Optimal. Leaf size=41 \[ \frac {2-51 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac {16-13 x}{18 \sqrt {2+3 x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1828, 651}
\begin {gather*} \frac {2-51 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac {16-13 x}{18 \sqrt {3 x^2+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 651
Rule 1828
Rubi steps
\begin {align*} \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx &=\frac {2-51 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac {1}{6} \int \frac {-\frac {26}{3}-16 x}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac {2-51 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac {16-13 x}{18 \sqrt {2+3 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 30, normalized size = 0.73 \begin {gather*} \frac {-94+27 x-144 x^2+117 x^3}{54 \left (2+3 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 51, normalized size = 1.24
method | result | size |
gosper | \(\frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\) | \(27\) |
trager | \(\frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\) | \(27\) |
risch | \(\frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\) | \(27\) |
default | \(-\frac {8 x^{2}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {47}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {17 x}{18 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {13 x}{18 \sqrt {3 x^{2}+2}}\) | \(51\) |
meijerg | \(\frac {\sqrt {2}\, x \left (3 x^{2}+3\right )}{24 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {5 \sqrt {2}\, x^{3}}{12 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {5 \sqrt {2}\, \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{18 \sqrt {\pi }}+\frac {8 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (18 x^{2}+8\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{27 \sqrt {\pi }}\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 50, normalized size = 1.22 \begin {gather*} \frac {13 \, x}{18 \, \sqrt {3 \, x^{2} + 2}} - \frac {8 \, x^{2}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {17 \, x}{18 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {47}{27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 40, normalized size = 0.98 \begin {gather*} \frac {{\left (117 \, x^{3} - 144 \, x^{2} + 27 \, x - 94\right )} \sqrt {3 \, x^{2} + 2}}{54 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 180 vs.
\(2 (36) = 72\).
time = 19.00, size = 180, normalized size = 4.39 \begin {gather*} \frac {10 x^{3}}{18 x^{2} \sqrt {3 x^{2} + 2} + 12 \sqrt {3 x^{2} + 2}} + \frac {x^{3}}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} - \frac {72 x^{2}}{81 x^{2} \sqrt {3 x^{2} + 2} + 54 \sqrt {3 x^{2} + 2}} + \frac {x}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} - \frac {32}{81 x^{2} \sqrt {3 x^{2} + 2} + 54 \sqrt {3 x^{2} + 2}} - \frac {5}{27 x^{2} \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.83, size = 25, normalized size = 0.61 \begin {gather*} \frac {9 \, {\left ({\left (13 \, x - 16\right )} x + 3\right )} x - 94}{54 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.11, size = 185, normalized size = 4.51 \begin {gather*} \frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {17}{16}+\frac {\sqrt {6}\,1{}\mathrm {i}}{48}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {17}{24}+\frac {\sqrt {6}\,1{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {17}{16}+\frac {\sqrt {6}\,1{}\mathrm {i}}{48}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {17}{24}+\frac {\sqrt {6}\,1{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-192+\sqrt {6}\,69{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (192+\sqrt {6}\,69{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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