Optimal. Leaf size=85 \[ \frac {1}{108} (109-18 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac {559}{864} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {559 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1728 \sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {779, 612, 621, 206} \begin {gather*} \frac {1}{108} (109-18 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac {559}{864} (6 x+5) \sqrt {3 x^2+5 x+2}-\frac {559 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1728 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rubi steps
\begin {align*} \int (5-x) (3+2 x) \sqrt {2+5 x+3 x^2} \, dx &=\frac {1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}+\frac {559}{72} \int \sqrt {2+5 x+3 x^2} \, dx\\ &=\frac {559}{864} (5+6 x) \sqrt {2+5 x+3 x^2}+\frac {1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {559 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{1728}\\ &=\frac {559}{864} (5+6 x) \sqrt {2+5 x+3 x^2}+\frac {1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {559}{864} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {559}{864} (5+6 x) \sqrt {2+5 x+3 x^2}+\frac {1}{108} (109-18 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {559 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1728 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 0.79 \begin {gather*} \frac {-559 \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 (432 x^3-1896 x^2-7426 x-4539\right )}{5184} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 69, normalized size = 0.81 \begin {gather*} \frac {1}{864} \sqrt {3 x^2+5 x+2} \left (-432 x^3+1896 x^2+7426 x+4539\right )-\frac {559 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{864 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 68, normalized size = 0.80 \begin {gather*} -\frac {1}{864} \, {\left (432 \, x^{3} - 1896 \, x^{2} - 7426 \, x - 4539\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {559}{10368} \, \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 = 0.25, size = 64, normalized size = 0.75 \begin {gather*} -\frac {1}{864} \, {\left (2 \, {\left (12 \, {\left (18 \, x - 79\right )} x - 3713\right )} x - 4539\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {559}{5184} \, \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 = 79, normalized size = 0.93 \begin {gather*} -\frac {\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}} x}{6}-\frac {559 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{5184}+\frac {109 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{108}+\frac {559 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{864} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 87, normalized size = 1.02 \begin {gather*} -\frac {1}{6} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {109}{108} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {559}{144} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {559}{5184} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {2795}{864} \, \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 = 0.70, size = 119, normalized size = 1.40 \begin {gather*} \frac {46\,\left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x+2}}{3}-\frac {23\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{108}+\frac {109\,\sqrt {3\,x^2+5\,x+2}\,\left (72\,x^2+30\,x-27\right )}{2592}-\frac {x\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{6}+\frac {545\,\sqrt {3}\,\ln \left (2\,\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (6\,x+5\right )}{3}\right )}{5184} \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 (- 7 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 2 x^{2} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 15 \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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