Optimal. Leaf size=54 \[ \sqrt {x+2} \sqrt {x+3}-\sinh ^{-1}\left (\sqrt {x+2}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x+2}}{\sqrt {x+3}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1958, 154, 157, 54, 215, 93, 207} \[ \sqrt {x+2} \sqrt {x+3}-\sinh ^{-1}\left (\sqrt {x+2}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x+2}}{\sqrt {x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 154
Rule 157
Rule 207
Rule 215
Rule 1958
Rubi steps
\begin {align*} \int \frac {x}{(1+x) \sqrt {\frac {2+x}{3+x}}} \, dx &=\int \frac {x \sqrt {3+x}}{(1+x) \sqrt {2+x}} \, dx\\ &=\sqrt {2+x} \sqrt {3+x}+\int \frac {-\frac {5}{2}-\frac {x}{2}}{(1+x) \sqrt {2+x} \sqrt {3+x}} \, dx\\ &=\sqrt {2+x} \sqrt {3+x}-\frac {1}{2} \int \frac {1}{\sqrt {2+x} \sqrt {3+x}} \, dx-2 \int \frac {1}{(1+x) \sqrt {2+x} \sqrt {3+x}} \, dx\\ &=\sqrt {2+x} \sqrt {3+x}-4 \operatorname {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\frac {\sqrt {2+x}}{\sqrt {3+x}}\right )-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {2+x}\right )\\ &=\sqrt {2+x} \sqrt {3+x}-\sinh ^{-1}\left (\sqrt {2+x}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {2+x}}{\sqrt {3+x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 106, normalized size = 1.96 \[ \frac {\sqrt {x+3} \left (x^2+5 x+6\right )+2 \sqrt {2} \sqrt {x+2} \sqrt {-(x+3)^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x+2}}{\sqrt {-x-3}}\right )-\sqrt {x+2} (x+3) \sinh ^{-1}\left (\sqrt {x+2}\right )}{\sqrt {\frac {x+2}{x+3}} (x+3)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 83, normalized size = 1.54 \[ {\left (x + 3\right )} \sqrt {\frac {x + 2}{x + 3}} + \sqrt {2} \log \left (\frac {2 \, \sqrt {2} {\left (x + 3\right )} \sqrt {\frac {x + 2}{x + 3}} + 3 \, x + 7}{x + 1}\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {x + 2}{x + 3}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 2}{x + 3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 107, normalized size = 1.98 \[ -\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sqrt {\frac {x + 2}{x + 3}} \right |}}{2 \, {\left (\sqrt {2} + 2 \, \sqrt {\frac {x + 2}{x + 3}}\right )}}\right ) - \frac {\sqrt {\frac {x + 2}{x + 3}}}{\frac {x + 2}{x + 3} - 1} - \frac {1}{2} \, \log \left (\sqrt {\frac {x + 2}{x + 3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left | \sqrt {\frac {x + 2}{x + 3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 81, normalized size = 1.50 \[ \frac {\left (x +2\right ) \left (2 \sqrt {2}\, \arctanh \left (\frac {\left (3 x +7\right ) \sqrt {2}}{4 \sqrt {x^{2}+5 x +6}}\right )-\ln \left (x +\frac {5}{2}+\sqrt {x^{2}+5 x +6}\right )+2 \sqrt {x^{2}+5 x +6}\right )}{2 \sqrt {\frac {x +2}{x +3}}\, \sqrt {\left (x +3\right ) \left (x +2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.22, size = 103, normalized size = 1.91 \[ -\sqrt {2} \log \left (-\frac {\sqrt {2} - 2 \, \sqrt {\frac {x + 2}{x + 3}}}{\sqrt {2} + 2 \, \sqrt {\frac {x + 2}{x + 3}}}\right ) - \frac {\sqrt {\frac {x + 2}{x + 3}}}{\frac {x + 2}{x + 3} - 1} - \frac {1}{2} \, \log \left (\sqrt {\frac {x + 2}{x + 3}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 2}{x + 3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 62, normalized size = 1.15 \[ 2\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sqrt {\frac {x+2}{x+3}}\right )-\frac {\sqrt {\frac {x+2}{x+3}}}{\frac {x+2}{x+3}-1}-\mathrm {atanh}\left (\sqrt {\frac {x+2}{x+3}}\right ) \]
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 \[ \int \frac {x}{\sqrt {\frac {x + 2}{x + 3}} \left (x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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