Optimal. Leaf size=108 \[ -\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{41472 \sqrt {3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {793, 626, 635,
212} \begin {gather*} \frac {1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac {839 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac {839 (6 x+5) \sqrt {3 x^2+5 x+2}}{20736}+\frac {839 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{41472 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 793
Rubi steps
\begin {align*} \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx &=\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839}{108} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {839 \int \sqrt {2+5 x+3 x^2} \, dx}{1728}\\ &=-\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{41472}\\ &=-\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{20736}\\ &=-\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{41472 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 76, normalized size = 0.70 \begin {gather*} \frac {-3 \sqrt {2+5 x+3 x^2} \left (-561921-2406950 x-3567288 x^2-2032560 x^3-210816 x^4+103680 x^5\right )+4195 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{311040} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 98, normalized size = 0.91
method | result | size |
risch | \(-\frac {\left (103680 x^{5}-210816 x^{4}-2032560 x^{3}-3567288 x^{2}-2406950 x -561921\right ) \sqrt {3 x^{2}+5 x +2}}{103680}+\frac {839 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{124416}\) | \(70\) |
trager | \(\left (-x^{5}+\frac {61}{30} x^{4}+\frac {941}{48} x^{3}+\frac {148637}{4320} x^{2}+\frac {240695}{10368} x +\frac {187307}{34560}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {839 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +5 \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{124416}\) | \(81\) |
default | \(-\frac {x \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{9}+\frac {161 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{270}+\frac {839 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{2592}-\frac {839 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{20736}+\frac {839 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{124416}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 116, normalized size = 1.07 \begin {gather*} -\frac {1}{9} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {161}{270} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {839}{432} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {4195}{2592} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {839}{3456} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {839}{124416} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {4195}{20736} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.05, size = 78, normalized size = 0.72 \begin {gather*} -\frac {1}{103680} \, {\left (103680 \, x^{5} - 210816 \, x^{4} - 2032560 \, x^{3} - 3567288 \, x^{2} - 2406950 \, x - 561921\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {839}{248832} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \end {gather*}
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 \begin {gather*} - \int \left (- 89 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 76 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 11 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 6 x^{4} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 30 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.65, size = 74, normalized size = 0.69 \begin {gather*} -\frac {1}{103680} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (30 \, x - 61\right )} x - 4705\right )} x - 148637\right )} x - 1203475\right )} x - 561921\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {839}{124416} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end {gather*}
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 \begin {gather*} -\int \left (2\,x+3\right )\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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