Optimal. Leaf size=164 \[ \frac{c x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-\frac{c}{a x}}}+\frac{c x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{12 a \sqrt{c-\frac{c}{a x}}}-\frac{c x \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2 \sqrt{c-\frac{c}{a x}}}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{8 a^3} \]
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Rubi [A] time = 0.342206, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6178, 863, 873, 875, 208} \[ \frac{c x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-\frac{c}{a x}}}+\frac{c x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{12 a \sqrt{c-\frac{c}{a x}}}-\frac{c x \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2 \sqrt{c-\frac{c}{a x}}}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 6178
Rule 863
Rule 873
Rule 875
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x^4 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 \sqrt{c-\frac{c}{a x}}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{6 a}\\ &=\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^2}{12 a \sqrt{c-\frac{c}{a x}}}+\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 \sqrt{c-\frac{c}{a x}}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{8 a^2}\\ &=-\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x}{8 a^2 \sqrt{c-\frac{c}{a x}}}+\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^2}{12 a \sqrt{c-\frac{c}{a x}}}+\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 \sqrt{c-\frac{c}{a x}}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{16 a^3}\\ &=-\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x}{8 a^2 \sqrt{c-\frac{c}{a x}}}+\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^2}{12 a \sqrt{c-\frac{c}{a x}}}+\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 \sqrt{c-\frac{c}{a x}}}-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{a^2}+\frac{c^2 x^2}{a^2}} \, dx,x,\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{8 a^5}\\ &=-\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x}{8 a^2 \sqrt{c-\frac{c}{a x}}}+\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^2}{12 a \sqrt{c-\frac{c}{a x}}}+\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 \sqrt{c-\frac{c}{a x}}}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{8 a^3}\\ \end{align*}
Mathematica [A] time = 0.463492, size = 147, normalized size = 0.9 \[ \frac{\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^2 x^2+2 a x-3\right ) \sqrt{c-\frac{c}{a x}}}{a x-1}+3 \sqrt{c} \log \left (2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )-3 \sqrt{c} \log (1-a x)}{48 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.175, size = 121, normalized size = 0.7 \begin{align*}{\frac{x}{48}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 16\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+4\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-6\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \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{c - \frac{c}{a x}} x^{2}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22936, size = 717, normalized size = 4.37 \begin{align*} \left [\frac{3 \,{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (8 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{96 \,{\left (a^{4} x - a^{3}\right )}}, -\frac{3 \,{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \,{\left (8 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{48 \,{\left (a^{4} x - a^{3}\right )}}\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{\sqrt{c - \frac{c}{a x}} x^{2}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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