Optimal. Leaf size=44 \[ \frac {3}{16 (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}}-\frac {1}{12 \left (4 x^2+12 x+9\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {640, 607} \begin {gather*} \frac {3}{16 (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}}-\frac {1}{12 \left (4 x^2+12 x+9\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 607
Rule 640
Rubi steps
\begin {align*} \int \frac {x}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx &=-\frac {1}{12 \left (9+12 x+4 x^2\right )^{3/2}}-\frac {3}{2} \int \frac {1}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx\\ &=-\frac {1}{12 \left (9+12 x+4 x^2\right )^{3/2}}+\frac {3}{16 (3+2 x) \left (9+12 x+4 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.61 \begin {gather*} \frac {-8 x-3}{48 (2 x+3)^3 \sqrt {(2 x+3)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 49, normalized size = 1.11 \begin {gather*} -\frac {2}{3 \left (\sqrt {4 x^2+12 x+9}-2 x-3\right )^3}-\frac {3}{\left (\sqrt {4 x^2+12 x+9}-2 x-3\right )^4} \end {gather*}
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.40, size = 29, normalized size = 0.66 \begin {gather*} -\frac {8 \, x + 3}{48 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 22, normalized size = 0.50 \begin {gather*} -\frac {\left (2 x +3\right ) \left (8 x +3\right )}{48 \left (\left (2 x +3\right )^{2}\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 24, normalized size = 0.55 \begin {gather*} -\frac {1}{12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}}} + \frac {3}{16 \, {\left (2 \, x + 3\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 26, normalized size = 0.59 \begin {gather*} -\frac {\left (8\,x+3\right )\,\sqrt {4\,x^2+12\,x+9}}{48\,{\left (2\,x+3\right )}^5} \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}{\left (\left (2 x + 3\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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