Optimal. Leaf size=128 \[ -\frac {3 (47 x+37)}{10 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^2}+\frac {10551 x+9146}{50 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )}+\frac {6853}{125 \sqrt {2 x+3}}+\frac {7451}{75 (2 x+3)^{3/2}}+310 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {45603}{125} \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 8, 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 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^2}+\frac {10551 x+9146}{50 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )}+\frac {6853}{125 \sqrt {2 x+3}}+\frac {7451}{75 (2 x+3)^{3/2}}+310 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {45603}{125} \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)^{5/2} \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}-\frac {1}{10} \int \frac {1550+1269 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac {9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \frac {60505+52755 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {7451}{75 (3+2 x)^{3/2}}-\frac {3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac {9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+\frac {1}{250} \int \frac {150515+111765 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {7451}{75 (3+2 x)^{3/2}}+\frac {6853}{125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac {9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+\frac {\int \frac {296545+102795 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx}{1250}\\ &=\frac {7451}{75 (3+2 x)^{3/2}}+\frac {6853}{125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac {9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+\frac {1}{625} \operatorname {Subst}\left (\int \frac {284705+102795 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {7451}{75 (3+2 x)^{3/2}}+\frac {6853}{125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac {9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-930 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )+\frac {136809}{125} \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {7451}{75 (3+2 x)^{3/2}}+\frac {6853}{125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac {9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+310 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {45603}{125} \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 = 0.29, size = 91, normalized size = 0.71 \begin {gather*} \frac {\frac {5 \left (740124 x^5+4247856 x^4+9453447 x^3+10168583 x^2+5278129 x+1057511\right )}{(2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^2}-273618 \sqrt {15} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{3750}+310 \tanh ^{-1}\left (\sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 120, normalized size = 0.94 \begin {gather*} \frac {185031 (2 x+3)^5-651537 (2 x+3)^4+619101 (2 x+3)^3-10115 (2 x+3)^2-114080 (2 x+3)-10400}{375 (2 x+3)^{3/2} \left (3 (2 x+3)^2-8 (2 x+3)+5\right )^2}+310 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {45603}{125} \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 = 0.43, size = 220, normalized size = 1.72 \begin {gather*} \frac {136809 \, \sqrt {5} \sqrt {3} {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 581250 \, {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 581250 \, {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) + 5 \, {\left (740124 \, x^{5} + 4247856 \, x^{4} + 9453447 \, x^{3} + 10168583 \, x^{2} + 5278129 \, x + 1057511\right )} \sqrt {2 \, x + 3}}{3750 \, {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 134, normalized size = 1.05 \begin {gather*} \frac {45603}{1250} \, \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 {64 \, {\left (921 \, x + 1414\right )}}{1875 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}} + \frac {396801 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 1551207 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 1922011 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 737605 \, \sqrt {2 \, x + 3}}{625 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 155 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 155 \, \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 = 142, normalized size = 1.11 \begin {gather*} -\frac {45603 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{625}-155 \ln \left (-1+\sqrt {2 x +3}\right )+155 \ln \left (\sqrt {2 x +3}+1\right )+\frac {\frac {171801 \left (2 x +3\right )^{\frac {3}{2}}}{625}-\frac {60021 \sqrt {2 x +3}}{125}}{\left (6 x +4\right )^{2}}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {20}{\sqrt {2 x +3}+1}-\frac {416}{375 \left (2 x +3\right )^{\frac {3}{2}}}-\frac {9824}{625 \sqrt {2 x +3}}+\frac {3}{\left (-1+\sqrt {2 x +3}\right )^{2}}+\frac {20}{-1+\sqrt {2 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 152, normalized size = 1.19 \begin {gather*} \frac {45603}{1250} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {185031 \, {\left (2 \, x + 3\right )}^{5} - 651537 \, {\left (2 \, x + 3\right )}^{4} + 619101 \, {\left (2 \, x + 3\right )}^{3} - 10115 \, {\left (2 \, x + 3\right )}^{2} - 228160 \, x - 352640}{375 \, {\left (9 \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - 48 \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + 94 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 80 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 25 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}}\right )}} + 155 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 155 \, \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 = 0.08, size = 118, normalized size = 0.92 \begin {gather*} 310\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {45603\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{625}-\frac {\frac {45632\,x}{675}+\frac {2023\,{\left (2\,x+3\right )}^2}{675}-\frac {68789\,{\left (2\,x+3\right )}^3}{375}+\frac {24131\,{\left (2\,x+3\right )}^4}{125}-\frac {6853\,{\left (2\,x+3\right )}^5}{125}+\frac {70528}{675}}{\frac {25\,{\left (2\,x+3\right )}^{3/2}}{9}-\frac {80\,{\left (2\,x+3\right )}^{5/2}}{9}+\frac {94\,{\left (2\,x+3\right )}^{7/2}}{9}-\frac {16\,{\left (2\,x+3\right )}^{9/2}}{3}+{\left (2\,x+3\right )}^{11/2}} \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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