Optimal. Leaf size=82 \[ \frac {x^{m+1} \sqrt {c-a^2 c x^2}}{(m+1) \sqrt {1-a^2 x^2}}+\frac {a x^{m+2} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.17, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6153, 6150, 43} \[ \frac {x^{m+1} \sqrt {c-a^2 c x^2}}{(m+1) \sqrt {1-a^2 x^2}}+\frac {a x^{m+2} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{\tanh ^{-1}(a x)} x^m \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int x^m (1+a x) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (x^m+a x^{1+m}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(1+m) \sqrt {1-a^2 x^2}}+\frac {a x^{2+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.60 \[ \frac {x^{m+1} \sqrt {c-a^2 c x^2} \left (\frac {a x}{m+2}+\frac {1}{m+1}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 80, normalized size = 0.98 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (a m + a\right )} x^{2} + {\left (m + 2\right )} x\right )} x^{m}}{{\left (a^{2} m^{2} + 3 \, a^{2} m + 2 \, a^{2}\right )} x^{2} - m^{2} - 3 \, m - 2} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )} x^{m}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 52, normalized size = 0.63 \[ \frac {x^{1+m} \left (a m x +a x +m +2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{\left (2+m \right ) \left (1+m \right ) \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 30, normalized size = 0.37 \[ \frac {a \sqrt {c} x^{2} x^{m}}{m + 2} + \frac {\sqrt {c} x x^{m}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 51, normalized size = 0.62 \[ \frac {x^{m+1}\,\sqrt {c-a^2\,c\,x^2}\,\left (m+a\,x+a\,m\,x+2\right )}{\sqrt {1-a^2\,x^2}\,\left (m^2+3\,m+2\right )} \]
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 \[ \int \frac {x^{m} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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