Optimal. Leaf size=36 \[ \frac {\sqrt {x^2+2 x-3}}{2 (1-x)}+\sqrt {x^2+2 x-3} \]
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Rubi [A] time = 0.13, 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 = {1593, 1586, 1638, 650} \[ \frac {\sqrt {x^2+2 x-3}}{2 (1-x)}+\sqrt {x^2+2 x-3} \]
Antiderivative was successfully verified.
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Rule 650
Rule 1586
Rule 1593
Rule 1638
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 = 0.01, size = 26, normalized size = 0.72 \[ \frac {2 x^2+3 x-9}{2 \sqrt {x^2+2 x-3}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 26, normalized size = 0.72 \[ \frac {(2 x-3) \sqrt {x^2+2 x-3}}{2 (x-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 22, normalized size = 0.61 \[ \frac {\sqrt {x^{2} + 2 \, x - 3} {\left (2 \, x - 3\right )}}{2 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 30, normalized size = 0.83 \[ \sqrt {x^{2} + 2 \, x - 3} + \frac {2}{x - \sqrt {x^{2} + 2 \, x - 3} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 21, normalized size = 0.58
method | result | size |
gosper | \(\frac {\left (-3+2 x \right ) \left (3+x \right )}{2 \sqrt {x^{2}+2 x -3}}\) | \(21\) |
trager | \(\frac {\left (-3+2 x \right ) \sqrt {x^{2}+2 x -3}}{2 x -2}\) | \(23\) |
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 = 1.42, size = 28, normalized size = 0.78 \[ \sqrt {x^{2} + 2 \, x - 3} - \frac {\sqrt {x^{2} + 2 \, x - 3}}{2 \, {\left (x - 1\right )}} \]
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 \[ \frac {\left (x-\frac {3}{2}\right )\,\sqrt {x^2+2\,x-3}}{x-1} \]
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 \[ \int \frac {x^{2}}{\sqrt {\left (x - 1\right ) \left (x + 3\right )} \left (x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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