Optimal. Leaf size=83 \[ \frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(1+m) \sqrt {1-a^2 x^2}}-\frac {a x^{2+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 83, 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 = {6288, 6285, 45}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
Rule 6288
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 = 50, normalized size = 0.60 \begin {gather*} \frac {x^{1+m} \left (\frac {1}{1+m}-\frac {a x}{2+m}\right ) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 68, normalized size = 0.82
method | result | size |
gosper | \(\frac {x^{1+m} \left (a m x +a x -m -2\right ) \sqrt {-a^{2} c \,x^{2}+c}\, \sqrt {-a^{2} x^{2}+1}}{\left (2+m \right ) \left (1+m \right ) \left (a x -1\right ) \left (a x +1\right )}\) | \(68\) |
risch | \(\frac {\sqrt {-\frac {c \left (-a^{2} x^{2}+1\right )}{a^{2} x^{2}-1}}\, \left (a^{2} x^{2}-1\right ) \sqrt {c}\, \left (a m x +a x -m -2\right ) x \,x^{m}}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2+m \right ) \left (1+m \right )}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 63, normalized size = 0.76 \begin {gather*} -\frac {{\left (a \sqrt {c} {\left (m + 1\right )} x^{2} - \sqrt {c} {\left (m + 2\right )} x\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} x^{m}}{{\left (m^{2} + 3 \, m + 2\right )} a^{2} x^{2} - m^{2} - 3 \, m - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 80, normalized size = 0.96 \begin {gather*} \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} \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 {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, 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 [B]
time = 1.23, size = 52, normalized size = 0.63 \begin {gather*} \frac {x\,x^m\,\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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