Optimal. Leaf size=104 \[ -\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}} (c-a c x)^{3/2}}{5 a c}-\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-a c x}}{5 a}-\frac {8 c x \sqrt {1-\frac {1}{a^2 x^2}}}{5 a \sqrt {c-a c x}} \]
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Rubi [A] time = 0.19, antiderivative size = 137, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6176, 6181, 78, 45, 37} \[ \frac {12 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{5 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {6 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{5 a \sqrt {1-\frac {1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Rule 37
Rule 45
Rule 78
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{-\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 {1-\frac {x}{a}}{x^{7/2} \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}} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}+\frac {\left (9 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 a \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {6 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{5 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}-\frac {\left (6 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {12 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{5 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {6 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{5 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 58, normalized size = 0.56 \[ \frac {2 \sqrt {\frac {1}{a x}+1} \left (a^2 x^2-3 a x+6\right ) \sqrt {c-a c x}}{5 a^2 \sqrt {1-\frac {1}{a x}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.66, size = 60, normalized size = 0.58 \[ \frac {2 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \, a x + 6\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{3} x - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 69, normalized size = 0.66 \[ -\frac {4 \, \sqrt {-a c x - c} {\left | c \right |}}{a^{2} c} - \frac {2 \, {\left ({\left (a c x + c\right )}^{2} \sqrt {-a c x - c} {\left | c \right |} + 5 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c {\left | c \right |}\right )}}{5 \, a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 55, normalized size = 0.53 \[ \frac {2 \left (a x +1\right ) \left (a^{2} x^{2}-3 a x +6\right ) \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}}{5 a^{2} \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 69, normalized size = 0.66 \[ \frac {2 \, {\left (a^{3} \sqrt {-c} x^{3} - 2 \, a^{2} \sqrt {-c} x^{2} + 3 \, a \sqrt {-c} x + 6 \, \sqrt {-c}\right )} {\left (a x - 1\right )}}{5 \, {\left (a^{3} x - a^{2}\right )} \sqrt {a x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 57, normalized size = 0.55 \[ \frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (a^3\,x^3-2\,a^2\,x^2+3\,a\,x+6\right )}{5\,a^2\,\left (a\,x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 90.92, size = 105, normalized size = 1.01 \[ - \frac {4 i c x \sqrt {\frac {1}{a c x + c}}}{5 a} - \frac {12 i c \sqrt {\frac {1}{a c x + c}}}{5 a^{2}} - \frac {2 i \left (- a c x + c\right )^{2} \sqrt {\frac {1}{a c x + c}}}{5 a^{2} c} + \frac {2 i \left (- a c x + c\right )^{3} \sqrt {\frac {1}{a c x + c}}}{5 a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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