Optimal. Leaf size=73 \[ \frac {41 x+26}{210 \left (3 x^2+2\right )^{3/2}}+\frac {2137 x+312}{7350 \sqrt {3 x^2+2}}-\frac {104 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1225 \sqrt {35}} \]
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Rubi [A] time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {823, 12, 725, 206} \begin {gather*} \frac {41 x+26}{210 \left (3 x^2+2\right )^{3/2}}+\frac {2137 x+312}{7350 \sqrt {3 x^2+2}}-\frac {104 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1225 \sqrt {35}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 725
Rule 823
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx &=\frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}-\frac {1}{630} \int \frac {-1206-492 x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac {312+2137 x}{7350 \sqrt {2+3 x^2}}+\frac {\int \frac {11232}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{132300}\\ &=\frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac {312+2137 x}{7350 \sqrt {2+3 x^2}}+\frac {104 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{1225}\\ &=\frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac {312+2137 x}{7350 \sqrt {2+3 x^2}}-\frac {104 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{1225}\\ &=\frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac {312+2137 x}{7350 \sqrt {2+3 x^2}}-\frac {104 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1225 \sqrt {35}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.86 \begin {gather*} \frac {\frac {35 \left (6411 x^3+936 x^2+5709 x+1534\right )}{\left (3 x^2+2\right )^{3/2}}-624 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{257250} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 103, normalized size = 1.41 \begin {gather*} \frac {312 \, \sqrt {35} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (6411 \, x^{3} + 936 \, x^{2} + 5709 \, x + 1534\right )} \sqrt {3 \, x^{2} + 2}}{257250 \, {\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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giac [A] time = 0.21, size = 93, normalized size = 1.27 \begin {gather*} \frac {104}{42875} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {3 \, {\left ({\left (2137 \, x + 312\right )} x + 1903\right )} x + 1534}{7350 \, {\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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maple [B] time = 0.05, size = 122, normalized size = 1.67 \begin {gather*} -\frac {x}{12 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {x}{12 \sqrt {3 x^{2}+2}}+\frac {39 x}{140 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {1833 x}{4900 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {104 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{42875}+\frac {13}{105 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {52}{1225 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 81, normalized size = 1.11 \begin {gather*} \frac {104}{42875} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {2137 \, x}{7350 \, \sqrt {3 \, x^{2} + 2}} + \frac {52}{1225 \, \sqrt {3 \, x^{2} + 2}} + \frac {41 \, x}{210 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {13}{105 \, {\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 = 0.14, size = 218, normalized size = 2.99 \begin {gather*} \frac {\sqrt {35}\,\left (104\,\ln \left (x+\frac {3}{2}\right )-104\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{42875}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {123}{560}+\frac {\sqrt {6}\,39{}\mathrm {i}}{560}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {41}{280}+\frac {\sqrt {6}\,13{}\mathrm {i}}{280}\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 {123}{560}+\frac {\sqrt {6}\,39{}\mathrm {i}}{560}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {41}{280}+\frac {\sqrt {6}\,13{}\mathrm {i}}{280}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-3744+\sqrt {6}\,7113{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1058400\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (3744+\sqrt {6}\,7113{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1058400\,\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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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