Optimal. Leaf size=126 \[ \frac {\sqrt {\frac {1}{a x}+1} x^{m+1} \sqrt {c-\frac {c}{a x}}}{(m+1) \sqrt {1-\frac {1}{a x}}}-\frac {(4 m+3) x^m \sqrt {c-\frac {c}{a x}} \, _2F_1\left (\frac {1}{2},-m;1-m;-\frac {1}{a x}\right )}{2 a m (m+1) \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.27, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6182, 6181, 79, 64} \[ \frac {\sqrt {\frac {1}{a x}+1} x^{m+1} \sqrt {c-\frac {c}{a x}}}{(m+1) \sqrt {1-\frac {1}{a x}}}-\frac {(4 m+3) x^m \sqrt {c-\frac {c}{a x}} \, _2F_1\left (\frac {1}{2},-m;1-m;-\frac {1}{a x}\right )}{2 a m (m+1) \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 64
Rule 79
Rule 6181
Rule 6182
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx &=\frac {\sqrt {c-\frac {c}{a x}} \int e^{-\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^m \, dx}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\left (\sqrt {c-\frac {c}{a x}} \left (\frac {1}{x}\right )^m x^m\right ) \operatorname {Subst}\left (\int \frac {x^{-2-m} \left (1-\frac {x}{a}\right )}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^{1+m}}{(1+m) \sqrt {1-\frac {1}{a x}}}+\frac {\left ((3+4 m) \sqrt {c-\frac {c}{a x}} \left (\frac {1}{x}\right )^m x^m\right ) \operatorname {Subst}\left (\int \frac {x^{-1-m}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a (1+m) \sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^{1+m}}{(1+m) \sqrt {1-\frac {1}{a x}}}-\frac {(3+4 m) \sqrt {c-\frac {c}{a x}} x^m \, _2F_1\left (\frac {1}{2},-m;1-m;-\frac {1}{a x}\right )}{2 a m (1+m) \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 93, normalized size = 0.74 \[ \frac {x^m \sqrt {c-\frac {c}{a x}} \left (2 a m x \sqrt {\frac {1}{a x}+1}-(4 m+3) \, _2F_1\left (\frac {1}{2},-m;1-m;-\frac {1}{a x}\right )\right )}{2 a m (m+1) \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}, x\right ) \]
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 \[ \int \sqrt {c - \frac {c}{a x}} x^{m} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int x^{m} \sqrt {c -\frac {c}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}\, dx \]
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 \[ \int \sqrt {c - \frac {c}{a x}} x^{m} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
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 \[ \int x^m\,\sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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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