Integrand size = 27, antiderivative size = 181 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=\frac {7 (167495-349806 x) \sqrt {2+5 x+3 x^2}}{36864}+\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {12443893 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{73728 \sqrt {3}}+\frac {44625 \sqrt {5} \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1024} \]
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Time = 0.08 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {826, 828, 857, 635, 212, 738} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=-\frac {12443893 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{73728 \sqrt {3}}+\frac {44625 \sqrt {5} \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1024}-\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}+\frac {7 (121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{240 (2 x+3)}+\frac {7 (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}}{4608}+\frac {7 (167495-349806 x) \sqrt {3 x^2+5 x+2}}{36864} \]
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Rule 212
Rule 635
Rule 738
Rule 826
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {7}{96} \int \frac {(-404-484 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx \\ & = \frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}+\frac {7}{768} \int \frac {(-19488-23192 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx \\ & = \frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {7 \int \frac {(2361672+2798448 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx}{73728} \\ & = \frac {7 (167495-349806 x) \sqrt {2+5 x+3 x^2}}{36864}+\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}+\frac {7 \int \frac {-145828656-170659104 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{3538944} \\ & = \frac {7 (167495-349806 x) \sqrt {2+5 x+3 x^2}}{36864}+\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {12443893 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{73728}+\frac {223125 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{1024} \\ & = \frac {7 (167495-349806 x) \sqrt {2+5 x+3 x^2}}{36864}+\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {12443893 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{36864}-\frac {223125}{512} \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right ) \\ & = \frac {7 (167495-349806 x) \sqrt {2+5 x+3 x^2}}{36864}+\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {12443893 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{73728 \sqrt {3}}+\frac {44625 \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1024} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.68 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=\frac {-\frac {3 \sqrt {2+5 x+3 x^2} \left (-91912653-89867034 x-19284852 x^2-12848072 x^3-15112992 x^4-6830784 x^5-926208 x^6+414720 x^7\right )}{(3+2 x)^2}+48195000 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-62219465 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{552960} \]
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Time = 0.36 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {1244160 x^{9}-705024 x^{8}-24293952 x^{7}-81345312 x^{6}-127770744 x^{5}-152320900 x^{4}-391721506 x^{3}-763642833 x^{2}-639297333 x -183825306}{184320 \left (3+2 x \right )^{2} \sqrt {3 x^{2}+5 x +2}}-\frac {12443893 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{221184}-\frac {44625 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1024}\) | \(127\) |
trager | \(-\frac {\left (414720 x^{7}-926208 x^{6}-6830784 x^{5}-15112992 x^{4}-12848072 x^{3}-19284852 x^{2}-89867034 x -91912653\right ) \sqrt {3 x^{2}+5 x +2}}{184320 \left (3+2 x \right )^{2}}+\frac {44625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{1024}-\frac {12443893 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{221184}\) | \(148\) |
default | \(\frac {27 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{10 \left (x +\frac {3}{2}\right )}+\frac {51 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{8}-\frac {1127 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{480}-\frac {20293 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{4608}-\frac {408107 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{36864}-\frac {12443893 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{221184}+\frac {357 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{32}+\frac {2975 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{128}+\frac {44625 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{1024}-\frac {44625 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1024}-\frac {27 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{20}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{40 \left (x +\frac {3}{2}\right )^{2}}\) | \(253\) |
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Time = 0.30 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.96 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=\frac {62219465 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 48195000 \, \sqrt {5} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 12 \, {\left (414720 \, x^{7} - 926208 \, x^{6} - 6830784 \, x^{5} - 15112992 \, x^{4} - 12848072 \, x^{3} - 19284852 \, x^{2} - 89867034 \, x - 91912653\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{2211840 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \]
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\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.20 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=\frac {39}{40} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{10 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1127}{80} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {7}{12} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{4 \, {\left (2 \, x + 3\right )}} - \frac {20293}{768} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {5635}{4608} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {408107}{6144} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {12443893}{221184} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {44625}{1024} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {1172465}{36864} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
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Time = 0.36 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.54 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=-\frac {1}{184320} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (30 \, x - 157\right )} x - 725\right )} x - 67409\right )} x + 1173065\right )} x - 8219517\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {44625}{1024} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {12443893}{221184} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {25 \, {\left (5878 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 22241 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 75807 \, \sqrt {3} x + 27061 \, \sqrt {3} - 75807 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{512 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \]
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Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^3} \,d x \]
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