Optimal. Leaf size=211 \[ -\frac {4 \sqrt {2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{a^{5/2} \sqrt {1-\frac {1}{a x}}}+\frac {4 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^2 \left (\frac {1}{a x}+1\right )^{5/2} \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {2 x \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.24, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6176, 6181, 96, 94, 93, 206} \[ \frac {4 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{a^{5/2} \sqrt {1-\frac {1}{a x}}}+\frac {2 x^2 \left (\frac {1}{a x}+1\right )^{5/2} \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {2 x \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 206
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{3 \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 {\left (1+\frac {x}{a}\right )^{3/2}}{x^{7/2} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a 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^{5/2} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{a \sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} 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^{3/2} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x^2 \sqrt {c-a c x}}{5 \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^3 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x^2 \sqrt {c-a c x}}{5 \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^3 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x^2 \sqrt {c-a c x}}{5 \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 )}{a^{5/2} \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 114, normalized size = 0.54 \[ \frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {\frac {1}{a x}+1} \left (3 a^2 x^2+11 a x+38\right )-30 \sqrt {2} \sqrt {\frac {1}{x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )\right )}{15 a^{5/2} \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 272, normalized size = 1.29 \[ \left [\frac {2 \, {\left (15 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 49 \, a x + 38\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{15 \, {\left (a^{3} x - a^{2}\right )}}, -\frac {2 \, {\left (30 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 49 \, a x + 38\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{15 \, {\left (a^{3} x - a^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.18, size = 137, normalized size = 0.65 \[ -\frac {\frac {60 i \, \sqrt {2} \sqrt {-c} \arctan \left (-i\right ) - 104 \, \sqrt {2} \sqrt {-c}}{a \mathrm {sgn}\relax (c)} + \frac {2 \, {\left (30 \, \sqrt {2} c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right ) - 3 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} + 5 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c - 30 \, \sqrt {-a c x - c} c^{2}\right )}}{a c^{2} \mathrm {sgn}\left (-a c x - c\right )}}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 125, normalized size = 0.59 \[ -\frac {2 \left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-3 x^{2} a^{2} \sqrt {-c \left (a x +1\right )}+30 \sqrt {c}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-11 x a \sqrt {-c \left (a x +1\right )}-38 \sqrt {-c \left (a x +1\right )}\right )}{15 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x +1\right )}\, a^{2}} \]
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 = -0.00 \[ \int \frac {x\,\sqrt {c-a\,c\,x}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \sqrt {- c \left (a x - 1\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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