Optimal. Leaf size=81 \[ -\frac {(139 x+121) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}+30 \sqrt {2 x+3}-130 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+100 \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 81, 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 = {818, 824, 826, 1166, 207} \begin {gather*} -\frac {(139 x+121) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}+30 \sqrt {2 x+3}-130 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+100 \sqrt {\frac {5}{3}} \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 818
Rule 824
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {1}{3} \int \frac {\sqrt {3+2 x} (-60+135 x)}{2+5 x+3 x^2} \, dx\\ &=30 \sqrt {3+2 x}-\frac {(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {1}{9} \int \frac {-1080-495 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=30 \sqrt {3+2 x}-\frac {(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {2}{9} \operatorname {Subst}\left (\int \frac {-675-495 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=30 \sqrt {3+2 x}-\frac {(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+390 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-500 \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=30 \sqrt {3+2 x}-\frac {(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}-130 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+100 \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 92, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {2 x+3} \left (8 x^2+209 x+183\right )+390 \left (3 x^2+5 x+2\right ) \tanh ^{-1}\left (\sqrt {2 x+3}\right )-100 \sqrt {15} \left (3 x^2+5 x+2\right ) \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{9 x^2+15 x+6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 98, normalized size = 1.21 \begin {gather*} -\frac {2 \left (4 (2 x+3)^{5/2}+185 (2 x+3)^{3/2}-225 \sqrt {2 x+3}\right )}{3 \left (3 (2 x+3)^2-8 (2 x+3)+5\right )}-130 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+100 \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 124, normalized size = 1.53 \begin {gather*} \frac {50 \, \sqrt {5} \sqrt {3} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 195 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 195 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - {\left (8 \, x^{2} + 209 \, x + 183\right )} \sqrt {2 \, x + 3}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 111, normalized size = 1.37 \begin {gather*} -\frac {50}{3} \, \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 {8}{9} \, \sqrt {2 \, x + 3} - \frac {2 \, {\left (587 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 695 \, \sqrt {2 \, x + 3}\right )}}{9 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 65 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 65 \, \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 = 95, normalized size = 1.17 \begin {gather*} \frac {100 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{3}+65 \ln \left (-1+\sqrt {2 x +3}\right )-65 \ln \left (\sqrt {2 x +3}+1\right )-\frac {8 \sqrt {2 x +3}}{9}-\frac {850 \sqrt {2 x +3}}{27 \left (2 x +\frac {4}{3}\right )}-\frac {6}{\sqrt {2 x +3}+1}-\frac {6}{-1+\sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 107, normalized size = 1.32 \begin {gather*} -\frac {50}{3} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {8}{9} \, \sqrt {2 \, x + 3} - \frac {2 \, {\left (587 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 695 \, \sqrt {2 \, x + 3}\right )}}{9 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 65 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 65 \, \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 = 0.08, size = 81, normalized size = 1.00 \begin {gather*} -\frac {\frac {1390\,\sqrt {2\,x+3}}{27}-\frac {1174\,{\left (2\,x+3\right )}^{3/2}}{27}}{\frac {16\,x}{3}-{\left (2\,x+3\right )}^2+\frac {19}{3}}-\frac {8\,\sqrt {2\,x+3}}{9}+\mathrm {atan}\left (\sqrt {2\,x+3}\,1{}\mathrm {i}\right )\,130{}\mathrm {i}-\frac {\sqrt {15}\,\mathrm {atan}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}\,1{}\mathrm {i}}{5}\right )\,100{}\mathrm {i}}{3} \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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