Optimal. Leaf size=74 \[ \frac {x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}-\frac {a x^3 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 74, 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} \[ \frac {x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}-\frac {a x^3 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-a^2 x^2}} \]
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 \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{-\tanh ^{-1}(a x)} x \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int x (1-a x) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (x-a x^2\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}-\frac {a x^3 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.57 \[ -\frac {x^2 (2 a x-3) \sqrt {c-a^2 c x^2}}{6 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 50, normalized size = 0.68 \[ \frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x^{3} - 3 \, x^{2}\right )}}{6 \, {\left (a^{2} x^{2} - 1\right )}} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 51, normalized size = 0.69 \[ \frac {x^{2} \left (2 a x -3\right ) \sqrt {-a^{2} c \,x^{2}+c}\, \sqrt {-a^{2} x^{2}+1}}{6 \left (a x -1\right ) \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 41, normalized size = 0.55 \[ -\frac {{\left (2 \, a \sqrt {c} x^{3} - 3 \, \sqrt {c} x^{2}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )}}{6 \, {\left (a^{2} x^{2} - 1\right )}} \]
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 \[ \int \frac {x\,\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \]
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 \[ \int \frac {x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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