Optimal. Leaf size=98 \[ \frac {2}{3} x \sqrt {c-a c x} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \, _2F_1\left (-\frac {3}{2},\frac {n-1}{2};-\frac {1}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6176, 6181, 132} \[ \frac {2}{3} x \sqrt {c-a c x} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \, _2F_1\left (-\frac {3}{2},\frac {n-1}{2};-\frac {1}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right ) \]
Antiderivative was successfully verified.
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Rule 132
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{n \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 )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2}{3} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (-1+n)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x \sqrt {c-a c x} \, _2F_1\left (-\frac {3}{2},\frac {1}{2} (-1+n);-\frac {1}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 98, normalized size = 1.00 \[ \frac {2 (a x+1) \sqrt {c-a c x} \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \left (\frac {a x-1}{a x+1}\right )^{\frac {n-1}{2}} \, _2F_1\left (-\frac {3}{2},\frac {n-1}{2};-\frac {1}{2};\frac {2}{a x+1}\right )}{3 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \sqrt {-a c x +c}\, dx \]
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 \[ \int \sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\sqrt {c-a\,c\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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