Optimal. Leaf size=116 \[ \frac{2}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}+\frac{1}{378} (102 x+109) \left (3 x^2-x+2\right )^{5/2}-\frac{71 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{2592}-\frac{1633 (1-6 x) \sqrt{3 x^2-x+2}}{20736}-\frac{37559 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{41472 \sqrt{3}} \]
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Rubi [A] time = 0.0822707, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1653, 779, 612, 619, 215} \[ \frac{2}{21} (2 x+1)^2 \left (3 x^2-x+2\right )^{5/2}+\frac{1}{378} (102 x+109) \left (3 x^2-x+2\right )^{5/2}-\frac{71 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{2592}-\frac{1633 (1-6 x) \sqrt{3 x^2-x+2}}{20736}-\frac{37559 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{41472 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 779
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int (1+2 x) \left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{84} \int (1+2 x) (40+204 x) \left (2-x+3 x^2\right )^{3/2} \, dx\\ &=\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{378} (109+102 x) \left (2-x+3 x^2\right )^{5/2}+\frac{71}{108} \int \left (2-x+3 x^2\right )^{3/2} \, dx\\ &=-\frac{71 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{2592}+\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{378} (109+102 x) \left (2-x+3 x^2\right )^{5/2}+\frac{1633 \int \sqrt{2-x+3 x^2} \, dx}{1728}\\ &=-\frac{1633 (1-6 x) \sqrt{2-x+3 x^2}}{20736}-\frac{71 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{2592}+\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{378} (109+102 x) \left (2-x+3 x^2\right )^{5/2}+\frac{37559 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{41472}\\ &=-\frac{1633 (1-6 x) \sqrt{2-x+3 x^2}}{20736}-\frac{71 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{2592}+\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{378} (109+102 x) \left (2-x+3 x^2\right )^{5/2}+\frac{\left (1633 \sqrt{\frac{23}{3}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{41472}\\ &=-\frac{1633 (1-6 x) \sqrt{2-x+3 x^2}}{20736}-\frac{71 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{2592}+\frac{2}{21} (1+2 x)^2 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{378} (109+102 x) \left (2-x+3 x^2\right )^{5/2}-\frac{37559 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{41472 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.035977, size = 70, normalized size = 0.6 \[ \frac{6 \sqrt{3 x^2-x+2} \left (497664 x^6+518400 x^5+653184 x^4+744336 x^3+531384 x^2+275410 x+203337\right )+262913 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{870912} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 100, normalized size = 0.9 \begin{align*}{\frac{8\,{x}^{2}}{21} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{41\,x}{63} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{145}{378} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{-71+426\,x}{2592} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{-1633+9798\,x}{20736}\sqrt{3\,{x}^{2}-x+2}}+{\frac{37559\,\sqrt{3}}{124416}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50264, size = 163, normalized size = 1.41 \begin{align*} \frac{8}{21} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{41}{63} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x + \frac{145}{378} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} + \frac{71}{432} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x - \frac{71}{2592} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} + \frac{1633}{3456} \, \sqrt{3 \, x^{2} - x + 2} x + \frac{37559}{124416} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{1633}{20736} \, \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 = 1.56758, size = 277, normalized size = 2.39 \begin{align*} \frac{1}{145152} \,{\left (497664 \, x^{6} + 518400 \, x^{5} + 653184 \, x^{4} + 744336 \, x^{3} + 531384 \, x^{2} + 275410 \, x + 203337\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{37559}{248832} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\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 (2 x + 1\right ) \left (3 x^{2} - x + 2\right )^{\frac{3}{2}} \left (4 x^{2} + 3 x + 1\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17022, size = 105, normalized size = 0.91 \begin{align*} \frac{1}{145152} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (24 \,{\left (2 \,{\left (24 \, x + 25\right )} x + 63\right )} x + 1723\right )} x + 22141\right )} x + 137705\right )} x + 203337\right )} \sqrt{3 \, x^{2} - x + 2} - \frac{37559}{124416} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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