Optimal. Leaf size=54 \[ \frac {376 (8 x+7)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 (2 x+3)^2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {804, 636} \[ \frac {376 (8 x+7)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 (2 x+3)^2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 636
Rule 804
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (3+2 x)^2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {188}{3} \int \frac {3+2 x}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (3+2 x)^2 (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {376 (7+8 x)}{3 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 0.61 \[ \frac {2 \left (4372 x^3+10932 x^2+8925 x+2371\right )}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 51, normalized size = 0.94 \[ \frac {2 \, {\left (4372 \, x^{3} + 10932 \, x^{2} + 8925 \, x + 2371\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 28, normalized size = 0.52 \[ \frac {2 \, {\left ({\left (4 \, {\left (1093 \, x + 2733\right )} x + 8925\right )} x + 2371\right )}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.70 \[ \frac {2 \left (4372 x^{3}+10932 x^{2}+8925 x +2371\right ) \left (x +1\right ) \left (3 x +2\right )}{3 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 76, normalized size = 1.41 \[ \frac {8744 \, x}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {4 \, x^{2}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {21860}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {1114 \, x}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {1042}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 48, normalized size = 0.89 \[ \frac {36062\,x+8744\,x\,\left (3\,x^2+5\,x+2\right )+21872\,x^2+14226}{\sqrt {3\,x^2+5\,x+2}\,\left (27\,x^2+45\,x+18\right )} \]
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 \[ - \int \left (- \frac {51 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {8 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {4 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {45}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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