Optimal. Leaf size=107 \[ \frac{2 c^2 (1-a x)^{3/2} \sqrt{a x+1}}{(c-a c x)^{3/2}}+\frac{8 c^2 (1-a x)^{3/2}}{\sqrt{a x+1} (c-a c x)^{3/2}}-\frac{2 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{(c-a c x)^{3/2}} \]
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Rubi [A] time = 0.128615, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6130, 23, 87, 63, 208} \[ \frac{2 c^2 (1-a x)^{3/2} \sqrt{a x+1}}{(c-a c x)^{3/2}}+\frac{8 c^2 (1-a x)^{3/2}}{\sqrt{a x+1} (c-a c x)^{3/2}}-\frac{2 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{(c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 23
Rule 87
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx &=\int \frac{(1-a x)^{3/2} \sqrt{c-a c x}}{x (1+a x)^{3/2}} \, dx\\ &=\frac{(1-a x)^{3/2} \int \frac{(c-a c x)^2}{x (1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=\frac{(1-a x)^{3/2} \int \left (-\frac{4 a c^2}{(1+a x)^{3/2}}+\frac{a c^2}{\sqrt{1+a x}}+\frac{c^2}{x \sqrt{1+a x}}\right ) \, dx}{(c-a c x)^{3/2}}\\ &=\frac{8 c^2 (1-a x)^{3/2}}{\sqrt{1+a x} (c-a c x)^{3/2}}+\frac{2 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{(c-a c x)^{3/2}}+\frac{\left (c^2 (1-a x)^{3/2}\right ) \int \frac{1}{x \sqrt{1+a x}} \, dx}{(c-a c x)^{3/2}}\\ &=\frac{8 c^2 (1-a x)^{3/2}}{\sqrt{1+a x} (c-a c x)^{3/2}}+\frac{2 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{(c-a c x)^{3/2}}+\frac{\left (2 c^2 (1-a x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a}+\frac{x^2}{a}} \, dx,x,\sqrt{1+a x}\right )}{a (c-a c x)^{3/2}}\\ &=\frac{8 c^2 (1-a x)^{3/2}}{\sqrt{1+a x} (c-a c x)^{3/2}}+\frac{2 c^2 (1-a x)^{3/2} \sqrt{1+a x}}{(c-a c x)^{3/2}}-\frac{2 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt{1+a x}\right )}{(c-a c x)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0347293, size = 51, normalized size = 0.48 \[ \frac{2 c \sqrt{1-a x} \left (\text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},a x+1\right )+a x+4\right )}{\sqrt{a x+1} \sqrt{c-a c x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.108, size = 78, normalized size = 0.7 \begin{align*} 2\,{\frac{\sqrt{-c \left ( ax-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}{ \left ( ax-1 \right ) \left ( ax+1 \right ) c} \left ( \sqrt{c}{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ) \sqrt{c \left ( ax+1 \right ) }-acx-5\,c \right ) } \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 (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{-a c x + c}}{{\left (a x + 1\right )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60018, size = 466, normalized size = 4.36 \begin{align*} \left [\frac{{\left (a^{2} x^{2} - 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right ) - 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x + 5\right )}}{a^{2} x^{2} - 1}, -\frac{2 \,{\left ({\left (a^{2} x^{2} - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}{\left (a x + 5\right )}\right )}}{a^{2} x^{2} - 1}\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 (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{x \left (a x + 1\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22053, size = 127, normalized size = 1.19 \begin{align*} 2 \,{\left (\frac{\arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{4}{\sqrt{a c x + c}} + \frac{\sqrt{a c x + c}}{c}\right )}{\left | c \right |} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{c}{\left | c \right |} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) + 6 \, \sqrt{-c}{\left | c \right |}\right )}}{\sqrt{-c} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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