Optimal. Leaf size=67 \[ -\frac {7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}-\frac {556-1461 x}{54 \sqrt {2+3 x^2}}-\frac {8 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {833, 792, 221}
\begin {gather*} -\frac {7 (2-7 x) (2 x+3)^2}{18 \left (3 x^2+2\right )^{3/2}}-\frac {556-1461 x}{54 \sqrt {3 x^2+2}}-\frac {8 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 792
Rule 833
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{5/2}} \, dx &=-\frac {7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}+\frac {1}{18} \int \frac {(314-24 x) (3+2 x)}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}-\frac {556-1461 x}{54 \sqrt {2+3 x^2}}-\frac {8}{9} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}-\frac {556-1461 x}{54 \sqrt {2+3 x^2}}-\frac {8 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 61, normalized size = 0.91 \begin {gather*} -\frac {1490-3741 x+72 x^2-4971 x^3}{54 \left (2+3 x^2\right )^{3/2}}+\frac {8 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{9 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 77, normalized size = 1.15
method | result | size |
risch | \(\frac {4971 x^{3}-72 x^{2}+3741 x -1490}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {8 \arcsinh \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{27}\) | \(40\) |
trager | \(\frac {4971 x^{3}-72 x^{2}+3741 x -1490}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {8 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{27}\) | \(57\) |
default | \(\frac {8 x^{3}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {547 x}{18 \sqrt {3 x^{2}+2}}-\frac {8 \arcsinh \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{27}-\frac {4 x^{2}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {745}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {17 x}{2 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\) | \(77\) |
meijerg | \(\frac {45 \sqrt {2}\, x \left (3 x^{2}+3\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {4 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (18 x^{2}+8\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{27 \sqrt {\pi }}+\frac {21 \sqrt {2}\, x^{3}}{4 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {27 \sqrt {2}\, \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{2 \sqrt {\pi }}-\frac {16 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (30 x^{2}+15\right )}{20 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {3 \sqrt {\pi }\, \arcsinh \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{81 \sqrt {\pi }}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 91, normalized size = 1.36 \begin {gather*} \frac {8}{27} \, x {\left (\frac {9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {4}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}\right )} - \frac {8}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {1609 \, x}{54 \, \sqrt {3 \, x^{2} + 2}} - \frac {4 \, x^{2}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {17 \, x}{2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {745}{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 = 3.43, size = 81, normalized size = 1.21 \begin {gather*} \frac {8 \, \sqrt {3} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + {\left (4971 \, x^{3} - 72 \, x^{2} + 3741 \, x - 1490\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {243 x}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {126 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {4 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {8 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {135}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.91, size = 48, normalized size = 0.72 \begin {gather*} \frac {8}{27} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {3 \, {\left ({\left (1657 \, x - 24\right )} x + 1247\right )} x - 1490}{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 = 1.70, size = 200, normalized size = 2.99 \begin {gather*} -\frac {8\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {427}{48}+\frac {\sqrt {6}\,721{}\mathrm {i}}{48}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {427}{72}+\frac {\sqrt {6}\,721{}\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 {427}{48}+\frac {\sqrt {6}\,721{}\mathrm {i}}{48}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {427}{72}+\frac {\sqrt {6}\,721{}\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 (-96+\sqrt {6}\,2067{}\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 (96+\sqrt {6}\,2067{}\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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