Optimal. Leaf size=57 \[ \frac {2 \sqrt {c-\frac {c}{a x}} x^{1+m} \, _2F_1\left (-\frac {1}{2},\frac {1}{2}+m;\frac {3}{2}+m;-a x\right )}{(1+2 m) \sqrt {1-a x}} \]
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Rubi [A]
time = 0.16, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6269, 6263,
862, 66} \begin {gather*} \frac {2 x^{m+1} \sqrt {c-\frac {c}{a x}} \, _2F_1\left (-\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};-a x\right )}{(2 m+1) \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 862
Rule 6263
Rule 6269
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^m \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int e^{\tanh ^{-1}(a x)} x^{-\frac {1}{2}+m} \sqrt {1-a x} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{-\frac {1}{2}+m} \sqrt {1-a^2 x^2}}{\sqrt {1-a x}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int x^{-\frac {1}{2}+m} \sqrt {1+a x} \, dx}{\sqrt {1-a x}}\\ &=\frac {2 \sqrt {c-\frac {c}{a x}} x^{1+m} \, _2F_1\left (-\frac {1}{2},\frac {1}{2}+m;\frac {3}{2}+m;-a x\right )}{(1+2 m) \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 56, normalized size = 0.98 \begin {gather*} \frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \, _2F_1\left (-\frac {1}{2},\frac {1}{2}+m;\frac {3}{2}+m;-a x\right )}{\left (\frac {1}{2}+m\right ) \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +1\right ) x^{m} \sqrt {c -\frac {c}{a x}}}{\sqrt {-a^{2} x^{2}+1}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \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}}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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