Optimal. Leaf size=74 \[ \frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3}{2 \sqrt {1-a^2 x^2}}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x^4}{3 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 74, 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 = {6295, 6285, 45}
\begin {gather*} \frac {x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 \sqrt {1-a^2 x^2}}-\frac {a x^4 \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
Rule 6295
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 \begin {gather*} -\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^3 (-3+2 a x)}{6 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 54, normalized size = 0.73
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x^{3} \sqrt {-a^{2} x^{2}+1}\, \left (2 a x -3\right )}{6 a^{2} x^{2}-6}\) | \(54\) |
gosper | \(\frac {x^{3} \left (2 a x -3\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {-a^{2} x^{2}+1}}{6 \left (a x -1\right ) \left (a x +1\right )}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.29, size = 43, normalized size = 0.58 \begin {gather*} \frac {{\left (2 i \, a \sqrt {c} x^{3} - 3 i \, \sqrt {c} x^{2}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )}}{6 \, {\left (a^{3} x^{2} - a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 58, normalized size = 0.78 \begin {gather*} \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 )}} \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^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\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 = 0.97, size = 67, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (\frac {x^4\,\sqrt {1-a^2\,x^2}}{3\,a}-\frac {x^3\,\sqrt {1-a^2\,x^2}}{2\,a^2}\right )}{\frac {1}{a^2}-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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