Optimal. Leaf size=95 \[ \frac{64 c^2 x \sqrt{1-\frac{1}{a^2 x^2}}}{15 \sqrt{c-a c x}}+\frac{16}{15} c x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{2}{5} x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2} \]
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Rubi [A] time = 0.185946, antiderivative size = 137, normalized size of antiderivative = 1.44, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6176, 6181, 89, 78, 37} \[ \frac{86 \sqrt{\frac{1}{a x}+1} (c-a c x)^{3/2}}{15 a^2 x \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{2 x \sqrt{\frac{1}{a x}+1} (c-a c x)^{3/2}}{5 \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{28 \sqrt{\frac{1}{a x}+1} (c-a c x)^{3/2}}{15 a \left (1-\frac{1}{a x}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Rule 6176
Rule 6181
Rule 89
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\frac{(c-a c x)^{3/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^{7/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{3/2}}{5 \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{\left (2 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{-\frac{7}{a}+\frac{5 x}{2 a^2}}{x^{5/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5 \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{28 \sqrt{1+\frac{1}{a x}} (c-a c x)^{3/2}}{15 a \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{2 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{3/2}}{5 \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{\left (43 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{15 a^2 \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{28 \sqrt{1+\frac{1}{a x}} (c-a c x)^{3/2}}{15 a \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{86 \sqrt{1+\frac{1}{a x}} (c-a c x)^{3/2}}{15 a^2 \left (1-\frac{1}{a x}\right )^{3/2} x}+\frac{2 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{3/2}}{5 \left (1-\frac{1}{a x}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0335499, size = 60, normalized size = 0.63 \[ -\frac{2 c \sqrt{\frac{1}{a x}+1} \left (3 a^2 x^2-14 a x+43\right ) \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 56, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 3\,{a}^{2}{x}^{2}-14\,ax+43 \right ) }{15\,a \left ( ax-1 \right ) ^{2}} \left ( -acx+c \right ) ^{{\frac{3}{2}}}\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 = 1.10012, size = 97, normalized size = 1.02 \begin{align*} -\frac{2 \,{\left (3 \, a^{3} \sqrt{-c} c x^{3} - 11 \, a^{2} \sqrt{-c} c x^{2} + 29 \, a \sqrt{-c} c x + 43 \, \sqrt{-c} c\right )}{\left (a x - 1\right )}}{15 \,{\left (a^{2} x - a\right )} \sqrt{a x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8899, size = 147, normalized size = 1.55 \begin{align*} -\frac{2 \,{\left (3 \, a^{3} c x^{3} - 11 \, a^{2} c x^{2} + 29 \, a c x + 43 \, c\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{2} x - a\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 = 1.16384, size = 84, normalized size = 0.88 \begin{align*} \frac{2 \,{\left (3 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 20 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c + 60 \, \sqrt{-a c x - c} c^{2}\right )}{\left | c \right |}}{15 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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