Optimal. Leaf size=57 \[ \frac {2 x^{m+1} \sqrt {c-\frac {c}{a x}} \, _2F_1\left (-\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};-a x\right )}{(2 m+1) \sqrt {1-a x}} \]
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Rubi [A] time = 0.23, 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 = {6134, 6128, 848, 64} \[ \frac {2 x^{m+1} \sqrt {c-\frac {c}{a x}} \, _2F_1\left (-\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};-a x\right )}{(2 m+1) \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 64
Rule 848
Rule 6128
Rule 6134
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 = 0.03, size = 56, normalized size = 0.98 \[ \frac {x^{m+1} \sqrt {c-\frac {c}{a x}} \, _2F_1\left (-\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};-a x\right )}{\left (m+\frac {1}{2}\right ) \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x^{m} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, 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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^m\,\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{m} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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