Optimal. Leaf size=47 \[ \frac {280 (6 x+5)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {638, 613} \begin {gather*} \frac {280 (6 x+5)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 638
Rubi steps
\begin {align*} \int \frac {5-x}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {140}{3} \int \frac {1}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {280 (5+6 x)}{3 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 31, normalized size = 0.66 \begin {gather*} \frac {2 \left (840 x^3+2100 x^2+1715 x+457\right )}{\left (3 x^2+5 x+2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 43, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {3 x^2+5 x+2} \left (840 x^3+2100 x^2+1715 x+457\right )}{(x+1)^2 (3 x+2)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 51, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (840 \, x^{3} + 2100 \, x^{2} + 1715 \, x + 457\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 29, normalized size = 0.62 \begin {gather*} \frac {2 \, {\left (35 \, {\left (12 \, {\left (2 \, x + 5\right )} x + 49\right )} x + 457\right )}}{{\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 [A] time = 0.00, size = 38, normalized size = 0.81 \begin {gather*} \frac {2 \left (840 x^{3}+2100 x^{2}+1715 x +457\right ) \left (x +1\right ) \left (3 x +2\right )}{\left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 59, normalized size = 1.26 \begin {gather*} \frac {560 \, x}{\sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {1400}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {70 \, x}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {58}{3 \, {\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 [B] time = 2.42, size = 36, normalized size = 0.77 \begin {gather*} \frac {2310\,x+560\,x\,\left (3\,x^2+5\,x+2\right )+1400\,x^2+914}{{\left (3\,x^2+5\,x+2\right )}^{3/2}} \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 \frac {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}}\, dx - \int \left (- \frac {5}{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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