Optimal. Leaf size=42 \[ \frac {1}{20} \left (4 x^2+12 x+9\right )^{5/2}-\frac {3}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 42, 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, 609} \begin {gather*} \frac {1}{20} \left (4 x^2+12 x+9\right )^{5/2}-\frac {3}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rule 640
Rubi steps
\begin {align*} \int x \left (9+12 x+4 x^2\right )^{3/2} \, dx &=\frac {1}{20} \left (9+12 x+4 x^2\right )^{5/2}-\frac {3}{2} \int \left (9+12 x+4 x^2\right )^{3/2} \, dx\\ &=-\frac {3}{16} (3+2 x) \left (9+12 x+4 x^2\right )^{3/2}+\frac {1}{20} \left (9+12 x+4 x^2\right )^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.88 \begin {gather*} \frac {x^2 \sqrt {(2 x+3)^2} \left (16 x^3+90 x^2+180 x+135\right )}{20 x+30} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (9+12 x+4 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 21, normalized size = 0.50 \begin {gather*} \frac {8}{5} \, x^{5} + 9 \, x^{4} + 18 \, x^{3} + \frac {27}{2} \, x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 53, normalized size = 1.26 \begin {gather*} \frac {8}{5} \, x^{5} \mathrm {sgn}\left (2 \, x + 3\right ) + 9 \, x^{4} \mathrm {sgn}\left (2 \, x + 3\right ) + 18 \, x^{3} \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {27}{2} \, x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) - \frac {243}{80} \, \mathrm {sgn}\left (2 \, x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 37, normalized size = 0.88 \begin {gather*} \frac {\left (16 x^{3}+90 x^{2}+180 x +135\right ) \left (\left (2 x +3\right )^{2}\right )^{\frac {3}{2}} x^{2}}{10 \left (2 x +3\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 44, normalized size = 1.05 \begin {gather*} \frac {1}{20} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}} - \frac {3}{8} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} x - \frac {9}{16} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 24, normalized size = 0.57 \begin {gather*} \frac {{\left (4\,x^2+12\,x+9\right )}^{3/2}\,\left (16\,x^2+18\,x-9\right )}{80} \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 x \left (\left (2 x + 3\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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