Optimal. Leaf size=55 \[ -\frac {2}{3} \sqrt {2 x+3}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {34}{3} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {824, 826, 1166, 207} \[ -\frac {2}{3} \sqrt {2 x+3}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {34}{3} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 824
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) \sqrt {3+2 x}}{2+5 x+3 x^2} \, dx &=-\frac {2}{3} \sqrt {3+2 x}+\frac {1}{3} \int \frac {49+31 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {2}{3} \sqrt {3+2 x}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {5+31 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {2}{3} \sqrt {3+2 x}-36 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {170}{3} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {2}{3} \sqrt {3+2 x}+12 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {34}{3} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 1.00 \[ -\frac {2}{3} \sqrt {2 x+3}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {34}{3} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 71, normalized size = 1.29 \[ \frac {17}{9} \, \sqrt {5} \sqrt {3} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) - \frac {2}{3} \, \sqrt {2 \, x + 3} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 74, normalized size = 1.35 \[ \frac {17}{9} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {2}{3} \, \sqrt {2 \, x + 3} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 53, normalized size = 0.96 \[ -\frac {34 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{9}-6 \ln \left (-1+\sqrt {2 x +3}\right )+6 \ln \left (\sqrt {2 x +3}+1\right )-\frac {2 \sqrt {2 x +3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 70, normalized size = 1.27 \[ \frac {17}{9} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {2}{3} \, \sqrt {2 \, x + 3} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 38, normalized size = 0.69 \[ 12\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {34\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{9}-\frac {2\,\sqrt {2\,x+3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.20, size = 102, normalized size = 1.85 \[ - \frac {2 \sqrt {2 x + 3}}{3} + \frac {170 \left (\begin {cases} - \frac {\sqrt {15} \operatorname {acoth}{\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} \right )}}{15} & \text {for}\: 2 x + 3 > \frac {5}{3} \\- \frac {\sqrt {15} \operatorname {atanh}{\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} \right )}}{15} & \text {for}\: 2 x + 3 < \frac {5}{3} \end {cases}\right )}{3} - 6 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 6 \log {\left (\sqrt {2 x + 3} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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