Optimal. Leaf size=42 \[ -\frac {1}{8} (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}+\frac {1}{28} \left (9+12 x+4 x^2\right )^{7/2} \]
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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 = {654, 623}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
Rule 654
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 = 0.01, size = 47, normalized size = 1.12 \begin {gather*} \frac {x^2 \sqrt {(3+2 x)^2} \left (1701+3780 x+3780 x^2+2016 x^3+560 x^4+64 x^5\right )}{42+28 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 47, normalized size = 1.12
method | result | size |
gosper | \(\frac {x^{2} \left (64 x^{5}+560 x^{4}+2016 x^{3}+3780 x^{2}+3780 x +1701\right ) \left (\left (2 x +3\right )^{2}\right )^{\frac {5}{2}}}{14 \left (2 x +3\right )^{5}}\) | \(47\) |
default | \(\frac {x^{2} \left (64 x^{5}+560 x^{4}+2016 x^{3}+3780 x^{2}+3780 x +1701\right ) \left (\left (2 x +3\right )^{2}\right )^{\frac {5}{2}}}{14 \left (2 x +3\right )^{5}}\) | \(47\) |
risch | \(\frac {32 \sqrt {\left (2 x +3\right )^{2}}\, x^{7}}{7 \left (2 x +3\right )}+\frac {40 \sqrt {\left (2 x +3\right )^{2}}\, x^{6}}{2 x +3}+\frac {144 \sqrt {\left (2 x +3\right )^{2}}\, x^{5}}{2 x +3}+\frac {270 \sqrt {\left (2 x +3\right )^{2}}\, x^{4}}{2 x +3}+\frac {270 \sqrt {\left (2 x +3\right )^{2}}\, x^{3}}{2 x +3}+\frac {243 \sqrt {\left (2 x +3\right )^{2}}\, x^{2}}{2 \left (2 x +3\right )}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 44, normalized size = 1.05 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.84, size = 31, normalized size = 0.74 \begin {gather*} \frac {32}{7} \, x^{7} + 40 \, x^{6} + 144 \, x^{5} + 270 \, x^{4} + 270 \, x^{3} + \frac {243}{2} \, x^{2} \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 {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (34) = 68\).
time = 0.64, size = 75, normalized size = 1.79 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \int x\,{\left (4\,x^2+12\,x+9\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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