Optimal. Leaf size=92 \[ -\frac {2 (139 x+121) (2 x+3)^2}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 (2834 x+2481)}{9 \sqrt {3 x^2+5 x+2}}-\frac {8 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{9 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {818, 777, 621, 206} \begin {gather*} -\frac {2 (139 x+121) (2 x+3)^2}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 (2834 x+2481)}{9 \sqrt {3 x^2+5 x+2}}-\frac {8 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{9 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 777
Rule 818
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (3+2 x)^2 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(-359-6 x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (3+2 x)^2 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2481+2834 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {8}{9} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x)^2 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2481+2834 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {16}{9} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {2 (3+2 x)^2 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2481+2834 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {8 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{9 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 67, normalized size = 0.73 \begin {gather*} \frac {2 \left (16448 x^3+41074 x^2+33443 x+8835\right )}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {8 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )}{9 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.43, size = 81, normalized size = 0.88 \begin {gather*} \frac {2 \sqrt {3 x^2+5 x+2} \left (16448 x^3+41074 x^2+33443 x+8835\right )}{9 (x+1)^2 (3 x+2)^2}-\frac {16 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{9 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 112, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (2 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 3 \, {\left (16448 \, x^{3} + 41074 \, x^{2} + 33443 \, x + 8835\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{27 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 63, normalized size = 0.68 \begin {gather*} \frac {8}{27} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {2 \, {\left ({\left (2 \, {\left (8224 \, x + 20537\right )} x + 33443\right )} x + 8835\right )}}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 161, normalized size = 1.75 \begin {gather*} \frac {8 x^{3}}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {32 x^{2}}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {607 x}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {8 x}{9 \sqrt {3 x^{2}+5 x +2}}-\frac {8 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{27}-\frac {10855}{486 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {4033 \left (6 x +5\right )}{486 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {32864 x}{27}+\frac {82160}{81}}{\sqrt {3 x^{2}+5 x +2}}-\frac {20}{27 \sqrt {3 x^{2}+5 x +2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.23, size = 197, normalized size = 2.14 \begin {gather*} \frac {8}{27} \, x {\left (\frac {1410 \, x}{\sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {9 \, x^{2}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {1175}{\sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {55 \, x}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {46}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}\right )} - \frac {8}{27} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {3760}{27} \, \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {42272 \, x}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {4 \, x^{2}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {11680}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {2318 \, x}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {2030}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{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 \frac {{\left (2\,x+3\right )}^3\,\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 (- \frac {243 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {126 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {4 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {8 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {135}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \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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