Optimal. Leaf size=140 \[ \frac{16 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^3 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}-\frac{8 x^2 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.215309, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {6176, 6181, 45, 37} \[ \frac{16 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^3 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}-\frac{8 x^2 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 45
Rule 37
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^{5/2} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^{9/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{\left (4 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^{7/2}} \, dx,x,\frac{1}{x}\right )}{7 a \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \left (1+\frac{1}{a x}\right )^{3/2} x^2 \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^{5/2}} \, dx,x,\frac{1}{x}\right )}{35 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{16 \left (1+\frac{1}{a x}\right )^{3/2} x \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{8 \left (1+\frac{1}{a x}\right )^{3/2} x^2 \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0331096, size = 64, normalized size = 0.46 \[ \frac{2 \sqrt{\frac{1}{a x}+1} (a x+1) \left (15 a^2 x^2-12 a x+8\right ) \sqrt{c-a c x}}{105 a^3 \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 49, normalized size = 0.4 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 15\,{a}^{2}{x}^{2}-12\,ax+8 \right ) }{105\,{a}^{3}}\sqrt{-acx+c}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10896, size = 74, normalized size = 0.53 \begin{align*} \frac{2 \,{\left (15 \, a^{3} \sqrt{-c} x^{3} + 3 \, a^{2} \sqrt{-c} x^{2} - 4 \, a \sqrt{-c} x + 8 \, \sqrt{-c}\right )} \sqrt{a x + 1}}{105 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58551, size = 151, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (15 \, a^{4} x^{4} + 18 \, a^{3} x^{3} - a^{2} x^{2} + 4 \, a x + 8\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{105 \,{\left (a^{4} x - a^{3}\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 [A] time = 1.24733, size = 196, normalized size = 1.4 \begin{align*} \frac{2 \,{\left (\frac{22 \, \sqrt{2} \sqrt{-c} c}{a^{2} \mathrm{sgn}\left (c\right )} + \frac{15 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} - 84 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c - 175 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{2} + 14 \,{\left (3 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 10 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c\right )} c}{a^{2} c^{2} \mathrm{sgn}\left (-a c x - c\right )}\right )}}{105 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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