Optimal. Leaf size=133 \[ -\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}+\frac {(2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}}{1890}+\frac {1129 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac {1129 (6 x+5) \sqrt {3 x^2+5 x+2}}{20736}+\frac {1129 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{41472 \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {832, 779, 612, 621, 206} \begin {gather*} -\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}+\frac {(2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}}{1890}+\frac {1129 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac {1129 (6 x+5) \sqrt {3 x^2+5 x+2}}{20736}+\frac {1129 \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 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {1}{21} \int (3+2 x) \left (\frac {721}{2}+237 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac {1129}{108} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}-\frac {1129 \int \sqrt {2+5 x+3 x^2} \, dx}{1728}\\ &=-\frac {1129 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac {1129 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{41472}\\ &=-\frac {1129 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac {1129 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{20736}\\ &=-\frac {1129 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac {1129 \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 = 0.05, size = 82, normalized size = 0.62 \begin {gather*} \frac {39515 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-6 \sqrt {3 x^2+5 x+2} \left (1244160 x^6-311040 x^5-27084672 x^4-79049520 x^3-94861176 x^2-51971350 x-10669737\right )}{4354560} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.71, size = 84, normalized size = 0.63 \begin {gather*} \frac {1129 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{20736 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-1244160 x^6+311040 x^5+27084672 x^4+79049520 x^3+94861176 x^2+51971350 x+10669737\right )}{725760} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 83, normalized size = 0.62 \begin {gather*} -\frac {1}{725760} \, {\left (1244160 \, x^{6} - 311040 \, x^{5} - 27084672 \, x^{4} - 79049520 \, x^{3} - 94861176 \, x^{2} - 51971350 \, x - 10669737\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1129}{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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giac [A] time = 0.21, size = 79, normalized size = 0.59 \begin {gather*} -\frac {1}{725760} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (90 \, {\left (4 \, x - 1\right )} x - 7837\right )} x - 182985\right )} x - 3952549\right )} x - 25985675\right )} x - 10669737\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1129}{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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maple [A] time = 0.05, size = 115, normalized size = 0.86 \begin {gather*} -\frac {4 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x^{2}}{21}+\frac {43 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x}{63}+\frac {1129 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{124416}+\frac {5017 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{1890}+\frac {1129 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{2592}-\frac {1129 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{20736} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 133, normalized size = 1.00 \begin {gather*} -\frac {4}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{2} + \frac {43}{63} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {5017}{1890} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {1129}{432} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {5645}{2592} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {1129}{3456} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {1129}{124416} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {5645}{20736} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \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 )}^2\,\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 327 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 406 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 185 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 4 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 12 x^{5} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 90 \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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