Optimal. Leaf size=81 \[ \frac {x^{m+1} \sqrt {c-\frac {c}{a^2 x^2}}}{m \sqrt {1-a^2 x^2}}-\frac {a x^{m+2} \sqrt {c-\frac {c}{a^2 x^2}}}{(m+1) \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.23, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6160, 6150, 43} \[ \frac {x^{m+1} \sqrt {c-\frac {c}{a^2 x^2}}}{m \sqrt {1-a^2 x^2}}-\frac {a x^{m+2} \sqrt {c-\frac {c}{a^2 x^2}}}{(m+1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^m \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int e^{-\tanh ^{-1}(a x)} x^{-1+m} \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int x^{-1+m} (1-a x) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \left (x^{-1+m}-a x^m\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^{1+m}}{m \sqrt {1-a^2 x^2}}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x^{2+m}}{(1+m) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 0.64 \[ \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {x^m}{m}-\frac {a x^{m+1}}{m+1}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 78, normalized size = 0.96 \[ \frac {\sqrt {-a^{2} x^{2} + 1} {\left (a m x^{2} - {\left (m + 1\right )} x\right )} x^{m} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{{\left (a^{2} m^{2} + a^{2} m\right )} x^{2} - m^{2} - m} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{m}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 69, normalized size = 0.85 \[ \frac {x^{1+m} \left (a x m -m -1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {-a^{2} x^{2}+1}}{\left (1+m \right ) m \left (a x -1\right ) \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.39, size = 55, normalized size = 0.68 \[ \frac {{\left (i \, a \sqrt {c} m x + \sqrt {c} {\left (-i \, m - i\right )}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} x^{m}}{{\left (m^{2} + m\right )} a^{3} x^{2} - {\left (m^{2} + m\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 78, normalized size = 0.96 \[ -\frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (\frac {x^m\,x^2\,\sqrt {1-a^2\,x^2}}{a\,\left (m+1\right )}-\frac {x\,x^m\,\sqrt {1-a^2\,x^2}}{a^2\,m}\right )}{\frac {1}{a^2}-x^2} \]
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 \[ \int \frac {x^{m} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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