Optimal. Leaf size=74 \[ \frac {x^2 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {x \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.19, 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 = {6192, 6193, 43} \[ \frac {x^2 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {x \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{-\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^2 \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int x (-1+a x) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (-x+a x^2\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=-\frac {x \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^2 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 0.58 \[ \frac {x (2 a x-3) \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 25, normalized size = 0.34 \[ \frac {{\left (2 \, a x^{3} - 3 \, x^{2}\right )} \sqrt {-a^{2} c}}{6 \, a} \]
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 \[ \int \sqrt {-a^{2} c x^{2} + c} x \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 47, normalized size = 0.64 \[ \frac {x^{2} \left (2 a x -3\right ) \sqrt {-a^{2} c \,x^{2}+c}\, \sqrt {\frac {a x -1}{a x +1}}}{6 a x -6} \]
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 \[ \int \sqrt {-a^{2} c x^{2} + c} x \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
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 \[ \int x\,\sqrt {c-a^2\,c\,x^2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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 \[ \int x \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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