Optimal. Leaf size=103 \[ -\frac{1}{15} \left (3 x^2+5 x+2\right )^{5/2}+\frac{35}{144} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac{35 (6 x+5) \sqrt{3 x^2+5 x+2}}{1152}+\frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{2304 \sqrt{3}} \]
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Rubi [A] time = 0.0288204, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {640, 612, 621, 206} \[ -\frac{1}{15} \left (3 x^2+5 x+2\right )^{5/2}+\frac{35}{144} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac{35 (6 x+5) \sqrt{3 x^2+5 x+2}}{1152}+\frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{2304 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (5-x) \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac{1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac{35}{6} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac{35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{15} \left (2+5 x+3 x^2\right )^{5/2}-\frac{35}{96} \int \sqrt{2+5 x+3 x^2} \, dx\\ &=-\frac{35 (5+6 x) \sqrt{2+5 x+3 x^2}}{1152}+\frac{35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac{35 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{2304}\\ &=-\frac{35 (5+6 x) \sqrt{2+5 x+3 x^2}}{1152}+\frac{35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac{35 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{1152}\\ &=-\frac{35 (5+6 x) \sqrt{2+5 x+3 x^2}}{1152}+\frac{35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac{35 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{2304 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0611855, size = 72, normalized size = 0.7 \[ \frac{175 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )-6 \sqrt{3 x^2+5 x+2} \left (3456 x^4-13680 x^3-48792 x^2-43070 x-11589\right )}{34560} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 83, normalized size = 0.8 \begin{align*}{\frac{175+210\,x}{144} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{175+210\,x}{1152}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{35\,\sqrt{3}}{6912}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{1}{15} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49352, size = 136, normalized size = 1.32 \begin{align*} -\frac{1}{15} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{35}{24} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{175}{144} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{35}{192} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{35}{6912} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{175}{1152} \, \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 = 1.34983, size = 232, normalized size = 2.25 \begin{align*} -\frac{1}{5760} \,{\left (3456 \, x^{4} - 13680 \, x^{3} - 48792 \, x^{2} - 43070 \, x - 11589\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{35}{13824} \, \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 - 23 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 10 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 3 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 10 \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 = 1.13778, size = 93, normalized size = 0.9 \begin{align*} -\frac{1}{5760} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (24 \, x - 95\right )} x - 2033\right )} x - 21535\right )} x - 11589\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{35}{6912} \, \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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