Optimal. Leaf size=66 \[ -\frac {\sqrt {2 x+3} (35 x+29)}{3 x^2+5 x+2}-82 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {316 \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{\sqrt {15}} \]
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Rubi [A] time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {820, 826, 1166, 207} \begin {gather*} -\frac {\sqrt {2 x+3} (35 x+29)}{3 x^2+5 x+2}-82 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {316 \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{\sqrt {15}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 820
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {(5-x) \sqrt {3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {\sqrt {3+2 x} (29+35 x)}{2+5 x+3 x^2}-\int \frac {76+35 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {\sqrt {3+2 x} (29+35 x)}{2+5 x+3 x^2}-2 \operatorname {Subst}\left (\int \frac {47+35 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+5 x+3 x^2}+246 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-316 \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+5 x+3 x^2}-82 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {316 \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )}{\sqrt {15}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 66, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2 x+3} (35 x+29)}{3 x^2+5 x+2}-82 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {316 \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{\sqrt {15}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 83, normalized size = 1.26 \begin {gather*} -\frac {2 \left (35 (2 x+3)^{3/2}-47 \sqrt {2 x+3}\right )}{3 (2 x+3)^2-8 (2 x+3)+5}-82 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {316 \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{\sqrt {15}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 113, normalized size = 1.71 \begin {gather*} \frac {158 \, \sqrt {15} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac {\sqrt {15} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 615 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 615 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 15 \, {\left (35 \, x + 29\right )} \sqrt {2 \, x + 3}}{15 \, {\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 = 0.18, size = 102, normalized size = 1.55 \begin {gather*} -\frac {158}{15} \, \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 {2 \, {\left (35 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 47 \, \sqrt {2 \, x + 3}\right )}}{3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19} - 41 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 41 \, \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 = 86, normalized size = 1.30 \begin {gather*} \frac {316 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{15}+41 \ln \left (-1+\sqrt {2 x +3}\right )-41 \ln \left (\sqrt {2 x +3}+1\right )-\frac {34 \sqrt {2 x +3}}{3 \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 = 1.12, size = 98, normalized size = 1.48 \begin {gather*} -\frac {158}{15} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {2 \, {\left (35 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 47 \, \sqrt {2 \, x + 3}\right )}}{3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19} - 41 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 41 \, \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 = 2.40, size = 66, normalized size = 1.00 \begin {gather*} \frac {316\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{15}-\frac {\frac {94\,\sqrt {2\,x+3}}{3}-\frac {70\,{\left (2\,x+3\right )}^{3/2}}{3}}{\frac {16\,x}{3}-{\left (2\,x+3\right )}^2+\frac {19}{3}}-82\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 122.22, size = 212, normalized size = 3.21 \begin {gather*} 340 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: x \geq - \frac {3}{2} \wedge x < - \frac {2}{3} \end {cases}\right ) - 282 \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 ) + 41 \log {\left (\sqrt {2 x + 3} - 1 \right )} - 41 \log {\left (\sqrt {2 x + 3} + 1 \right )} - \frac {6}{\sqrt {2 x + 3} + 1} - \frac {6}{\sqrt {2 x + 3} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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