Optimal. Leaf size=102 \[ -\frac{3 \sqrt{2 x+3} (47 x+37)}{10 \left (3 x^2+5 x+2\right )^2}+\frac{\sqrt{2 x+3} (11739 x+9734)}{50 \left (3 x^2+5 x+2\right )}+542 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{17463}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
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Rubi [A] time = 0.063425, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {822, 826, 1166, 207} \[ -\frac{3 \sqrt{2 x+3} (47 x+37)}{10 \left (3 x^2+5 x+2\right )^2}+\frac{\sqrt{2 x+3} (11739 x+9734)}{50 \left (3 x^2+5 x+2\right )}+542 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{17463}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 822
Rule 826
Rule 1166
Rule 207
Rubi steps
\begin{align*} \int \frac{5-x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac{3 \sqrt{3+2 x} (37+47 x)}{10 \left (2+5 x+3 x^2\right )^2}-\frac{1}{10} \int \frac{1106+705 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{3 \sqrt{3+2 x} (37+47 x)}{10 \left (2+5 x+3 x^2\right )^2}+\frac{\sqrt{3+2 x} (9734+11739 x)}{50 \left (2+5 x+3 x^2\right )}+\frac{1}{50} \int \frac{25289+11739 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{3 \sqrt{3+2 x} (37+47 x)}{10 \left (2+5 x+3 x^2\right )^2}+\frac{\sqrt{3+2 x} (9734+11739 x)}{50 \left (2+5 x+3 x^2\right )}+\frac{1}{25} \operatorname{Subst}\left (\int \frac{15361+11739 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)}{10 \left (2+5 x+3 x^2\right )^2}+\frac{\sqrt{3+2 x} (9734+11739 x)}{50 \left (2+5 x+3 x^2\right )}-1626 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )+\frac{52389}{25} \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)}{10 \left (2+5 x+3 x^2\right )^2}+\frac{\sqrt{3+2 x} (9734+11739 x)}{50 \left (2+5 x+3 x^2\right )}+542 \tanh ^{-1}\left (\sqrt{3+2 x}\right )-\frac{17463}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.108466, size = 81, normalized size = 0.79 \[ \frac{1}{50} \left (\frac{\sqrt{2 x+3} \left (35217 x^3+87897 x^2+71443 x+18913\right )}{\left (3 x^2+5 x+2\right )^2}+27100 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-34926 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 124, normalized size = 1.2 \begin{align*} 486\,{\frac{1}{ \left ( 6\,x+4 \right ) ^{2}} \left ({\frac{571\, \left ( 3+2\,x \right ) ^{3/2}}{450}}-{\frac{121\,\sqrt{3+2\,x}}{54}} \right ) }-{\frac{17463\,\sqrt{15}}{125}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+44\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+271\,\ln \left ( 1+\sqrt{3+2\,x} \right ) +3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+44\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-271\,\ln \left ( -1+\sqrt{3+2\,x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50646, size = 181, normalized size = 1.77 \begin{align*} \frac{17463}{250} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{35217 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 141159 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 181867 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 74725 \, \sqrt{2 \, x + 3}}{25 \,{\left (9 \,{\left (2 \, x + 3\right )}^{4} - 48 \,{\left (2 \, x + 3\right )}^{3} + 94 \,{\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 271 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 271 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.91248, size = 485, normalized size = 4.75 \begin{align*} \frac{17463 \, \sqrt{5} \sqrt{3}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-\frac{\sqrt{5} \sqrt{3} \sqrt{2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 67750 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 67750 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 5 \,{\left (35217 \, x^{3} + 87897 \, x^{2} + 71443 \, x + 18913\right )} \sqrt{2 \, x + 3}}{250 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08575, size = 162, normalized size = 1.59 \begin{align*} \frac{17463}{250} \, \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{35217 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 141159 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 181867 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 74725 \, \sqrt{2 \, x + 3}}{25 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 271 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 271 \, \log \left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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