Optimal. Leaf size=85 \[ -\frac {3 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )}-\frac {506}{25 \sqrt {2 x+3}}-34 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {1356}{25} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, 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 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )}-\frac {506}{25 \sqrt {2 x+3}}-34 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {1356}{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)^{3/2} \left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {3 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}-\frac {1}{5} \int \frac {508+423 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {506}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}-\frac {1}{25} \int \frac {1184+759 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {506}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}-\frac {2}{25} \operatorname {Subst}\left (\int \frac {91+759 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {506}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+102 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-\frac {4068}{25} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {506}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}-34 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {1356}{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 = 0.08, size = 85, normalized size = 1.00 \begin {gather*} -\frac {3 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )}-\frac {506}{25 \sqrt {2 x+3}}-34 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {1356}{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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IntegrateAlgebraic [A] time = 0.25, size = 93, normalized size = 1.09 \begin {gather*} -\frac {2 \left (759 (2 x+3)^2-1319 (2 x+3)+260\right )}{25 \sqrt {2 x+3} \left (3 (2 x+3)^2-8 (2 x+3)+5\right )}-34 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {1356}{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 = 0.41, size = 144, normalized size = 1.69 \begin {gather*} \frac {678 \, \sqrt {5} \sqrt {3} {\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 2125 \, {\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 2125 \, {\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 5 \, {\left (1518 \, x^{2} + 3235 \, x + 1567\right )} \sqrt {2 \, x + 3}}{125 \, {\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 111, normalized size = 1.31 \begin {gather*} -\frac {678}{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 {2 \, {\left (759 \, {\left (2 \, x + 3\right )}^{2} - 2638 \, x - 3697\right )}}{25 \, {\left (3 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - 8 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + 5 \, \sqrt {2 \, x + 3}\right )}} - 17 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 17 \, \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 = 95, normalized size = 1.12 \begin {gather*} \frac {1356 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{125}+17 \ln \left (-1+\sqrt {2 x +3}\right )-17 \ln \left (\sqrt {2 x +3}+1\right )-\frac {102 \sqrt {2 x +3}}{25 \left (2 x +\frac {4}{3}\right )}-\frac {6}{\sqrt {2 x +3}+1}-\frac {104}{25 \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 = 1.23, size = 107, normalized size = 1.26 \begin {gather*} -\frac {678}{125} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {2 \, {\left (759 \, {\left (2 \, x + 3\right )}^{2} - 2638 \, x - 3697\right )}}{25 \, {\left (3 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - 8 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + 5 \, \sqrt {2 \, x + 3}\right )}} - 17 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 17 \, \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 = 72, normalized size = 0.85 \begin {gather*} \frac {1356\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{125}-34\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )+\frac {\frac {5276\,x}{75}-\frac {506\,{\left (2\,x+3\right )}^2}{25}+\frac {7394}{75}}{\frac {5\,\sqrt {2\,x+3}}{3}-\frac {8\,{\left (2\,x+3\right )}^{3/2}}{3}+{\left (2\,x+3\right )}^{5/2}} \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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