Optimal. Leaf size=224 \[ \frac{23 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{1-\frac{1}{a x}}}+\frac{a \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{2 x \sqrt{1-\frac{1}{a x}}}+\frac{7 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.264502, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {6176, 6181, 101, 154, 157, 54, 215, 93, 206} \[ \frac{23 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{1-\frac{1}{a x}}}+\frac{a \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{2 x \sqrt{1-\frac{1}{a x}}}+\frac{7 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 101
Rule 154
Rule 157
Rule 54
Rule 215
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx &=\frac{\sqrt{c-a c x} \int \frac{e^{3 \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{x} \left (1+\frac{x}{a}\right )^{3/2}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x}-\frac{\left (a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}} \left (\frac{1}{2}+\frac{7 x}{2 a}\right )}{\sqrt{x} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{7 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{9}{4 a}-\frac{23 x}{4 a^2}}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{7 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (23 a \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 )}{8 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{7 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (23 a \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 )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{7 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x}+\frac{23 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.14794, size = 132, normalized size = 0.59 \[ \frac{\sqrt{c-a c x} \left (\frac{23 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\left (\frac{1}{x}\right )^{3/2}}-\frac{16 \sqrt{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\left (\frac{1}{x}\right )^{3/2}}+\sqrt{\frac{1}{a x}+1} (9 a x+2)\right )}{4 x^2 \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.211, size = 144, normalized size = 0.6 \begin{align*}{\frac{ax-1}{ \left ( 4\,ax+4 \right ){x}^{2}}\sqrt{-c \left ( ax-1 \right ) } \left ( -16\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{2}{a}^{2}c+23\,c\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{2}{a}^{2}+9\,xa\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}+2\,\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \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}}{x^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69091, size = 979, normalized size = 4.37 \begin{align*} \left [\frac{16 \, \sqrt{2}{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 23 \,{\left (a^{3} x^{3} - a^{2} x^{2}\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 (9 \, a^{2} x^{2} + 11 \, a x + 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{8 \,{\left (a x^{3} - x^{2}\right )}}, -\frac{16 \, \sqrt{2}{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) - 23 \,{\left (a^{3} x^{3} - a^{2} x^{2}\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 (9 \, a^{2} x^{2} + 11 \, a x + 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{4 \,{\left (a x^{3} - x^{2}\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 [C] time = 1.31402, size = 248, normalized size = 1.11 \begin{align*} -\frac{1}{4} \, a^{2} c^{2}{\left (\frac{16 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{c^{\frac{3}{2}} \mathrm{sgn}\left (-a c x - c\right )} - \frac{23 \, \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{c^{\frac{3}{2}} \mathrm{sgn}\left (-a c x - c\right )} + \frac{9 \,{\left (-a c x - c\right )}^{\frac{3}{2}} + 7 \, \sqrt{-a c x - c} c}{a^{2} c^{3} x^{2} \mathrm{sgn}\left (-a c x - c\right )}\right )} - \frac{16 i \, \sqrt{2} a^{2} \sqrt{-c} \arctan \left (-i\right ) - 23 i \, a^{2} \sqrt{-c} \arctan \left (-i \, \sqrt{2}\right ) - 11 \, \sqrt{2} a^{2} \sqrt{-c}}{4 \, \mathrm{sgn}\left (c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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