Optimal. Leaf size=38 \[ 12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {34 \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{\sqrt {15}} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {826, 1166, 207} \begin {gather*} 12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {34 \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 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {5-x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {13-x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=34 \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 )\\ &=12 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {34 \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )}{\sqrt {15}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.00 \begin {gather*} 12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {34 \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 = 0.14, size = 38, normalized size = 1.00 \begin {gather*} 12 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {34 \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 [A] time = 0.40, size = 56, normalized size = 1.47 \begin {gather*} \frac {17}{15} \, \sqrt {15} \log \left (-\frac {\sqrt {15} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 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 [B] time = 0.17, size = 65, normalized size = 1.71 \begin {gather*} \frac {17}{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 ) + 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 = 44, normalized size = 1.16 \begin {gather*} -\frac {34 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{15}-6 \ln \left (-1+\sqrt {2 x +3}\right )+6 \ln \left (\sqrt {2 x +3}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.18, size = 61, normalized size = 1.61 \begin {gather*} \frac {17}{15} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + 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 = 29, normalized size = 0.76 \begin {gather*} 12\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {34\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 82.71, size = 95, normalized size = 2.50 \begin {gather*} 34 \left (\begin {cases} - \frac {\sqrt {15} \operatorname {acoth}{\left (\frac {\sqrt {15}}{3 \sqrt {2 x + 3}} \right )}}{15} & \text {for}\: \frac {1}{2 x + 3} > \frac {3}{5} \\- \frac {\sqrt {15} \operatorname {atanh}{\left (\frac {\sqrt {15}}{3 \sqrt {2 x + 3}} \right )}}{15} & \text {for}\: \frac {1}{2 x + 3} < \frac {3}{5} \end {cases}\right ) - 6 \log {\left (-1 + \frac {1}{\sqrt {2 x + 3}} \right )} + 6 \log {\left (1 + \frac {1}{\sqrt {2 x + 3}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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