Optimal. Leaf size=141 \[ -\frac {3 (47 x+37)}{10 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^2}+\frac {9957 x+8852}{50 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac {24409}{3125 \sqrt {2 x+3}}+\frac {102697}{1875 (2 x+3)^{3/2}}+\frac {56399}{625 (2 x+3)^{5/2}}+266 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {806841 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{3125} \]
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Rubi [A] time = 0.12, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, 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)^{5/2} \left (3 x^2+5 x+2\right )^2}+\frac {9957 x+8852}{50 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac {24409}{3125 \sqrt {2 x+3}}+\frac {102697}{1875 (2 x+3)^{3/2}}+\frac {56399}{625 (2 x+3)^{5/2}}+266 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {806841 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{3125} \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)^{7/2} \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}-\frac {1}{10} \int \frac {1772+1551 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {1}{50} \int \frac {76349+69699 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {56399}{625 (3+2 x)^{5/2}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {1}{250} \int \frac {202447+169197 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {\int \frac {474341+308091 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx}{1250}\\ &=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {\int \frac {758023-73227 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx}{6250}\\ &=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {1735727-73227 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )}{3125}\\ &=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+\frac {2420523 \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )}{3125}-798 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {56399}{625 (3+2 x)^{5/2}}+\frac {102697}{1875 (3+2 x)^{3/2}}-\frac {24409}{3125 \sqrt {3+2 x}}-\frac {3 (37+47 x)}{10 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2}+\frac {8852+9957 x}{50 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+266 \tanh ^{-1}\left (\sqrt {3+2 x}\right )-\frac {806841 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )}{3125}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 121, normalized size = 0.86 \begin {gather*} \frac {-\frac {28125 (47 x+37)}{\left (3 x^2+5 x+2\right )^2}+\frac {1875 (9957 x+8852)}{3 x^2+5 x+2}+2 (2 x+3) \left (21 (2 x+3) \left (593750 \sqrt {2 x+3} \tanh ^{-1}\left (\sqrt {2 x+3}\right )-115263 \sqrt {30 x+45} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )-17435\right )+2567425\right )+8459850}{93750 (2 x+3)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 129, normalized size = 0.91 \begin {gather*} \frac {-659043 (2 x+3)^6+8136261 (2 x+3)^5-23916753 (2 x+3)^4+24720095 (2 x+3)^3-6945760 (2 x+3)^2-728800 (2 x+3)-156000}{9375 (2 x+3)^{5/2} \left (3 (2 x+3)^2-8 (2 x+3)+5\right )^2}+266 \tanh ^{-1}\left (\sqrt {2 x+3}\right )-\frac {806841 \sqrt {\frac {3}{5}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{3125} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 245, normalized size = 1.74 \begin {gather*} \frac {2420523 \, \sqrt {5} \sqrt {3} {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (-\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 12468750 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 12468750 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 5 \, {\left (5272344 \, x^{6} + 14906052 \, x^{5} - 18312714 \, x^{4} - 114099329 \, x^{3} - 160041829 \, x^{2} - 94082723 \, x - 20250051\right )} \sqrt {2 \, x + 3}}{93750 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 143, normalized size = 1.01 \begin {gather*} \frac {806841}{31250} \, \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 {202995 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 745077 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 831169 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 259087 \, \sqrt {2 \, x + 3}}{625 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} - \frac {32 \, {\left (12861 \, {\left (2 \, x + 3\right )}^{2} + 3070 \, x + 4800\right )}}{9375 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}} + 133 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 133 \, \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 = 0.03, size = 151, normalized size = 1.07 \begin {gather*} -\frac {806841 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{15625}-133 \ln \left (-1+\sqrt {2 x +3}\right )+133 \ln \left (\sqrt {2 x +3}+1\right )+\frac {\frac {22599 \left (2 x +3\right )^{\frac {3}{2}}}{125}-\frac {196587 \sqrt {2 x +3}}{625}}{\left (6 x +4\right )^{2}}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {8}{\sqrt {2 x +3}+1}-\frac {416}{625 \left (2 x +3\right )^{\frac {5}{2}}}-\frac {9824}{1875 \left (2 x +3\right )^{\frac {3}{2}}}-\frac {137184}{3125 \sqrt {2 x +3}}+\frac {3}{\left (-1+\sqrt {2 x +3}\right )^{2}}+\frac {8}{-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 = 161, normalized size = 1.14 \begin {gather*} \frac {806841}{31250} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {659043 \, {\left (2 \, x + 3\right )}^{6} - 8136261 \, {\left (2 \, x + 3\right )}^{5} + 23916753 \, {\left (2 \, x + 3\right )}^{4} - 24720095 \, {\left (2 \, x + 3\right )}^{3} + 6945760 \, {\left (2 \, x + 3\right )}^{2} + 1457600 \, x + 2342400}{9375 \, {\left (9 \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - 48 \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + 94 \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - 80 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + 25 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}}\right )}} + 133 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 133 \, \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 = 127, normalized size = 0.90 \begin {gather*} 266\,\mathrm {atanh}\left (\sqrt {2\,x+3}\right )-\frac {806841\,\sqrt {15}\,\mathrm {atanh}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}}{5}\right )}{15625}-\frac {\frac {58304\,x}{3375}+\frac {1389152\,{\left (2\,x+3\right )}^2}{16875}-\frac {4944019\,{\left (2\,x+3\right )}^3}{16875}+\frac {2657417\,{\left (2\,x+3\right )}^4}{9375}-\frac {301343\,{\left (2\,x+3\right )}^5}{3125}+\frac {24409\,{\left (2\,x+3\right )}^6}{3125}+\frac {31232}{1125}}{\frac {25\,{\left (2\,x+3\right )}^{5/2}}{9}-\frac {80\,{\left (2\,x+3\right )}^{7/2}}{9}+\frac {94\,{\left (2\,x+3\right )}^{9/2}}{9}-\frac {16\,{\left (2\,x+3\right )}^{11/2}}{3}+{\left (2\,x+3\right )}^{13/2}} \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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