Optimal. Leaf size=158 \[ \frac{2}{27} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^4+\frac{77}{81} x^3 \left (3 x^2-x+2\right )^{5/2}+\frac{913}{486} x^2 \left (3 x^2-x+2\right )^{5/2}-\frac{11 (283-5850 x) \left (3 x^2-x+2\right )^{5/2}}{58320}+\frac{54593 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{559872}+\frac{1255639 (1-6 x) \sqrt{3 x^2-x+2}}{4478976}+\frac{28879697 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}} \]
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Rubi [A] time = 0.200635, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1653, 12, 779, 612, 619, 215} \[ \frac{2}{27} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^4+\frac{77}{81} x^3 \left (3 x^2-x+2\right )^{5/2}+\frac{913}{486} x^2 \left (3 x^2-x+2\right )^{5/2}-\frac{11 (283-5850 x) \left (3 x^2-x+2\right )^{5/2}}{58320}+\frac{54593 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{559872}+\frac{1255639 (1-6 x) \sqrt{3 x^2-x+2}}{4478976}+\frac{28879697 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 12
Rule 779
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}+\frac{1}{108} \int 308 x (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2} \, dx\\ &=\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{27} \int x (1+2 x)^3 \left (2-x+3 x^2\right )^{3/2} \, dx\\ &=\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{648} \int x \left (2-x+3 x^2\right )^{3/2} \left (24+96 x+332 x^2\right ) \, dx\\ &=\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}+\frac{11 \int x (-824+3510 x) \left (2-x+3 x^2\right )^{3/2} \, dx}{1944}\\ &=-\frac{11 (283-5850 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}-\frac{54593 \int \left (2-x+3 x^2\right )^{3/2} \, dx}{23328}\\ &=\frac{54593 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{11 (283-5850 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}-\frac{1255639 \int \sqrt{2-x+3 x^2} \, dx}{373248}\\ &=\frac{1255639 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}+\frac{54593 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{11 (283-5850 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}-\frac{28879697 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{8957952}\\ &=\frac{1255639 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}+\frac{54593 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{11 (283-5850 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}-\frac{\left (1255639 \sqrt{\frac{23}{3}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{8957952}\\ &=\frac{1255639 (1-6 x) \sqrt{2-x+3 x^2}}{4478976}+\frac{54593 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{559872}-\frac{11 (283-5850 x) \left (2-x+3 x^2\right )^{5/2}}{58320}+\frac{913}{486} x^2 \left (2-x+3 x^2\right )^{5/2}+\frac{77}{81} x^3 \left (2-x+3 x^2\right )^{5/2}+\frac{2}{27} (1+2 x)^4 \left (2-x+3 x^2\right )^{5/2}+\frac{28879697 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{8957952 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0498844, size = 80, normalized size = 0.51 \[ \frac{6 \sqrt{3 x^2-x+2} \left (238878720 x^8+510105600 x^7+635765760 x^6+711210240 x^5+649452672 x^4+421626672 x^3+201289704 x^2+84014278 x+12499587\right )-144398485 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{134369280} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 134, normalized size = 0.9 \begin{align*}{\frac{32\,{x}^{4}}{27} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{269\,{x}^{3}}{81} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1777\,{x}^{2}}{486} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1099\,x}{648} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{-1255639+7533834\,x}{4478976}\sqrt{3\,{x}^{2}-x+2}}-{\frac{28879697\,\sqrt{3}}{26873856}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }-{\frac{-54593+327558\,x}{559872} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{1207}{58320} \left ( 3\,{x}^{2}-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 = 1.54024, size = 209, normalized size = 1.32 \begin{align*} \frac{32}{27} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x^{4} + \frac{269}{81} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x^{3} + \frac{1777}{486} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x^{2} + \frac{1099}{648} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} x + \frac{1207}{58320} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{5}{2}} - \frac{54593}{93312} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x + \frac{54593}{559872} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} - \frac{1255639}{746496} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{28879697}{26873856} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) + \frac{1255639}{4478976} \, \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 = 1.40745, size = 354, normalized size = 2.24 \begin{align*} \frac{1}{22394880} \,{\left (238878720 \, x^{8} + 510105600 \, x^{7} + 635765760 \, x^{6} + 711210240 \, x^{5} + 649452672 \, x^{4} + 421626672 \, x^{3} + 201289704 \, x^{2} + 84014278 \, x + 12499587\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{28879697}{53747712} \, \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 )^{3} \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 = 1.17993, size = 119, normalized size = 0.75 \begin{align*} \frac{1}{22394880} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (30 \,{\left (36 \,{\left (2 \,{\left (96 \, x + 205\right )} x + 511\right )} x + 20579\right )} x + 563761\right )} x + 2927963\right )} x + 8387071\right )} x + 42007139\right )} x + 12499587\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{28879697}{26873856} \, \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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