Optimal. Leaf size=72 \[ -\frac {\sqrt {2 x+3} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}-106 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {248}{3} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 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 = {818, 826, 1166, 207} \begin {gather*} -\frac {\sqrt {2 x+3} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}-106 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {248}{3} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 818
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {\sqrt {3+2 x} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {1}{3} \int \frac {-302-143 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {\sqrt {3+2 x} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {-175-143 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {\sqrt {3+2 x} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+318 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-\frac {1240}{3} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {\sqrt {3+2 x} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}-106 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {248}{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 = 0.05, size = 70, normalized size = 0.97 \begin {gather*} \frac {1}{9} \left (-\frac {3 \sqrt {2 x+3} (139 x+121)}{3 x^2+5 x+2}-954 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+248 \sqrt {15} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 89, normalized size = 1.24 \begin {gather*} -\frac {2 \left (139 (2 x+3)^{3/2}-175 \sqrt {2 x+3}\right )}{3 \left (3 (2 x+3)^2-8 (2 x+3)+5\right )}-106 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {248}{3} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 119, normalized size = 1.65 \begin {gather*} \frac {124 \, \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 ) - 477 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 477 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 3 \, {\left (139 \, x + 121\right )} \sqrt {2 \, x + 3}}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 102, normalized size = 1.42 \begin {gather*} -\frac {124}{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 \, {\left (139 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 175 \, \sqrt {2 \, x + 3}\right )}}{3 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 53 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 53 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \end {gather*}
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 \begin {gather*} \frac {248 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{9}+53 \ln \left (-1+\sqrt {2 x +3}\right )-53 \ln \left (\sqrt {2 x +3}+1\right )-\frac {170 \sqrt {2 x +3}}{9 \left (2 x +\frac {4}{3}\right )}-\frac {6}{\sqrt {2 x +3}+1}-\frac {6}{-1+\sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 98, normalized size = 1.36 \begin {gather*} -\frac {124}{9} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {2 \, {\left (139 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 175 \, \sqrt {2 \, x + 3}\right )}}{3 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 53 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 53 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.41, size = 66, normalized size = 0.92 \begin {gather*} \frac {248\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{9}-\frac {\frac {350\,\sqrt {2\,x+3}}{9}-\frac {278\,{\left (2\,x+3\right )}^{3/2}}{9}}{\frac {16\,x}{3}-{\left (2\,x+3\right )}^2+\frac {19}{3}}-106\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right ) \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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