Optimal. Leaf size=60 \[ \frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \, _2F_1\left (-\frac {1}{2},-1-m;-m;-\frac {1}{a x}\right )}{(1+m) \sqrt {1-\frac {1}{a x}}} \]
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Rubi [A]
time = 0.16, 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 = {6317, 6316, 66}
\begin {gather*} \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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 6316
Rule 6317
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 ) \text {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 = 0.02, size = 60, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^m\,\sqrt {c-\frac {c}{a\,x}}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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