Optimal. Leaf size=98 \[ -\frac {3 (47 x+37)}{5 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )}-\frac {686}{25 \sqrt {2 x+3}}-\frac {262}{15 (2 x+3)^{3/2}}-10 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {936}{25} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {822, 828, 826, 1166, 207} \begin {gather*} -\frac {3 (47 x+37)}{5 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )}-\frac {686}{25 \sqrt {2 x+3}}-\frac {262}{15 (2 x+3)^{3/2}}-10 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {936}{25} \sqrt {\frac {3}{5}} \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 822
Rule 826
Rule 828
Rule 1166
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-\frac {1}{5} \int \frac {730+705 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {262}{15 (3+2 x)^{3/2}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-\frac {1}{25} \int \frac {2090+1965 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {262}{15 (3+2 x)^{3/2}}-\frac {686}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-\frac {1}{125} \int \frac {5770+5145 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {262}{15 (3+2 x)^{3/2}}-\frac {686}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-\frac {2}{125} \operatorname {Subst}\left (\int \frac {-3895+5145 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {262}{15 (3+2 x)^{3/2}}-\frac {686}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+30 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-\frac {2808}{25} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {262}{15 (3+2 x)^{3/2}}-\frac {686}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-10 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {936}{25} \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 = 0.11, size = 94, normalized size = 0.96 \begin {gather*} \frac {1}{75} \left (-\frac {45 (47 x+37)}{(2 x+3)^{3/2} \left (3 x^2+5 x+2\right )}-\frac {2058}{\sqrt {2 x+3}}-\frac {1310}{(2 x+3)^{3/2}}-750 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+2808 \sqrt {\frac {3}{5}} \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 = 0.21, size = 102, normalized size = 1.04 \begin {gather*} -\frac {2 \left (3087 (2 x+3)^3-6267 (2 x+3)^2+2020 (2 x+3)+260\right )}{75 (2 x+3)^{3/2} \left (3 (2 x+3)^2-8 (2 x+3)+5\right )}-10 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {936}{25} \sqrt {\frac {3}{5}} \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 = 0.40, size = 169, normalized size = 1.72 \begin {gather*} \frac {1404 \, \sqrt {5} \sqrt {3} {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 1875 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 1875 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 5 \, {\left (12348 \, x^{3} + 43032 \, x^{2} + 47767 \, x + 16633\right )} \sqrt {2 \, x + 3}}{375 \, {\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 116, normalized size = 1.18 \begin {gather*} -\frac {468}{125} \, \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 (903 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 1403 \, \sqrt {2 \, x + 3}\right )}}{125 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - \frac {16 \, {\left (609 \, x + 946\right )}}{375 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} - 5 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 5 \, \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 = 104, normalized size = 1.06 \begin {gather*} \frac {936 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{125}+5 \ln \left (-1+\sqrt {2 x +3}\right )-5 \ln \left (\sqrt {2 x +3}+1\right )-\frac {306 \sqrt {2 x +3}}{125 \left (2 x +\frac {4}{3}\right )}-\frac {6}{\sqrt {2 x +3}+1}-\frac {104}{75 \left (2 x +3\right )^{\frac {3}{2}}}-\frac {1624}{125 \sqrt {2 x +3}}-\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 = 1.17, size = 116, normalized size = 1.18 \begin {gather*} -\frac {468}{125} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {2 \, {\left (3087 \, {\left (2 \, x + 3\right )}^{3} - 6267 \, {\left (2 \, x + 3\right )}^{2} + 4040 \, x + 6320\right )}}{75 \, {\left (3 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 8 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 5 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}\right )}} - 5 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 5 \, \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 = 0.07, size = 82, normalized size = 0.84 \begin {gather*} \frac {936\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{125}-10\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {\frac {1616\,x}{45}-\frac {4178\,{\left (2\,x+3\right )}^2}{75}+\frac {686\,{\left (2\,x+3\right )}^3}{25}+\frac {2528}{45}}{\frac {5\,{\left (2\,x+3\right )}^{3/2}}{3}-\frac {8\,{\left (2\,x+3\right )}^{5/2}}{3}+{\left (2\,x+3\right )}^{7/2}} \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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