Optimal. Leaf size=60 \[ \frac {x^{m+1} \sqrt {c-\frac {c}{a x}} \, _2F_1\left (-\frac {1}{2},-m-1;-m;-\frac {1}{a x}\right )}{(m+1) \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.22, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6182, 6181, 64} \[ \frac {x^{m+1} \sqrt {c-\frac {c}{a x}} \, _2F_1\left (-\frac {1}{2},-m-1;-m;-\frac {1}{a x}\right )}{(m+1) \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 64
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 x^{-2-m} \sqrt {1+\frac {x}{a}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \, _2F_1\left (-\frac {1}{2},-1-m;-m;-\frac {1}{a x}\right )}{(1+m) \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 1.00 \[ \frac {x^{m+1} \sqrt {c-\frac {c}{a x}} \, _2F_1\left (-\frac {1}{2},-m-1;-m;-\frac {1}{a x}\right )}{(m+1) \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x + 1\right )} x^{m} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}, x\right ) \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a x}} x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
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 \[ \int \frac {x^{m} \sqrt {c -\frac {c}{a x}}}{\sqrt {\frac {a x -1}{a x +1}}}\, dx \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a x}} x^{m}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m\,\sqrt {c-\frac {c}{a\,x}}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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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