Optimal. Leaf size=33 \[ \frac {x^2}{2}+\sqrt {x+2} \sqrt {x+3}-5 \sinh ^{-1}\left (\sqrt {x+2}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {14, 80, 54, 215} \[ \frac {x^2}{2}+\sqrt {x+2} \sqrt {x+3}-5 \sinh ^{-1}\left (\sqrt {x+2}\right ) \]
Antiderivative was successfully verified.
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Rule 14
Rule 54
Rule 80
Rule 215
Rubi steps
\begin {align*} \int x \left (1+\frac {1}{\sqrt {2+x} \sqrt {3+x}}\right ) \, dx &=\int \left (x+\frac {x}{\sqrt {2+x} \sqrt {3+x}}\right ) \, dx\\ &=\frac {x^2}{2}+\int \frac {x}{\sqrt {2+x} \sqrt {3+x}} \, dx\\ &=\frac {x^2}{2}+\sqrt {2+x} \sqrt {3+x}-\frac {5}{2} \int \frac {1}{\sqrt {2+x} \sqrt {3+x}} \, dx\\ &=\frac {x^2}{2}+\sqrt {2+x} \sqrt {3+x}-5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {2+x}\right )\\ &=\frac {x^2}{2}+\sqrt {2+x} \sqrt {3+x}-5 \sinh ^{-1}\left (\sqrt {2+x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ \frac {x^2}{2}+\sqrt {x+2} \sqrt {x+3}-5 \sinh ^{-1}\left (\sqrt {x+2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 37, normalized size = 1.12 \[ \frac {1}{2} \, x^{2} + \sqrt {x + 3} \sqrt {x + 2} + \frac {5}{2} \, \log \left (2 \, \sqrt {x + 3} \sqrt {x + 2} - 2 \, x - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 39, normalized size = 1.18 \[ \frac {1}{2} \, {\left (x + 3\right )}^{2} + \sqrt {x + 3} \sqrt {x + 2} - 3 \, x + 5 \, \log \left (\sqrt {x + 3} - \sqrt {x + 2}\right ) - 9 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 58, normalized size = 1.76 \[ \frac {x^{2}}{2}-\frac {\sqrt {x +2}\, \sqrt {x +3}\, \left (5 \ln \left (x +\frac {5}{2}+\sqrt {x^{2}+5 x +6}\right )-2 \sqrt {x^{2}+5 x +6}\right )}{2 \sqrt {x^{2}+5 x +6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 36, normalized size = 1.09 \[ \frac {1}{2} \, x^{2} + \sqrt {x^{2} + 5 \, x + 6} - \frac {5}{2} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} + 5 \, x + 6} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.56, size = 180, normalized size = 5.45 \[ \frac {\frac {10\,\left (\sqrt {x+2}-\sqrt {2}\right )}{\sqrt {x+3}-\sqrt {3}}+\frac {10\,{\left (\sqrt {x+2}-\sqrt {2}\right )}^3}{{\left (\sqrt {x+3}-\sqrt {3}\right )}^3}-\frac {8\,\sqrt {6}\,{\left (\sqrt {x+2}-\sqrt {2}\right )}^2}{{\left (\sqrt {x+3}-\sqrt {3}\right )}^2}}{\frac {{\left (\sqrt {x+2}-\sqrt {2}\right )}^4}{{\left (\sqrt {x+3}-\sqrt {3}\right )}^4}-\frac {2\,{\left (\sqrt {x+2}-\sqrt {2}\right )}^2}{{\left (\sqrt {x+3}-\sqrt {3}\right )}^2}+1}-10\,\mathrm {atanh}\left (\frac {\sqrt {x+2}-\sqrt {2}}{\sqrt {x+3}-\sqrt {3}}\right )+\frac {x^2}{2} \]
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 \[ \int \frac {x \left (\sqrt {x + 2} \sqrt {x + 3} + 1\right )}{\sqrt {x + 2} \sqrt {x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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