Optimal. Leaf size=80 \[ -\frac{1}{9} \left (3 x^2+5 x+2\right )^{3/2}+\frac{35}{72} (6 x+5) \sqrt{3 x^2+5 x+2}-\frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{144 \sqrt{3}} \]
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Rubi [A] time = 0.0226837, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {640, 612, 621, 206} \[ -\frac{1}{9} \left (3 x^2+5 x+2\right )^{3/2}+\frac{35}{72} (6 x+5) \sqrt{3 x^2+5 x+2}-\frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{144 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (5-x) \sqrt{2+5 x+3 x^2} \, dx &=-\frac{1}{9} \left (2+5 x+3 x^2\right )^{3/2}+\frac{35}{6} \int \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{35}{72} (5+6 x) \sqrt{2+5 x+3 x^2}-\frac{1}{9} \left (2+5 x+3 x^2\right )^{3/2}-\frac{35}{144} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{35}{72} (5+6 x) \sqrt{2+5 x+3 x^2}-\frac{1}{9} \left (2+5 x+3 x^2\right )^{3/2}-\frac{35}{72} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{35}{72} (5+6 x) \sqrt{2+5 x+3 x^2}-\frac{1}{9} \left (2+5 x+3 x^2\right )^{3/2}-\frac{35 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{144 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.051881, size = 62, normalized size = 0.78 \[ \frac{1}{432} \left (-6 \sqrt{3 x^2+5 x+2} \left (24 x^2-170 x-159\right )-35 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 64, normalized size = 0.8 \begin{align*}{\frac{175+210\,x}{72}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{35\,\sqrt{3}}{432}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{1}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49398, size = 97, normalized size = 1.21 \begin{align*} -\frac{1}{9} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{35}{12} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{35}{432} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{175}{72} \, \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 = 1.36908, size = 188, normalized size = 2.35 \begin{align*} -\frac{1}{72} \,{\left (24 \, x^{2} - 170 \, x - 159\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{35}{864} \, \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 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 5 \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 = 1.18067, size = 80, normalized size = 1. \begin{align*} -\frac{1}{72} \,{\left (2 \,{\left (12 \, x - 85\right )} x - 159\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{35}{432} \, \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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