Optimal. Leaf size=137 \[ -\frac{1}{15} \sqrt{3 x^2+5 x+2} (2 x+3)^4+\frac{53}{60} \sqrt{3 x^2+5 x+2} (2 x+3)^3+\frac{391}{135} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{1}{648} (9650 x+27519) \sqrt{3 x^2+5 x+2}+\frac{28051 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1296 \sqrt{3}} \]
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Rubi [A] time = 0.0856792, antiderivative size = 137, 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 = {832, 779, 621, 206} \[ -\frac{1}{15} \sqrt{3 x^2+5 x+2} (2 x+3)^4+\frac{53}{60} \sqrt{3 x^2+5 x+2} (2 x+3)^3+\frac{391}{135} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{1}{648} (9650 x+27519) \sqrt{3 x^2+5 x+2}+\frac{28051 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1296 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)^4}{\sqrt{2+5 x+3 x^2}} \, dx &=-\frac{1}{15} (3+2 x)^4 \sqrt{2+5 x+3 x^2}+\frac{1}{15} \int \frac{(3+2 x)^3 \left (\frac{497}{2}+159 x\right )}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{53}{60} (3+2 x)^3 \sqrt{2+5 x+3 x^2}-\frac{1}{15} (3+2 x)^4 \sqrt{2+5 x+3 x^2}+\frac{1}{180} \int \frac{(3+2 x)^2 \left (\frac{11691}{2}+4692 x\right )}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{391}{135} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{53}{60} (3+2 x)^3 \sqrt{2+5 x+3 x^2}-\frac{1}{15} (3+2 x)^4 \sqrt{2+5 x+3 x^2}+\frac{\int \frac{(3+2 x) \left (\frac{170205}{2}+72375 x\right )}{\sqrt{2+5 x+3 x^2}} \, dx}{1620}\\ &=\frac{391}{135} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{53}{60} (3+2 x)^3 \sqrt{2+5 x+3 x^2}-\frac{1}{15} (3+2 x)^4 \sqrt{2+5 x+3 x^2}+\frac{1}{648} (27519+9650 x) \sqrt{2+5 x+3 x^2}+\frac{28051 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{1296}\\ &=\frac{391}{135} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{53}{60} (3+2 x)^3 \sqrt{2+5 x+3 x^2}-\frac{1}{15} (3+2 x)^4 \sqrt{2+5 x+3 x^2}+\frac{1}{648} (27519+9650 x) \sqrt{2+5 x+3 x^2}+\frac{28051}{648} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{391}{135} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{53}{60} (3+2 x)^3 \sqrt{2+5 x+3 x^2}-\frac{1}{15} (3+2 x)^4 \sqrt{2+5 x+3 x^2}+\frac{1}{648} (27519+9650 x) \sqrt{2+5 x+3 x^2}+\frac{28051 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{1296 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0597789, size = 72, normalized size = 0.53 \[ \frac{140255 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )-6 \sqrt{3 x^2+5 x+2} \left (3456 x^4-2160 x^3-93912 x^2-268750 x-281829\right )}{19440} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 111, normalized size = 0.8 \begin{align*} -{\frac{16\,{x}^{4}}{15}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{2\,{x}^{3}}{3}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{3913\,{x}^{2}}{135}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{26875\,x}{324}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{93943}{1080}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{28051\,\sqrt{3}}{3888}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.07593, size = 147, normalized size = 1.07 \begin{align*} -\frac{16}{15} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{4} + \frac{2}{3} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{3} + \frac{3913}{135} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{2} + \frac{26875}{324} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{28051}{3888} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{93943}{1080} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12612, size = 236, normalized size = 1.72 \begin{align*} -\frac{1}{3240} \,{\left (3456 \, x^{4} - 2160 \, x^{3} - 93912 \, x^{2} - 268750 \, x - 281829\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{28051}{7776} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{999 x}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{864 x^{2}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{264 x^{3}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{16 x^{4}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{16 x^{5}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{405}{\sqrt{3 x^{2} + 5 x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16992, size = 93, normalized size = 0.68 \begin{align*} -\frac{1}{3240} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \, x - 5\right )} x - 3913\right )} x - 134375\right )} x - 281829\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{28051}{3888} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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