Optimal. Leaf size=158 \[ \frac {2}{27} \left (3 x^2-x+2\right )^{5/2} (2 x+1)^4+\frac {913}{486} x^2 \left (3 x^2-x+2\right )^{5/2}-\frac {11 (283-5850 x) \left (3 x^2-x+2\right )^{5/2}}{58320}+\frac {54593 (1-6 x) \left (3 x^2-x+2\right )^{3/2}}{559872}+\frac {1255639 (1-6 x) \sqrt {3 x^2-x+2}}{4478976}+\frac {77}{81} x^3 \left (3 x^2-x+2\right )^{5/2}+\frac {28879697 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{8957952 \sqrt {3}} \]
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Rubi [A] time = 0.20, antiderivative size = 158, normalized size of antiderivative = 1.00, 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 12
Rule 215
Rule 612
Rule 619
Rule 779
Rule 1653
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.91, size = 93, normalized size = 0.59 \[ \frac {1}{22394880} \, {\left (238878720 \, x^{8} + 510105600 \, x^{7} + 635765760 \, x^{6} + 711210240 \, x^{5} + 649452672 \, x^{4} + 421626672 \, x^{3} + 201289704 \, x^{2} + 84014278 \, x + 12499587\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {28879697}{53747712} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 88, normalized size = 0.56 \[ \frac {1}{22394880} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, {\left (36 \, {\left (2 \, {\left (96 \, x + 205\right )} x + 511\right )} x + 20579\right )} x + 563761\right )} x + 2927963\right )} x + 8387071\right )} x + 42007139\right )} x + 12499587\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {28879697}{26873856} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 134, normalized size = 0.85 \[ \frac {32 \left (3 x^{2}-x +2\right )^{\frac {5}{2}} x^{4}}{27}+\frac {269 \left (3 x^{2}-x +2\right )^{\frac {5}{2}} x^{3}}{81}+\frac {1777 \left (3 x^{2}-x +2\right )^{\frac {5}{2}} x^{2}}{486}+\frac {1099 \left (3 x^{2}-x +2\right )^{\frac {5}{2}} x}{648}-\frac {28879697 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{26873856}-\frac {54593 \left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{559872}-\frac {1255639 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{4478976}+\frac {1207 \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{58320} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 155, normalized size = 0.98 \[ \frac {32}{27} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x^{4} + \frac {269}{81} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x^{3} + \frac {1777}{486} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x^{2} + \frac {1099}{648} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x + \frac {1207}{58320} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} - \frac {54593}{93312} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + \frac {54593}{559872} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} - \frac {1255639}{746496} \, \sqrt {3 \, x^{2} - x + 2} x - \frac {28879697}{26873856} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) + \frac {1255639}{4478976} \, \sqrt {3 \, x^{2} - x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (2\,x+1\right )}^3\,{\left (3\,x^2-x+2\right )}^{3/2}\,\left (4\,x^2+3\,x+1\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (2 x + 1\right )^{3} \left (3 x^{2} - x + 2\right )^{\frac {3}{2}} \left (4 x^{2} + 3 x + 1\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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