Optimal. Leaf size=37 \[ \frac {43 x}{18 \sqrt {3 x^2+2}}-\frac {7 (2-7 x)}{18 \left (3 x^2+2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {778, 191} \[ \frac {43 x}{18 \sqrt {3 x^2+2}}-\frac {7 (2-7 x)}{18 \left (3 x^2+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 778
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)}{\left (2+3 x^2\right )^{5/2}} \, dx &=-\frac {7 (2-7 x)}{18 \left (2+3 x^2\right )^{3/2}}+\frac {43}{9} \int \frac {1}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {7 (2-7 x)}{18 \left (2+3 x^2\right )^{3/2}}+\frac {43 x}{18 \sqrt {2+3 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 0.68 \[ -\frac {-129 x^3-135 x+14}{18 \left (3 x^2+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 35, normalized size = 0.95 \[ \frac {{\left (129 \, x^{3} + 135 \, x - 14\right )} \sqrt {3 \, x^{2} + 2}}{18 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 23, normalized size = 0.62 \[ \frac {3 \, {\left (43 \, x^{2} + 45\right )} x - 14}{18 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 22, normalized size = 0.59 \[ \frac {129 x^{3}+135 x -14}{18 \left (3 x^{2}+2\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 36, normalized size = 0.97 \[ \frac {43 \, x}{18 \, \sqrt {3 \, x^{2} + 2}} + \frac {49 \, x}{18 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {7}{9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 161, normalized size = 4.35 \[ \frac {41\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{144\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {41\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{144\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {49}{16}+\frac {\sqrt {6}\,7{}\mathrm {i}}{16}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {49}{24}+\frac {\sqrt {6}\,7{}\mathrm {i}}{24}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {49}{16}+\frac {\sqrt {6}\,7{}\mathrm {i}}{16}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {49}{24}+\frac {\sqrt {6}\,7{}\mathrm {i}}{24}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 58.40, size = 122, normalized size = 3.30 \[ - \frac {2 x^{3}}{18 x^{2} \sqrt {3 x^{2} + 2} + 12 \sqrt {3 x^{2} + 2}} + \frac {15 x^{3}}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} + \frac {15 x}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} - \frac {7}{27 x^{2} \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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