Optimal. Leaf size=103 \[ -\frac {1}{15} \left (3 x^2+5 x+2\right )^{5/2}+\frac {35}{144} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {35 (6 x+5) \sqrt {3 x^2+5 x+2}}{1152}+\frac {35 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2304 \sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, 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}{15} \left (3 x^2+5 x+2\right )^{5/2}+\frac {35}{144} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {35 (6 x+5) \sqrt {3 x^2+5 x+2}}{1152}+\frac {35 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2304 \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 )^{3/2} \, dx &=-\frac {1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac {35}{6} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+5 x+3 x^2\right )^{5/2}-\frac {35}{96} \int \sqrt {2+5 x+3 x^2} \, dx\\ &=-\frac {35 (5+6 x) \sqrt {2+5 x+3 x^2}}{1152}+\frac {35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac {35 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{2304}\\ &=-\frac {35 (5+6 x) \sqrt {2+5 x+3 x^2}}{1152}+\frac {35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac {35 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{1152}\\ &=-\frac {35 (5+6 x) \sqrt {2+5 x+3 x^2}}{1152}+\frac {35}{144} (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+5 x+3 x^2\right )^{5/2}+\frac {35 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{2304 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 72, normalized size = 0.70 \begin {gather*} \frac {175 \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 (3456 x^4-13680 x^3-48792 x^2-43070 x-11589\right )}{34560} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.46, size = 74, normalized size = 0.72 \begin {gather*} \frac {35 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{1152 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-3456 x^4+13680 x^3+48792 x^2+43070 x+11589\right )}{5760} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 73, normalized size = 0.71 \begin {gather*} -\frac {1}{5760} \, {\left (3456 \, x^{4} - 13680 \, x^{3} - 48792 \, x^{2} - 43070 \, x - 11589\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {35}{13824} \, \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 = 0.19, size = 69, normalized size = 0.67 \begin {gather*} -\frac {1}{5760} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (24 \, x - 95\right )} x - 2033\right )} x - 21535\right )} x - 11589\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {35}{6912} \, \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 = 0.08, size = 83, normalized size = 0.81 \begin {gather*} \frac {35 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{6912}+\frac {35 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{144}-\frac {35 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{1152}-\frac {\left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 101, normalized size = 0.98 \begin {gather*} -\frac {1}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {35}{24} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {175}{144} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {35}{192} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {35}{6912} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {175}{1152} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.87, size = 130, normalized size = 1.26 \begin {gather*} \frac {35\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{6912}-\frac {5\,\left (6\,x+5\right )\,\sqrt {3\,x^2+5\,x+2}}{1152}+\frac {5\,\left (3\,x+\frac {5}{2}\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{12}-\frac {5\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{16}+\frac {5\,x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{24}+\frac {25\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{144}-\frac {{\left (3\,x^2+5\,x+2\right )}^{5/2}}{15} \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 (- 23 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 10 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 3 x^{3} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 10 \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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