Optimal. Leaf size=130 \[ \frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
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Rubi [A] time = 0.402853, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6167, 6151, 1807, 835, 807, 266, 63, 208} \[ \frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6151
Rule 1807
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^5} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^5} \, dx\\ &=-\left (c \int \frac{(1+a x)^2}{x^5 \sqrt{c-a^2 c x^2}} \, dx\right )\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{1}{4} \int \frac{-8 a c-7 a^2 c x}{x^4 \sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}-\frac{\int \frac{21 a^2 c^2+16 a^3 c^2 x}{x^3 \sqrt{c-a^2 c x^2}} \, dx}{12 c}\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{\int \frac{-32 a^3 c^3-21 a^4 c^3 x}{x^2 \sqrt{c-a^2 c x^2}} \, dx}{24 c^2}\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}-\frac{1}{8} \left (7 a^4 c\right ) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}-\frac{1}{16} \left (7 a^4 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}+\frac{1}{8} \left (7 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c-a^2 c x^2}\right )\\ &=\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}+\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.137362, size = 95, normalized size = 0.73 \[ \frac{\left (32 a^3 x^3+21 a^2 x^2+16 a x+6\right ) \sqrt{c-a^2 c x^2}}{24 x^4}+\frac{7}{8} a^4 \sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )-\frac{7}{8} a^4 \sqrt{c} \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.066, size = 287, normalized size = 2.2 \begin{align*}{\frac{1}{4\,c{x}^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{9\,{a}^{2}}{8\,c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{a}^{4}}{8}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) }-{\frac{7\,{a}^{4}}{8}\sqrt{-{a}^{2}c{x}^{2}+c}}+2\,{\frac{{a}^{3} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{3/2}}{cx}}+2\,{a}^{5}x\sqrt{-{a}^{2}c{x}^{2}+c}+2\,{\frac{{a}^{5}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c{x}^{2}+c}}} \right ) }+2\,{a}^{4}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }-2\,{\frac{{a}^{5}c}{\sqrt{{a}^{2}c}}\arctan \left ({\sqrt{{a}^{2}c}x{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \right ) }+{\frac{2\,a}{3\,c{x}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{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{\sqrt{-a^{2} c x^{2} + c}{\left (a x + 1\right )}}{{\left (a x - 1\right )} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61686, size = 416, normalized size = 3.2 \begin{align*} \left [\frac{21 \, a^{4} \sqrt{c} x^{4} \log \left (-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) + 2 \,{\left (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt{-a^{2} c x^{2} + c}}{48 \, x^{4}}, \frac{21 \, a^{4} \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) +{\left (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt{-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{x^{5} \left (a x - 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1829, size = 437, normalized size = 3.36 \begin{align*} -\frac{7 \, a^{4} c \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{4 \, \sqrt{-c}} + \frac{21 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{7} a^{4} c - 45 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{5} a^{4} c^{2} + 96 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{4} a^{3} \sqrt{-c} c^{2}{\left | a \right |} - 45 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{3} a^{4} c^{3} - 128 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt{-c} c^{3}{\left | a \right |} + 21 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )} a^{4} c^{4} + 32 \, a^{3} \sqrt{-c} c^{4}{\left | a \right |}}{12 \,{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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