Optimal. Leaf size=36 \[ \sqrt {-3+2 x+x^2}+\frac {\sqrt {-3+2 x+x^2}}{2 (1-x)} \]
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Rubi [A]
time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1607, 1600,
1652, 664} \begin {gather*} \frac {\sqrt {x^2+2 x-3}}{2 (1-x)}+\sqrt {x^2+2 x-3} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 1600
Rule 1607
Rule 1652
Rubi steps
\begin {align*} \int \frac {3 x^2+2 x^3}{\sqrt {-3+2 x+x^2} \left (-3+x+2 x^2\right )} \, dx &=\int \frac {x^2 (3+2 x)}{\sqrt {-3+2 x+x^2} \left (-3+x+2 x^2\right )} \, dx\\ &=\int \frac {x^2}{(-1+x) \sqrt {-3+2 x+x^2}} \, dx\\ &=\sqrt {-3+2 x+x^2}+\int \frac {1}{(-1+x) \sqrt {-3+2 x+x^2}} \, dx\\ &=\sqrt {-3+2 x+x^2}+\frac {\sqrt {-3+2 x+x^2}}{2 (1-x)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 26, normalized size = 0.72 \begin {gather*} \frac {(-3+2 x) \sqrt {-3+2 x+x^2}}{2 (-1+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 31, normalized size = 0.86
method | result | size |
gosper | \(\frac {\left (2 x -3\right ) \left (3+x \right )}{2 \sqrt {x^{2}+2 x -3}}\) | \(21\) |
trager | \(\frac {\left (2 x -3\right ) \sqrt {x^{2}+2 x -3}}{2 x -2}\) | \(23\) |
risch | \(\frac {2 x^{2}+3 x -9}{2 \sqrt {x^{2}+2 x -3}}\) | \(23\) |
default | \(\sqrt {x^{2}+2 x -3}-\frac {\sqrt {\left (-1+x \right )^{2}-4+4 x}}{2 \left (-1+x \right )}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.01, size = 28, normalized size = 0.78 \begin {gather*} \sqrt {x^{2} + 2 \, x - 3} - \frac {\sqrt {x^{2} + 2 \, x - 3}}{2 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 22, normalized size = 0.61 \begin {gather*} \frac {\sqrt {x^{2} + 2 \, x - 3} {\left (2 \, x - 3\right )}}{2 \, {\left (x - 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 (x - 1\right ) \left (x + 3\right )} \left (x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.83, size = 30, normalized size = 0.83 \begin {gather*} \sqrt {x^{2} + 2 \, x - 3} + \frac {2}{x - \sqrt {x^{2} + 2 \, x - 3} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 19, normalized size = 0.53 \begin {gather*} \frac {\left (x-\frac {3}{2}\right )\,\sqrt {x^2+2\,x-3}}{x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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