) \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (-n^5-2 n^4+2 n^3+8 n^2+59 n+90\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{3-n}{2}}}{2 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}-\frac{a n \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{2 c^2 x \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.646728, antiderivative size = 628, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6153, 6150, 129, 151, 155, 12, 131} \[ -\frac{a^2 \left (n^2+5\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{3-n}{2}} \, _2F_1\left (1,\frac{3-n}{2};\frac{5-n}{2};\frac{1-a x}{a x+1}\right )}{c^2 (3-n) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (n^2+4 n+5\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{2 c^2 (n+3) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (n^3+6 n^2+17 n+30\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-1)}}{2 c^2 (n+1) (n+3) \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (n^4+6 n^3+20 n^2+54 n+75\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1-n}{2}}}{2 c^2 (n+3) \left (1-n^2\right ) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (-n^5-2 n^4+2 n^3+8 n^2+59 n+90\right ) \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{3-n}{2}}}{2 c^2 \left (n^4-10 n^2+9\right ) \sqrt{c-a^2 c x^2}}-\frac{a n \sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{2 c^2 x \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} (a x+1)^{\frac{n-3}{2}} (1-a x)^{\frac{1}{2} (-n-3)}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Rule 6153
Rule 6150
Rule 129
Rule 151
Rule 155
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{x^3 \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^3 \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^3} \, 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}}{2 c^2 x^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}} \left (-a n-5 a^2 x\right )}{x^2} \, dx}{2 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}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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^2 \left (5+n^2\right )+4 a^3 n x\right )}{x} \, dx}{2 c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{a^2 \left (5+4 n+n^2\right ) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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^3 (3+n) \left (5+n^2\right )-3 a^4 \left (5+4 n+n^2\right ) x\right )}{x} \, dx}{2 a c^2 (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{a^2 \left (5+4 n+n^2\right ) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 x \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (30+17 n+6 n^2+n^3\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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^4 (1+n) (3+n) \left (5+n^2\right )+2 a^5 \left (30+17 n+6 n^2+n^3\right ) x\right )}{x} \, dx}{2 a^2 c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{a^2 \left (5+4 n+n^2\right ) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 x \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (30+17 n+6 n^2+n^3\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (75+54 n+20 n^2+6 n^3+n^4\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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^5 (1-n) (1+n) (3+n) \left (5+n^2\right )-a^6 \left (75+54 n+20 n^2+6 n^3+n^4\right ) x\right )}{x} \, dx}{2 a^3 c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{a^2 \left (5+4 n+n^2\right ) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 x \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (30+17 n+6 n^2+n^3\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (75+54 n+20 n^2+6 n^3+n^4\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (90+59 n+8 n^2+2 n^3-2 n^4-n^5\right ) (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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^6 (1-n) (3-n) (1+n) (3+n) \left (5+n^2\right ) (1-a x)^{\frac{1}{2}-\frac{n}{2}} (1+a x)^{-\frac{3}{2}+\frac{n}{2}}}{x} \, dx}{2 a^4 c^2 (1-n) (3-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}\\ &=\frac{a^2 \left (5+4 n+n^2\right ) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 x \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (30+17 n+6 n^2+n^3\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (75+54 n+20 n^2+6 n^3+n^4\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (90+59 n+8 n^2+2 n^3-2 n^4-n^5\right ) (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}+\frac{\left (a^2 \left (5+n^2\right ) \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}{2 c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{a^2 \left (5+4 n+n^2\right ) (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{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}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a n (1-a x)^{\frac{1}{2} (-3-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 x \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (30+17 n+6 n^2+n^3\right ) (1-a x)^{\frac{1}{2} (-1-n)} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 (1+n) (3+n) \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (75+54 n+20 n^2+6 n^3+n^4\right ) (1-a x)^{\frac{1-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 (1-n) (1+n) (3+n) \sqrt{c-a^2 c x^2}}+\frac{a^2 \left (90+59 n+8 n^2+2 n^3-2 n^4-n^5\right ) (1-a x)^{\frac{3-n}{2}} (1+a x)^{\frac{1}{2} (-3+n)} \sqrt{1-a^2 x^2}}{2 c^2 \left (9-10 n^2+n^4\right ) \sqrt{c-a^2 c x^2}}-\frac{a^2 \left (5+n^2\right ) (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 = 0.658955, size = 267, normalized size = 0.43 \[ \frac{\sqrt{1-a^2 x^2} (1-a x)^{\frac{1}{2} (-n-3)} (a x+1)^{\frac{n-3}{2}} \left (-\frac{a^2 (1-a x) \left (-(a x-1)^2 \left (2 \left (n^6-5 n^4-41 n^2+45\right ) \text{Hypergeometric2F1}\left (1,\frac{3}{2}-\frac{n}{2},\frac{5}{2}-\frac{n}{2},\frac{1-a x}{a x+1}\right )-n^6+n^5+8 n^4+2 n^3+35 n^2-87 n-270\right )+\left (n^4+6 n^3+20 n^2+54 n+75\right ) (n-3)^2 (a x-1)-(n-1) \left (n^3+6 n^2+17 n+30\right ) (n-3)^2\right )}{(n-3)^2 (n-1) (n+1) (n+3)}+\frac{a^2 \left (n^2+4 n+5\right )}{n+3}-\frac{a n}{x}-\frac{1}{x^2}\right )}{2 c^2 \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.204, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{x}^{3}} \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^{3}}\,{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^{9} - 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} - c^{3} x^{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{\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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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