Optimal. Leaf size=72 \[ -\frac {3 \sqrt {2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )}-58 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {384}{5} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 72, 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 = {822, 826, 1166, 207} \[ -\frac {3 \sqrt {2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )}-58 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {384}{5} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 822
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {3 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )}-\frac {1}{5} \int \frac {286+141 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {3 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )}-\frac {2}{5} \operatorname {Subst}\left (\int \frac {149+141 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {3 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )}+174 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-\frac {1152}{5} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {3 \sqrt {3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )}-58 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {384}{5} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 72, normalized size = 1.00 \[ -\frac {3 \sqrt {2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )}-58 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {384}{5} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 119, normalized size = 1.65 \[ \frac {192 \, \sqrt {5} \sqrt {3} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 725 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 725 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 15 \, {\left (47 \, x + 37\right )} \sqrt {2 \, x + 3}}{25 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 102, normalized size = 1.42 \[ -\frac {192}{25} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {6 \, {\left (47 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 67 \, \sqrt {2 \, x + 3}\right )}}{5 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 29 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 29 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 86, normalized size = 1.19 \[ \frac {384 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{25}+29 \ln \left (-1+\sqrt {2 x +3}\right )-29 \ln \left (\sqrt {2 x +3}+1\right )-\frac {34 \sqrt {2 x +3}}{5 \left (2 x +\frac {4}{3}\right )}-\frac {6}{\sqrt {2 x +3}+1}-\frac {6}{-1+\sqrt {2 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 98, normalized size = 1.36 \[ -\frac {192}{25} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {6 \, {\left (47 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 67 \, \sqrt {2 \, x + 3}\right )}}{5 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 29 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 29 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 66, normalized size = 0.92 \[ \frac {384\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{25}-\frac {\frac {134\,\sqrt {2\,x+3}}{5}-\frac {94\,{\left (2\,x+3\right )}^{3/2}}{5}}{\frac {16\,x}{3}-{\left (2\,x+3\right )}^2+\frac {19}{3}}-58\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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