Optimal. Leaf size=44 \[ \frac {1}{8 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac {1}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]
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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} \[ \frac {1}{8 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac {1}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]
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 )^{7/2}} \, dx &=-\frac {1}{20 \left (9+12 x+4 x^2\right )^{5/2}}-\frac {3}{2} \int \frac {1}{\left (9+12 x+4 x^2\right )^{7/2}} \, dx\\ &=-\frac {1}{20 \left (9+12 x+4 x^2\right )^{5/2}}+\frac {1}{8 (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.61 \[ \frac {-4 x-1}{40 (2 x+3)^5 \sqrt {(2 x+3)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 39, normalized size = 0.89 \[ -\frac {4 \, x + 1}{40 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]
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 \[ \mathit {sage}_{0} x \]
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 \[ -\frac {\left (2 x +3\right ) \left (4 x +1\right )}{40 \left (\left (2 x +3\right )^{2}\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 24, normalized size = 0.55 \[ -\frac {1}{20 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}}} + \frac {1}{8 \, {\left (2 \, x + 3\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 26, normalized size = 0.59 \[ -\frac {\left (4\,x+1\right )\,\sqrt {4\,x^2+12\,x+9}}{40\,{\left (2\,x+3\right )}^7} \]
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 \[ \int \frac {x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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