Optimal. Leaf size=116 \[ \frac {(4 m+3) x^{m+1} \sqrt {c-\frac {c}{a x}} \, _2F_1\left (\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};-a x\right )}{(m+1) (2 m+1) \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} x^{m+1} \sqrt {c-\frac {c}{a x}}}{(m+1) (1-a x)} \]
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Rubi [A] time = 0.29, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6134, 6128, 881, 848, 64} \[ \frac {(4 m+3) x^{m+1} \sqrt {c-\frac {c}{a x}} \, _2F_1\left (\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};-a x\right )}{(m+1) (2 m+1) \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} x^{m+1} \sqrt {c-\frac {c}{a x}}}{(m+1) (1-a x)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 848
Rule 881
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} (1-a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \sqrt {1-a^2 x^2}}{(1+m) (1-a x)}+\frac {\left ((3+4 m) \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{-\frac {1}{2}+m} \sqrt {1-a x}}{\sqrt {1-a^2 x^2}} \, dx}{2 (1+m) \sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \sqrt {1-a^2 x^2}}{(1+m) (1-a x)}+\frac {\left ((3+4 m) \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{-\frac {1}{2}+m}}{\sqrt {1+a x}} \, dx}{2 (1+m) \sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x^{1+m} \sqrt {1-a^2 x^2}}{(1+m) (1-a x)}+\frac {(3+4 m) \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+m) (1+2 m) \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 87, normalized size = 0.75 \[ -\frac {2 x^{m+1} \sqrt {c-\frac {c}{a x}} \left (a (4 m+3) x \, _2F_1\left (\frac {1}{2},m+\frac {3}{2};m+\frac {5}{2};-a x\right )-(2 m+3) \sqrt {a x+1}\right )}{\left (4 m^2+8 m+3\right ) \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, 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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a x}} x^{m}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sqrt {c -\frac {c}{a x}}\, \sqrt {-a^{2} x^{2}+1}}{a x +1}\, dx \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a x}} x^{m}}{a x + 1}\,{d x} \]
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 \[ \int \frac {x^m\,\sqrt {c-\frac {c}{a\,x}}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \]
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 \[ \int \frac {x^{m} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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