Optimal. Leaf size=92 \[ \frac{2 x^2 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{4 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.172632, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6176, 6181, 45, 37} \[ \frac{2 x^2 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{4 x \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}}} \]
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 \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^{3/2} \, 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^{7/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{\left (2 \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 )}{5 a \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{4 \left (1+\frac{1}{a x}\right )^{3/2} x \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \left (1+\frac{1}{a x}\right )^{3/2} x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.025395, size = 56, normalized size = 0.61 \[ \frac{2 \sqrt{\frac{1}{a x}+1} (a x+1) (3 a x-2) \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 41, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 3\,ax-2 \right ) }{15\,{a}^{2}}\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 = 1.0592, size = 55, normalized size = 0.6 \begin{align*} \frac{2 \,{\left (3 \, a^{2} \sqrt{-c} x^{2} + a \sqrt{-c} x - 2 \, \sqrt{-c}\right )} \sqrt{a x + 1}}{15 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57287, size = 131, normalized size = 1.42 \begin{align*} \frac{2 \,{\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} - a x - 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{3} x - a^{2}\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 = 1.18634, size = 101, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (\frac{2 \, \sqrt{2} \sqrt{-c} c^{2}}{\mathrm{sgn}\left (c\right )} + \frac{3 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 5 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c}{\mathrm{sgn}\left (-a c x - c\right )}\right )}}{15 \, a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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