Optimal. Leaf size=42 \[ \frac {1}{28} \left (4 x^2+12 x+9\right )^{7/2}-\frac {1}{8} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} \]
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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} \[ \frac {1}{28} \left (4 x^2+12 x+9\right )^{7/2}-\frac {1}{8} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} \]
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 )^{5/2} \, dx &=\frac {1}{28} \left (9+12 x+4 x^2\right )^{7/2}-\frac {3}{2} \int \left (9+12 x+4 x^2\right )^{5/2} \, dx\\ &=-\frac {1}{8} (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}+\frac {1}{28} \left (9+12 x+4 x^2\right )^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.12 \[ \frac {x^2 \sqrt {(2 x+3)^2} \left (64 x^5+560 x^4+2016 x^3+3780 x^2+3780 x+1701\right )}{28 x+42} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 31, normalized size = 0.74 \[ \frac {32}{7} \, x^{7} + 40 \, x^{6} + 144 \, x^{5} + 270 \, x^{4} + 270 \, x^{3} + \frac {243}{2} \, x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 75, normalized size = 1.79 \[ \frac {32}{7} \, x^{7} \mathrm {sgn}\left (2 \, x + 3\right ) + 40 \, x^{6} \mathrm {sgn}\left (2 \, x + 3\right ) + 144 \, x^{5} \mathrm {sgn}\left (2 \, x + 3\right ) + 270 \, x^{4} \mathrm {sgn}\left (2 \, x + 3\right ) + 270 \, x^{3} \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {243}{2} \, x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) - \frac {729}{56} \, \mathrm {sgn}\left (2 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 47, normalized size = 1.12 \[ \frac {\left (64 x^{5}+560 x^{4}+2016 x^{3}+3780 x^{2}+3780 x +1701\right ) \left (\left (2 x +3\right )^{2}\right )^{\frac {5}{2}} x^{2}}{14 \left (2 x +3\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.85, size = 44, normalized size = 1.05 \[ \frac {1}{28} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {7}{2}} - \frac {1}{4} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}} x - \frac {3}{8} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\left (4\,x^2+12\,x+9\right )}^{5/2} \,d x \]
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 \[ \int x \left (\left (2 x + 3\right )^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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