Optimal. Leaf size=47 \[ \frac {2 (3 x+4)}{169 \sqrt {3 x^2+8 x+1}}-\frac {3 x+4}{39 \left (3 x^2+8 x+1\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {614, 613} \[ \frac {2 (3 x+4)}{169 \sqrt {3 x^2+8 x+1}}-\frac {3 x+4}{39 \left (3 x^2+8 x+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 614
Rubi steps
\begin {align*} \int \frac {1}{\left (1+8 x+3 x^2\right )^{5/2}} \, dx &=-\frac {4+3 x}{39 \left (1+8 x+3 x^2\right )^{3/2}}-\frac {2}{13} \int \frac {1}{\left (1+8 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {4+3 x}{39 \left (1+8 x+3 x^2\right )^{3/2}}+\frac {2 (4+3 x)}{169 \sqrt {1+8 x+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.70 \[ \frac {(3 x+4) \left (18 x^2+48 x-7\right )}{507 \left (3 x^2+8 x+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 57, normalized size = 1.21 \[ \frac {3 \sqrt {3 x^2+8 x+1} \left (54 x^3+216 x^2+171 x-28\right )}{169 \left (-3 x+\sqrt {13}-4\right )^2 \left (3 x+\sqrt {13}+4\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 73, normalized size = 1.55 \[ -\frac {252 \, x^{4} + 1344 \, x^{3} + 1960 \, x^{2} - {\left (54 \, x^{3} + 216 \, x^{2} + 171 \, x - 28\right )} \sqrt {3 \, x^{2} + 8 \, x + 1} + 448 \, x + 28}{507 \, {\left (9 \, x^{4} + 48 \, x^{3} + 70 \, x^{2} + 16 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 27, normalized size = 0.57 \[ \frac {9 \, {\left (6 \, {\left (x + 4\right )} x + 19\right )} x - 28}{507 \, {\left (3 \, x^{2} + 8 \, x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 30, normalized size = 0.64
method | result | size |
gosper | \(\frac {54 x^{3}+216 x^{2}+171 x -28}{507 \left (3 x^{2}+8 x +1\right )^{\frac {3}{2}}}\) | \(30\) |
trager | \(\frac {54 x^{3}+216 x^{2}+171 x -28}{507 \left (3 x^{2}+8 x +1\right )^{\frac {3}{2}}}\) | \(30\) |
risch | \(\frac {54 x^{3}+216 x^{2}+171 x -28}{507 \left (3 x^{2}+8 x +1\right )^{\frac {3}{2}}}\) | \(30\) |
default | \(-\frac {6 x +8}{78 \left (3 x^{2}+8 x +1\right )^{\frac {3}{2}}}+\frac {6 x +8}{169 \sqrt {3 x^{2}+8 x +1}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 59, normalized size = 1.26 \[ \frac {6 \, x}{169 \, \sqrt {3 \, x^{2} + 8 \, x + 1}} + \frac {8}{169 \, \sqrt {3 \, x^{2} + 8 \, x + 1}} - \frac {x}{13 \, {\left (3 \, x^{2} + 8 \, x + 1\right )}^{\frac {3}{2}}} - \frac {4}{39 \, {\left (3 \, x^{2} + 8 \, x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 29, normalized size = 0.62 \[ \frac {\left (12\,x+16\right )\,\left (72\,x^2+192\,x-28\right )}{8112\,{\left (3\,x^2+8\,x+1\right )}^{3/2}} \]
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 \[ \int \frac {1}{\left (3 x^{2} + 8 x + 1\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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