Optimal. Leaf size=170 \[ \frac {2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.24, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6176, 6181, 98, 157, 54, 215, 93, 206} \[ \frac {2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 98
Rule 157
Rule 206
Rule 215
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx &=\frac {\sqrt {c-a c x} \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{\sqrt {x}} \, 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 {\left (1+\frac {x}{a}\right )^{3/2}}{x^{3/2} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {3}{2 a}-\frac {x}{2 a^2}}{\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 {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {\left (\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 )}{a \sqrt {1-\frac {1}{a x}}}-\frac {\left (4 \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 )}{a \sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {\left (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 )}{a \sqrt {1-\frac {1}{a x}}}-\frac {\left (8 \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 )}{a \sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {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 {a} \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 120, normalized size = 0.71 \[ \frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {\frac {1}{a x}+1}+\sqrt {\frac {1}{x}} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )-2 \sqrt {2} \sqrt {\frac {1}{x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.23, size = 352, normalized size = 2.07 \[ \left [\frac {2 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1}, -\frac {2 \, {\left (2 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a x - 1}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.21, size = 136, normalized size = 0.80 \[ -c {\left (\frac {4 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{\sqrt {c} \mathrm {sgn}\left (-a c x - c\right )} - \frac {2 \, \arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{\sqrt {c} \mathrm {sgn}\left (-a c x - c\right )} + \frac {-4 i \, \sqrt {2} \arctan \left (-i\right ) + 2 \, \sqrt {2} + 2 i \, \arctan \left (-i \, \sqrt {2}\right )}{\sqrt {-c} \mathrm {sgn}\relax (c)} - \frac {2 \, \sqrt {-a c x - c}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 107, normalized size = 0.63 \[ \frac {2 \left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-2 \sqrt {c}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+\sqrt {c}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right )+\sqrt {-c \left (a x +1\right )}\right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x +1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a c x + c}}{x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-a\,c\,x}}{x\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (a x - 1\right )}}{x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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