Optimal. Leaf size=117 \[ -\frac {2 (139 x+121) (2 x+3)^3}{3 \sqrt {3 x^2+5 x+2}}+\frac {1664}{27} \sqrt {3 x^2+5 x+2} (2 x+3)^2+\frac {10}{81} (1438 x+3369) \sqrt {3 x^2+5 x+2}+\frac {6265 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{81 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {818, 832, 779, 621, 206} \begin {gather*} -\frac {2 (139 x+121) (2 x+3)^3}{3 \sqrt {3 x^2+5 x+2}}+\frac {1664}{27} \sqrt {3 x^2+5 x+2} (2 x+3)^2+\frac {10}{81} (1438 x+3369) \sqrt {3 x^2+5 x+2}+\frac {6265 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{81 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 818
Rule 832
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac {2 (3+2 x)^3 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2}{3} \int \frac {(3+2 x)^2 (723+832 x)}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x)^3 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {1664}{27} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {2}{27} \int \frac {(3+2 x) (6625+7190 x)}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x)^3 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {1664}{27} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {10}{81} (3369+1438 x) \sqrt {2+5 x+3 x^2}+\frac {6265}{81} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x)^3 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {1664}{27} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {10}{81} (3369+1438 x) \sqrt {2+5 x+3 x^2}+\frac {12530}{81} \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)^3 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {1664}{27} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {10}{81} (3369+1438 x) \sqrt {2+5 x+3 x^2}+\frac {6265 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{81 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 81, normalized size = 0.69 \begin {gather*} -\frac {6 \left (72 x^4-102 x^3-3331 x^2+6920 x+9591\right )-6265 \sqrt {9 x^2+15 x+6} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )}{243 \sqrt {3 x^2+5 x+2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.60, size = 86, normalized size = 0.74 \begin {gather*} \frac {12530 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{81 \sqrt {3}}-\frac {2 \sqrt {3 x^2+5 x+2} \left (72 x^4-102 x^3-3331 x^2+6920 x+9591\right )}{81 (x+1) (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 97, normalized size = 0.83 \begin {gather*} \frac {6265 \, \sqrt {3} {\left (3 \, x^{2} + 5 \, x + 2\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 ) - 12 \, {\left (72 \, x^{4} - 102 \, x^{3} - 3331 \, x^{2} + 6920 \, x + 9591\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{486 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 67, normalized size = 0.57 \begin {gather*} -\frac {6265}{243} \, \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 ({\left (6 \, {\left (12 \, x - 17\right )} x - 3331\right )} x + 6920\right )} x + 9591\right )}}{81 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 130, normalized size = 1.11 \begin {gather*} -\frac {16 x^{4}}{9 \sqrt {3 x^{2}+5 x +2}}+\frac {68 x^{3}}{27 \sqrt {3 x^{2}+5 x +2}}+\frac {6662 x^{2}}{81 \sqrt {3 x^{2}+5 x +2}}-\frac {6265 x}{81 \sqrt {3 x^{2}+5 x +2}}+\frac {6265 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{243}-\frac {25739}{162 \sqrt {3 x^{2}+5 x +2}}-\frac {2525 \left (6 x +5\right )}{162 \sqrt {3 x^{2}+5 x +2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 109, normalized size = 0.93 \begin {gather*} -\frac {16 \, x^{4}}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {68 \, x^{3}}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {6662 \, x^{2}}{81 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {6265}{243} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {13840 \, x}{81 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {6394}{27 \, \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 \frac {{\left (2\,x+3\right )}^4\,\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 (- \frac {999 x}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {864 x^{2}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {264 x^{3}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {16 x^{4}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \frac {16 x^{5}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {405}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \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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