Optimal. Leaf size=115 \[ -\frac {3 (47 x+37)}{10 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^2}+\frac {2229 x+1888}{10 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )}+\frac {2667}{25 \sqrt {2 x+3}}+402 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {12717}{25} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {822, 828, 826, 1166, 207} \begin {gather*} -\frac {3 (47 x+37)}{10 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^2}+\frac {2229 x+1888}{10 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )}+\frac {2667}{25 \sqrt {2 x+3}}+402 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {12717}{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 822
Rule 826
Rule 828
Rule 1166
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}-\frac {1}{10} \int \frac {1328+987 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \frac {43485+33435 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+\frac {1}{250} \int \frac {90255+40005 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+\frac {1}{125} \operatorname {Subst}\left (\int \frac {60495+40005 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}-1206 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {38151}{25} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {2667}{25 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^2}+\frac {1888+2229 x}{10 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )}+402 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {12717}{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 = 0.14, size = 86, normalized size = 0.75 \begin {gather*} \frac {1}{50} \left (\frac {48006 x^4+193455 x^3+281403 x^2+175465 x+39661}{\sqrt {2 x+3} \left (3 x^2+5 x+2\right )^2}+20100 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-25434 \sqrt {\frac {3}{5}} \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 = 0.31, size = 111, normalized size = 0.97 \begin {gather*} \frac {24003 (2 x+3)^4-94581 (2 x+3)^3+117873 (2 x+3)^2-44015 (2 x+3)-2080}{25 \sqrt {2 x+3} \left (3 (2 x+3)^2-8 (2 x+3)+5\right )^2}+402 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {12717}{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 = 0.41, size = 195, normalized size = 1.70 \begin {gather*} \frac {12717 \, \sqrt {5} \sqrt {3} {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 50250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 50250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) + 5 \, {\left (48006 \, x^{4} + 193455 \, x^{3} + 281403 \, x^{2} + 175465 \, x + 39661\right )} \sqrt {2 \, x + 3}}{250 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 129, normalized size = 1.12 \begin {gather*} \frac {12717}{250} \, \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 {416}{125 \, \sqrt {2 \, x + 3}} + \frac {123759 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 492873 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 628469 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 253355 \, \sqrt {2 \, x + 3}}{125 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 201 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 201 \, \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 = 133, normalized size = 1.16 \begin {gather*} -\frac {12717 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{125}-201 \ln \left (-1+\sqrt {2 x +3}\right )+201 \ln \left (\sqrt {2 x +3}+1\right )+\frac {\frac {51759 \left (2 x +3\right )^{\frac {3}{2}}}{125}-\frac {18171 \sqrt {2 x +3}}{25}}{\left (6 x +4\right )^{2}}+\frac {32}{\sqrt {2 x +3}+1}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}-\frac {416}{125 \sqrt {2 x +3}}+\frac {32}{-1+\sqrt {2 x +3}}+\frac {3}{\left (-1+\sqrt {2 x +3}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 143, normalized size = 1.24 \begin {gather*} \frac {12717}{250} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {24003 \, {\left (2 \, x + 3\right )}^{4} - 94581 \, {\left (2 \, x + 3\right )}^{3} + 117873 \, {\left (2 \, x + 3\right )}^{2} - 88030 \, x - 134125}{25 \, {\left (9 \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - 48 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + 94 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - 80 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + 25 \, \sqrt {2 \, x + 3}\right )}} + 201 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 201 \, \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 = 109, normalized size = 0.95 \begin {gather*} 402\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {12717\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{125}-\frac {\frac {17606\,x}{45}-\frac {13097\,{\left (2\,x+3\right )}^2}{25}+\frac {10509\,{\left (2\,x+3\right )}^3}{25}-\frac {2667\,{\left (2\,x+3\right )}^4}{25}+\frac {5365}{9}}{\frac {25\,\sqrt {2\,x+3}}{9}-\frac {80\,{\left (2\,x+3\right )}^{3/2}}{9}+\frac {94\,{\left (2\,x+3\right )}^{5/2}}{9}-\frac {16\,{\left (2\,x+3\right )}^{7/2}}{3}+{\left (2\,x+3\right )}^{9/2}} \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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