Optimal. Leaf size=81 \[ -\frac {2}{15} (2 x+3)^{5/2}+\frac {62}{27} (2 x+3)^{3/2}+\frac {526}{27} \sqrt {2 x+3}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {850}{27} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {824, 826, 1166, 207} \begin {gather*} -\frac {2}{15} (2 x+3)^{5/2}+\frac {62}{27} (2 x+3)^{3/2}+\frac {526}{27} \sqrt {2 x+3}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {850}{27} \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 824
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{5/2}}{2+5 x+3 x^2} \, dx &=-\frac {2}{15} (3+2 x)^{5/2}+\frac {1}{3} \int \frac {(3+2 x)^{3/2} (49+31 x)}{2+5 x+3 x^2} \, dx\\ &=\frac {62}{27} (3+2 x)^{3/2}-\frac {2}{15} (3+2 x)^{5/2}+\frac {1}{9} \int \frac {\sqrt {3+2 x} (317+263 x)}{2+5 x+3 x^2} \, dx\\ &=\frac {526}{27} \sqrt {3+2 x}+\frac {62}{27} (3+2 x)^{3/2}-\frac {2}{15} (3+2 x)^{5/2}+\frac {1}{27} \int \frac {1801+1639 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {526}{27} \sqrt {3+2 x}+\frac {62}{27} (3+2 x)^{3/2}-\frac {2}{15} (3+2 x)^{5/2}+\frac {2}{27} \operatorname {Subst}\left (\int \frac {-1315+1639 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {526}{27} \sqrt {3+2 x}+\frac {62}{27} (3+2 x)^{3/2}-\frac {2}{15} (3+2 x)^{5/2}-36 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {4250}{27} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {526}{27} \sqrt {3+2 x}+\frac {62}{27} (3+2 x)^{3/2}-\frac {2}{15} (3+2 x)^{5/2}+12 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {850}{27} \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 = 0.07, size = 64, normalized size = 0.79 \begin {gather*} 12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {2}{405} \left (3 \sqrt {2 x+3} \left (36 x^2-202 x-1699\right )+2125 \sqrt {15} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 73, normalized size = 0.90 \begin {gather*} -\frac {2}{135} \sqrt {2 x+3} \left (9 (2 x+3)^2-155 (2 x+3)-1315\right )+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {850}{27} \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 = 81, normalized size = 1.00 \begin {gather*} \frac {425}{81} \, \sqrt {5} \sqrt {3} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) - \frac {2}{135} \, {\left (36 \, x^{2} - 202 \, x - 1699\right )} \sqrt {2 \, x + 3} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 92, normalized size = 1.14 \begin {gather*} -\frac {2}{15} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {62}{27} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {425}{81} \, \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 {526}{27} \, \sqrt {2 \, x + 3} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \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 = 0.01, size = 71, normalized size = 0.88 \begin {gather*} -\frac {850 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{81}-6 \ln \left (-1+\sqrt {2 x +3}\right )+6 \ln \left (\sqrt {2 x +3}+1\right )-\frac {2 \left (2 x +3\right )^{\frac {5}{2}}}{15}+\frac {62 \left (2 x +3\right )^{\frac {3}{2}}}{27}+\frac {526 \sqrt {2 x +3}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 88, normalized size = 1.09 \begin {gather*} -\frac {2}{15} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {62}{27} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {425}{81} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {526}{27} \, \sqrt {2 \, x + 3} + 6 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 6 \, \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 = 0.06, size = 62, normalized size = 0.77 \begin {gather*} \frac {526\,\sqrt {2\,x+3}}{27}+\frac {62\,{\left (2\,x+3\right )}^{3/2}}{27}-\frac {2\,{\left (2\,x+3\right )}^{5/2}}{15}-\mathrm {atan}\left (\sqrt {2\,x+3}\,1{}\mathrm {i}\right )\,12{}\mathrm {i}+\frac {\sqrt {15}\,\mathrm {atan}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}\,1{}\mathrm {i}}{5}\right )\,850{}\mathrm {i}}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 83.33, size = 126, normalized size = 1.56 \begin {gather*} - \frac {2 \left (2 x + 3\right )^{\frac {5}{2}}}{15} + \frac {62 \left (2 x + 3\right )^{\frac {3}{2}}}{27} + \frac {526 \sqrt {2 x + 3}}{27} + \frac {4250 \left (\begin {cases} - \frac {\sqrt {15} \operatorname {acoth}{\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} \right )}}{15} & \text {for}\: 2 x + 3 > \frac {5}{3} \\- \frac {\sqrt {15} \operatorname {atanh}{\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} \right )}}{15} & \text {for}\: 2 x + 3 < \frac {5}{3} \end {cases}\right )}{27} - 6 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 6 \log {\left (\sqrt {2 x + 3} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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