Optimal. Leaf size=115 \[ -\frac{3 (47 x+37)}{10 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^2}+\frac{2229 x+1888}{10 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )}+\frac{2667}{25 \sqrt{2 x+3}}+402 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{12717}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
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Rubi [A] time = 0.084689, antiderivative size = 115, normalized size of antiderivative = 1., 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} \[ -\frac{3 (47 x+37)}{10 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^2}+\frac{2229 x+1888}{10 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )}+\frac{2667}{25 \sqrt{2 x+3}}+402 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{12717}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 822
Rule 828
Rule 826
Rule 1166
Rule 207
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac{3 (37+47 x)}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2}-\frac{1}{10} \int \frac{1328+987 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{3 (37+47 x)}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac{1888+2229 x}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )}+\frac{1}{50} \int \frac{43485+33435 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac{2667}{25 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac{1888+2229 x}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )}+\frac{1}{250} \int \frac{90255+40005 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac{2667}{25 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac{1888+2229 x}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )}+\frac{1}{125} \operatorname{Subst}\left (\int \frac{60495+40005 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=\frac{2667}{25 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac{1888+2229 x}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )}-1206 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )+\frac{38151}{25} \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=\frac{2667}{25 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac{1888+2229 x}{10 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )}+402 \tanh ^{-1}\left (\sqrt{3+2 x}\right )-\frac{12717}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.146829, size = 86, normalized size = 0.75 \[ \frac{1}{50} \left (\frac{48006 x^4+193455 x^3+281403 x^2+175465 x+39661}{\sqrt{2 x+3} \left (3 x^2+5 x+2\right )^2}+20100 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-25434 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 133, normalized size = 1.2 \begin{align*} -{\frac{416}{125}{\frac{1}{\sqrt{3+2\,x}}}}+{\frac{486}{125\, \left ( 6\,x+4 \right ) ^{2}} \left ({\frac{213}{2} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3365}{18}\sqrt{3+2\,x}} \right ) }-{\frac{12717\,\sqrt{15}}{125}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }+32\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+201\,\ln \left ( 1+\sqrt{3+2\,x} \right ) +32\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}-201\,\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 = 1.71313, size = 193, normalized size = 1.68 \begin{align*} \frac{12717}{250} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{24003 \,{\left (2 \, x + 3\right )}^{4} - 94581 \,{\left (2 \, x + 3\right )}^{3} + 117873 \,{\left (2 \, x + 3\right )}^{2} - 88030 \, x - 134125}{25 \,{\left (9 \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - 48 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + 94 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - 80 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + 25 \, \sqrt{2 \, x + 3}\right )}} + 201 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 201 \, \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 = 1.91508, size = 575, normalized size = 5. \begin{align*} \frac{12717 \, \sqrt{5} \sqrt{3}{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (-\frac{\sqrt{5} \sqrt{3} \sqrt{2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 50250 \,{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 50250 \,{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 5 \,{\left (48006 \, x^{4} + 193455 \, x^{3} + 281403 \, x^{2} + 175465 \, x + 39661\right )} \sqrt{2 \, x + 3}}{250 \,{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\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 = 1.08534, size = 174, normalized size = 1.51 \begin{align*} \frac{12717}{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{416}{125 \, \sqrt{2 \, x + 3}} + \frac{123759 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 492873 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 628469 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 253355 \, \sqrt{2 \, x + 3}}{125 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 201 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 201 \, \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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