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 = 0.08, antiderivative size = 116, normalized size of antiderivative = 1.00, 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 215
Rule 612
Rule 619
Rule 779
Rule 1653
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*}
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Mathematica [A] time = 0.04, size = 70, normalized size = 0.60 \[ \frac {6 \sqrt {3 x^2-x+2} \left (497664 x^6+518400 x^5+653184 x^4+744336 x^3+531384 x^2+275410 x+203337\right )+262913 \sqrt {3} \sinh ^{-1}\left (\frac {6 x-1}{\sqrt {23}}\right )}{870912} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 83, normalized size = 0.72 \[ \frac {1}{145152} \, {\left (497664 \, x^{6} + 518400 \, x^{5} + 653184 \, x^{4} + 744336 \, x^{3} + 531384 \, x^{2} + 275410 \, x + 203337\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {37559}{248832} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 78, normalized size = 0.67 \[ \frac {1}{145152} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (24 \, {\left (2 \, {\left (24 \, x + 25\right )} x + 63\right )} x + 1723\right )} x + 22141\right )} x + 137705\right )} x + 203337\right )} \sqrt {3 \, x^{2} - x + 2} - \frac {37559}{124416} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 100, normalized size = 0.86 \[ \frac {8 \left (3 x^{2}-x +2\right )^{\frac {5}{2}} x^{2}}{21}+\frac {41 \left (3 x^{2}-x +2\right )^{\frac {5}{2}} x}{63}+\frac {37559 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{124416}+\frac {145 \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{378}+\frac {71 \left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{2592}+\frac {1633 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{20736} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 121, normalized size = 1.04 \[ \frac {8}{21} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x^{2} + \frac {41}{63} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x + \frac {145}{378} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} + \frac {71}{432} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x - \frac {71}{2592} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} + \frac {1633}{3456} \, \sqrt {3 \, x^{2} - x + 2} x + \frac {37559}{124416} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) - \frac {1633}{20736} \, \sqrt {3 \, x^{2} - x + 2} \]
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 \[ \int \left (2\,x+1\right )\,{\left (3\,x^2-x+2\right )}^{3/2}\,\left (4\,x^2+3\,x+1\right ) \,d x \]
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 \[ \int \left (2 x + 1\right ) \left (3 x^{2} - x + 2\right )^{\frac {3}{2}} \left (4 x^{2} + 3 x + 1\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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