Optimal. Leaf size=76 \[ \frac {x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.26, antiderivative size = 76, 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 = {6197, 6193, 43} \[ \frac {x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx &=\frac {\sqrt {c-\frac {c}{a^2 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}}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int x (-1+a x) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \left (-x+a x^2\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^2}{2 a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3}{3 \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 0.59 \[ \frac {x^2 (2 a x-3) \sqrt {c-\frac {c}{a^2 x^2}}}{6 a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 24, normalized size = 0.32 \[ \frac {{\left (2 \, a x^{3} - 3 \, x^{2}\right )} \sqrt {a^{2} c}}{6 \, a^{2}} \]
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 \[ \int \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 53, normalized size = 0.70 \[ \frac {x^{3} \left (2 a x -3\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {\frac {a x -1}{a x +1}}}{6 a x -6} \]
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 \[ \int \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 46, normalized size = 0.61 \[ \frac {x^3\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (2\,a\,x-3\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,\left (a\,x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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