Optimal. Leaf size=102 \[ -\frac {\sqrt {2 x+3} (35 x+29)}{2 \left (3 x^2+5 x+2\right )^2}+\frac {3 \sqrt {2 x+3} (1063 x+878)}{10 \left (3 x^2+5 x+2\right )}+730 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {4713}{5} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {820, 822, 826, 1166, 207} \begin {gather*} -\frac {\sqrt {2 x+3} (35 x+29)}{2 \left (3 x^2+5 x+2\right )^2}+\frac {3 \sqrt {2 x+3} (1063 x+878)}{10 \left (3 x^2+5 x+2\right )}+730 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {4713}{5} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 820
Rule 822
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {\sqrt {3+2 x} (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}-\frac {1}{2} \int \frac {286+175 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {\sqrt {3+2 x} (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac {3 \sqrt {3+2 x} (878+1063 x)}{10 \left (2+5 x+3 x^2\right )}+\frac {1}{10} \int \frac {6839+3189 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {\sqrt {3+2 x} (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac {3 \sqrt {3+2 x} (878+1063 x)}{10 \left (2+5 x+3 x^2\right )}+\frac {1}{5} \operatorname {Subst}\left (\int \frac {4111+3189 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {\sqrt {3+2 x} (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac {3 \sqrt {3+2 x} (878+1063 x)}{10 \left (2+5 x+3 x^2\right )}-2190 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {14139}{5} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {\sqrt {3+2 x} (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac {3 \sqrt {3+2 x} (878+1063 x)}{10 \left (2+5 x+3 x^2\right )}+730 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {4713}{5} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 81, normalized size = 0.79 \begin {gather*} \frac {1}{50} \left (\frac {5 \sqrt {2 x+3} \left (9567 x^3+23847 x^2+19373 x+5123\right )}{\left (3 x^2+5 x+2\right )^2}-9426 \sqrt {15} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )\right )+730 \tanh ^{-1}\left (\sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 102, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2 x+3} \left (9567 (2 x+3)^3-38409 (2 x+3)^2+49637 (2 x+3)-20555\right )}{5 \left (3 (2 x+3)^2-8 (2 x+3)+5\right )^2}+730 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {4713}{5} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 170, normalized size = 1.67 \begin {gather*} \frac {4713 \, \sqrt {5} \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 18250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 18250 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) + 5 \, {\left (9567 \, x^{3} + 23847 \, x^{2} + 19373 \, x + 5123\right )} \sqrt {2 \, x + 3}}{50 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 120, normalized size = 1.18 \begin {gather*} \frac {4713}{50} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) + \frac {9567 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 38409 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 49637 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 20555 \, \sqrt {2 \, x + 3}}{5 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 365 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 365 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 124, normalized size = 1.22 \begin {gather*} -\frac {4713 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{25}-365 \ln \left (-1+\sqrt {2 x +3}\right )+365 \ln \left (\sqrt {2 x +3}+1\right )+\frac {\frac {4527 \left (2 x +3\right )^{\frac {3}{2}}}{5}-1611 \sqrt {2 x +3}}{\left (6 x +4\right )^{2}}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {56}{\sqrt {2 x +3}+1}+\frac {3}{\left (-1+\sqrt {2 x +3}\right )^{2}}+\frac {56}{-1+\sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 134, normalized size = 1.31 \begin {gather*} \frac {4713}{50} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {9567 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 38409 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 49637 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 20555 \, \sqrt {2 \, x + 3}}{5 \, {\left (9 \, {\left (2 \, x + 3\right )}^{4} - 48 \, {\left (2 \, x + 3\right )}^{3} + 94 \, {\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 365 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 365 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.39, size = 101, normalized size = 0.99 \begin {gather*} 730\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )+\frac {\frac {4111\,\sqrt {2\,x+3}}{9}-\frac {49637\,{\left (2\,x+3\right )}^{3/2}}{45}+\frac {12803\,{\left (2\,x+3\right )}^{5/2}}{15}-\frac {1063\,{\left (2\,x+3\right )}^{7/2}}{5}}{\frac {160\,x}{9}-\frac {94\,{\left (2\,x+3\right )}^2}{9}+\frac {16\,{\left (2\,x+3\right )}^3}{3}-{\left (2\,x+3\right )}^4+\frac {215}{9}}-\frac {4713\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{25} \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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