Optimal. Leaf size=238 \[ -\frac{119 a^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{8 x \sqrt{1-\frac{1}{a x}}}+\frac{119 a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{8 \sqrt{1-\frac{1}{a x}}}+\frac{119 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{12 x^2 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{3 x^3 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.246345, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6176, 6181, 89, 80, 50, 54, 215} \[ -\frac{119 a^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{8 x \sqrt{1-\frac{1}{a x}}}+\frac{119 a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{8 \sqrt{1-\frac{1}{a x}}}+\frac{119 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{12 x^2 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{3 x^3 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 89
Rule 80
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx &=\frac{\sqrt{c-a c x} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x^{7/2}} \, 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{x^{3/2} \left (1-\frac{x}{a}\right )^2}{\left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^3}+\frac{\left (2 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{x^{3/2} \left (\frac{19}{2 a^2}-\frac{x}{2 a^3}\right )}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^3}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^3}+\frac{\left (119 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{x^{3/2}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{6 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^3}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^3}+\frac{119 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{12 \sqrt{1-\frac{1}{a x}} x^2}-\frac{\left (119 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^3}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^3}+\frac{119 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{12 \sqrt{1-\frac{1}{a x}} x^2}-\frac{119 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{8 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (119 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^3}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^3}+\frac{119 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{12 \sqrt{1-\frac{1}{a x}} x^2}-\frac{119 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{8 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (119 a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,\sqrt{\frac{1}{x}}\right )}{8 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x^3}-\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} x^3}+\frac{119 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{12 \sqrt{1-\frac{1}{a x}} x^2}-\frac{119 a^2 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{8 \sqrt{1-\frac{1}{a x}} x}+\frac{119 a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{8 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0910814, size = 98, normalized size = 0.41 \[ \frac{\sqrt{c-a c x} \left (-357 a^3 x^3-119 a^2 x^2+\frac{357 a^{7/2} \sqrt{\frac{1}{a x}+1} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\left (\frac{1}{x}\right )^{7/2}}+38 a x-8\right )}{24 a x^4 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.152, size = 114, normalized size = 0.5 \begin{align*} -{\frac{ax+1}{24\, \left ( ax-1 \right ) ^{2}{x}^{3}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{-c \left ( ax-1 \right ) } \left ( 357\,\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{3}{a}^{3}\sqrt{-c \left ( ax+1 \right ) }+357\,{x}^{3}{a}^{3}\sqrt{c}+119\,{x}^{2}{a}^{2}\sqrt{c}-38\,xa\sqrt{c}+8\,\sqrt{c} \right ){\frac{1}{\sqrt{c}}}} \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{-a c x + c} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63805, size = 632, normalized size = 2.66 \begin{align*} \left [\frac{357 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + a c x - 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) - 2 \,{\left (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{48 \,{\left (a x^{4} - x^{3}\right )}}, \frac{357 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) -{\left (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{24 \,{\left (a x^{4} - x^{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 [A] time = 1.24309, size = 150, normalized size = 0.63 \begin{align*} -\frac{1}{24} \, a^{3} c^{2}{\left (\frac{357 \, \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{c^{\frac{5}{2}}} + \frac{192}{\sqrt{-a c x - c} c^{2}} - \frac{165 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 376 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c + 219 \, \sqrt{-a c x - c} c^{2}}{a^{3} c^{5} x^{3}}\right )}{\left | c \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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