Optimal. Leaf size=103 \[ \frac{32}{27} \sqrt{3 x^2-x+2} x^2+\frac{412}{81} \sqrt{3 x^2-x+2} x+\frac{746}{81} \sqrt{3 x^2-x+2}+\frac{2 (12839-3871 x)}{1863 \sqrt{3 x^2-x+2}}+\frac{353 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{81 \sqrt{3}} \]
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Rubi [A] time = 0.12421, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1660, 1661, 640, 619, 215} \[ \frac{32}{27} \sqrt{3 x^2-x+2} x^2+\frac{412}{81} \sqrt{3 x^2-x+2} x+\frac{746}{81} \sqrt{3 x^2-x+2}+\frac{2 (12839-3871 x)}{1863 \sqrt{3 x^2-x+2}}+\frac{353 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{81 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 1661
Rule 640
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{2}{23} \int \frac{\frac{1127}{81}+\frac{7682 x}{27}+\frac{2852 x^2}{9}+\frac{368 x^3}{3}}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{32}{27} x^2 \sqrt{2-x+3 x^2}+\frac{2}{207} \int \frac{\frac{1127}{9}+2070 x+\frac{9476 x^2}{3}}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{412}{81} x \sqrt{2-x+3 x^2}+\frac{32}{27} x^2 \sqrt{2-x+3 x^2}+\frac{1}{621} \int \frac{-5566+17158 x}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{746}{81} \sqrt{2-x+3 x^2}+\frac{412}{81} x \sqrt{2-x+3 x^2}+\frac{32}{27} x^2 \sqrt{2-x+3 x^2}-\frac{353}{81} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{746}{81} \sqrt{2-x+3 x^2}+\frac{412}{81} x \sqrt{2-x+3 x^2}+\frac{32}{27} x^2 \sqrt{2-x+3 x^2}-\frac{353 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{81 \sqrt{69}}\\ &=\frac{2 (12839-3871 x)}{1863 \sqrt{2-x+3 x^2}}+\frac{746}{81} \sqrt{2-x+3 x^2}+\frac{412}{81} x \sqrt{2-x+3 x^2}+\frac{32}{27} x^2 \sqrt{2-x+3 x^2}+\frac{353 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{81 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0405456, size = 69, normalized size = 0.67 \[ \frac{6 \left (3312 x^4+13110 x^3+23207 x^2-2974 x+29997\right )-8119 \sqrt{9 x^2-3 x+6} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{5589 \sqrt{3 x^2-x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 115, normalized size = 1.1 \begin{align*}{\frac{32\,{x}^{4}}{9}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{380\,{x}^{3}}{27}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{2018\,{x}^{2}}{81}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{353\,\sqrt{3}}{243}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }+{\frac{353\,x}{81}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{-521+3126\,x}{414}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{557}{18}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5473, size = 131, normalized size = 1.27 \begin{align*} \frac{32 \, x^{4}}{9 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{380 \, x^{3}}{27 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{2018 \, x^{2}}{81 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{353}{243} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{5948 \, x}{1863 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{2222}{69 \, \sqrt{3 \, x^{2} - x + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.09804, size = 269, normalized size = 2.61 \begin{align*} \frac{8119 \, \sqrt{3}{\left (3 \, x^{2} - x + 2\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 12 \,{\left (3312 \, x^{4} + 13110 \, x^{3} + 23207 \, x^{2} - 2974 \, x + 29997\right )} \sqrt{3 \, x^{2} - x + 2}}{11178 \,{\left (3 \, x^{2} - x + 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 \frac{\left (2 x + 1\right )^{3} \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} - x + 2\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19926, size = 90, normalized size = 0.87 \begin{align*} \frac{353}{243} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) + \frac{2 \,{\left ({\left (23 \,{\left (6 \,{\left (24 \, x + 95\right )} x + 1009\right )} x - 2974\right )} x + 29997\right )}}{1863 \, \sqrt{3 \, x^{2} - x + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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