Optimal. Leaf size=74 \[ \frac {a x^4 \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-a^2 x^2}}+\frac {x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.21, 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 = {6160, 6150, 43} \[ \frac {a x^4 \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-a^2 x^2}}+\frac {x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^2 \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int e^{\tanh ^{-1}(a x)} x \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int x (1+a x) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \left (x+a x^2\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3}{2 \sqrt {1-a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x^4}{3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.57 \[ \frac {x^3 (2 a x+3) \sqrt {c-\frac {c}{a^2 x^2}}}{6 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 58, normalized size = 0.78 \[ -\frac {{\left (2 \, a x^{4} + 3 \, x^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left (a^{2} x^{2} - 1\right )}} \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{2}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 43, normalized size = 0.58 \[ \frac {x^{3} \left (2 a x +3\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{6 \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.39, size = 20, normalized size = 0.27 \[ -\frac {1}{3} i \, \sqrt {c} x^{3} - \frac {i \, \sqrt {c} x^{2}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 38, normalized size = 0.51 \[ \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (\frac {a\,x^4}{3}+\frac {x^3}{2}\right )}{\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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