Optimal. Leaf size=54 \[ \frac{1}{9} \left (9 x^2+12 x+8\right )^{3/2}-\frac{2}{3} (3 x+2) \sqrt{9 x^2+12 x+8}-\frac{8}{3} \sinh ^{-1}\left (\frac{3 x}{2}+1\right ) \]
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Rubi [A] time = 0.0180984, antiderivative size = 54, 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, 619, 215} \[ \frac{1}{9} \left (9 x^2+12 x+8\right )^{3/2}-\frac{2}{3} (3 x+2) \sqrt{9 x^2+12 x+8}-\frac{8}{3} \sinh ^{-1}\left (\frac{3 x}{2}+1\right ) \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int (-2+3 x) \sqrt{8+12 x+9 x^2} \, dx &=\frac{1}{9} \left (8+12 x+9 x^2\right )^{3/2}-4 \int \sqrt{8+12 x+9 x^2} \, dx\\ &=-\frac{2}{3} (2+3 x) \sqrt{8+12 x+9 x^2}+\frac{1}{9} \left (8+12 x+9 x^2\right )^{3/2}-8 \int \frac{1}{\sqrt{8+12 x+9 x^2}} \, dx\\ &=-\frac{2}{3} (2+3 x) \sqrt{8+12 x+9 x^2}+\frac{1}{9} \left (8+12 x+9 x^2\right )^{3/2}-\frac{2}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{144}}} \, dx,x,12+18 x\right )\\ &=-\frac{2}{3} (2+3 x) \sqrt{8+12 x+9 x^2}+\frac{1}{9} \left (8+12 x+9 x^2\right )^{3/2}-\frac{8}{3} \sinh ^{-1}\left (1+\frac{3 x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0203861, size = 40, normalized size = 0.74 \[ \frac{1}{9} \left (\left (9 x^2-6 x-4\right ) \sqrt{9 x^2+12 x+8}-24 \sinh ^{-1}\left (\frac{3 x}{2}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 43, normalized size = 0.8 \begin{align*} -{\frac{18\,x+12}{9}\sqrt{9\,{x}^{2}+12\,x+8}}-{\frac{8}{3}{\it Arcsinh} \left ( 1+{\frac{3\,x}{2}} \right ) }+{\frac{1}{9} \left ( 9\,{x}^{2}+12\,x+8 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49423, size = 70, normalized size = 1.3 \begin{align*} \frac{1}{9} \,{\left (9 \, x^{2} + 12 \, x + 8\right )}^{\frac{3}{2}} - 2 \, \sqrt{9 \, x^{2} + 12 \, x + 8} x - \frac{4}{3} \, \sqrt{9 \, x^{2} + 12 \, x + 8} - \frac{8}{3} \, \operatorname{arsinh}\left (\frac{3}{2} \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.26475, size = 123, normalized size = 2.28 \begin{align*} \frac{1}{9} \, \sqrt{9 \, x^{2} + 12 \, x + 8}{\left (9 \, x^{2} - 6 \, x - 4\right )} + \frac{8}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} + 12 \, x + 8} - 2\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 \left (3 x - 2\right ) \sqrt{9 x^{2} + 12 x + 8}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0945, size = 61, normalized size = 1.13 \begin{align*} \frac{1}{9} \,{\left (3 \,{\left (3 \, x - 2\right )} x - 4\right )} \sqrt{9 \, x^{2} + 12 \, x + 8} + \frac{8}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} + 12 \, x + 8} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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