Optimal. Leaf size=68 \[ -\frac {198}{25 \sqrt {2 x+3}}-\frac {26}{15 (2 x+3)^{3/2}}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {102}{25} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {828, 826, 1166, 207} \begin {gather*} -\frac {198}{25 \sqrt {2 x+3}}-\frac {26}{15 (2 x+3)^{3/2}}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {102}{25} \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 826
Rule 828
Rule 1166
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx &=-\frac {26}{15 (3+2 x)^{3/2}}+\frac {1}{5} \int \frac {-9-39 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {26}{15 (3+2 x)^{3/2}}-\frac {198}{25 \sqrt {3+2 x}}+\frac {1}{25} \int \frac {-147-297 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac {26}{15 (3+2 x)^{3/2}}-\frac {198}{25 \sqrt {3+2 x}}+\frac {2}{25} \operatorname {Subst}\left (\int \frac {597-297 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {26}{15 (3+2 x)^{3/2}}-\frac {198}{25 \sqrt {3+2 x}}+\frac {306}{25} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-36 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=-\frac {26}{15 (3+2 x)^{3/2}}-\frac {198}{25 \sqrt {3+2 x}}+12 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {102}{25} \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 = 0.08, size = 65, normalized size = 0.96 \begin {gather*} 12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {2 \left (2970 x+153 \sqrt {15} (2 x+3)^{3/2} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )+4780\right )}{375 (2 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 64, normalized size = 0.94 \begin {gather*} -\frac {2 (297 (2 x+3)+65)}{75 (2 x+3)^{3/2}}+12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {102}{25} \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 = 120, normalized size = 1.76 \begin {gather*} \frac {153 \, \sqrt {5} \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 2250 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 2250 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 20 \, {\left (297 \, x + 478\right )} \sqrt {2 \, x + 3}}{375 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 79, normalized size = 1.16 \begin {gather*} \frac {51}{125} \, \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 {4 \, {\left (297 \, x + 478\right )}}{75 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} + 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 = 62, normalized size = 0.91 \begin {gather*} -\frac {102 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{125}-6 \ln \left (-1+\sqrt {2 x +3}\right )+6 \ln \left (\sqrt {2 x +3}+1\right )-\frac {26}{15 \left (2 x +3\right )^{\frac {3}{2}}}-\frac {198}{25 \sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 75, normalized size = 1.10 \begin {gather*} \frac {51}{125} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {4 \, {\left (297 \, x + 478\right )}}{75 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} + 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 = 0.07, size = 43, normalized size = 0.63 \begin {gather*} 12\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {\frac {396\,x}{25}+\frac {1912}{75}}{{\left (2\,x+3\right )}^{3/2}}-\frac {102\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 105.82, size = 114, normalized size = 1.68 \begin {gather*} \frac {306 \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 )}{25} - 6 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 6 \log {\left (\sqrt {2 x + 3} + 1 \right )} - \frac {198}{25 \sqrt {2 x + 3}} - \frac {26}{15 \left (2 x + 3\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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