Optimal. Leaf size=126 \[ -\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac {35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {175 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{10368}+\frac {175 (6 x+5) \sqrt {3 x^2+5 x+2}}{82944}-\frac {175 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{165888 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {640, 612, 621, 206} \begin {gather*} -\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac {35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {175 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{10368}+\frac {175 (6 x+5) \sqrt {3 x^2+5 x+2}}{82944}-\frac {175 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{165888 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rubi steps
\begin {align*} \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}+\frac {35}{6} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175}{432} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}+\frac {175 \int \sqrt {2+5 x+3 x^2} \, dx}{6912}\\ &=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{165888}\\ &=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{82944}\\ &=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{165888 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 108, normalized size = 0.86 \begin {gather*} -\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac {35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {175 \left (\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 (144 x^3+360 x^2+290 x+75\right )\right )}{497664} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.65, size = 84, normalized size = 0.67 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-746496 x^6+1347840 x^5+13454208 x^4+26388720 x^3+23110872 x^2+9651790 x+1568541\right )}{580608}-\frac {175 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{82944 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 83, normalized size = 0.66 \begin {gather*} -\frac {1}{580608} \, {\left (746496 \, x^{6} - 1347840 \, x^{5} - 13454208 \, x^{4} - 26388720 \, x^{3} - 23110872 \, x^{2} - 9651790 \, x - 1568541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {175}{995328} \, \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 = 0.23, size = 79, normalized size = 0.63 \begin {gather*} -\frac {1}{580608} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (6 \, {\left (36 \, x - 65\right )} x - 3893\right )} x - 61085\right )} x - 962953\right )} x - 4825895\right )} x - 1568541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {175}{497664} \, \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 = 0.04, size = 102, normalized size = 0.81 \begin {gather*} -\frac {175 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{497664}+\frac {35 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{216}-\frac {175 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{10368}+\frac {175 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{82944}-\frac {\left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 130, normalized size = 1.03 \begin {gather*} -\frac {1}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {35}{36} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {175}{216} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {175}{1728} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {875}{10368} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {175}{13824} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {175}{497664} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {875}{82944} \, \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 (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,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 (- 96 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 165 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 113 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 15 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 9 x^{5} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 20 \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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