Optimal. Leaf size=54 \[ \frac {2 x \sqrt {c-\frac {c}{a x}} F_1\left (\frac {1}{2};\frac {n-1}{2},-\frac {n}{2};\frac {3}{2};a x,-a x\right )}{\sqrt {1-a x}} \]
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Rubi [A] time = 0.15, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6134, 6129, 133} \[ \frac {2 x \sqrt {c-\frac {c}{a x}} F_1\left (\frac {1}{2};\frac {n-1}{2},-\frac {n}{2};\frac {3}{2};a x,-a x\right )}{\sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {e^{n \tanh ^{-1}(a x)} \sqrt {1-a x}}{\sqrt {x}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1-a x)^{\frac {1}{2}-\frac {n}{2}} (1+a x)^{n/2}}{\sqrt {x}} \, dx}{\sqrt {1-a x}}\\ &=\frac {2 \sqrt {c-\frac {c}{a x}} x F_1\left (\frac {1}{2};\frac {1}{2} (-1+n),-\frac {n}{2};\frac {3}{2};a x,-a x\right )}{\sqrt {1-a x}}\\ \end {align*}
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Mathematica [F] time = 180.01, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \sqrt {\frac {a c x - c}{a x}}, x\right ) \]
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 \[ \int \sqrt {c - \frac {c}{a x}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \sqrt {c -\frac {c}{a x}}\, dx \]
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 \[ \int \sqrt {c - \frac {c}{a x}} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\sqrt {c-\frac {c}{a\,x}} \,d x \]
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 \[ \int \sqrt {- c \left (-1 + \frac {1}{a x}\right )} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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