Integrand size = 24, antiderivative size = 155 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {11763 \sqrt {1-2 x}}{78125}+\frac {1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac {1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac {(1-2 x)^{3/2} (734+24939 x)}{93750}-\frac {11763 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
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Time = 0.05 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {99, 154, 158, 152, 52, 65, 212} \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {11763 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125}-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{50 (5 x+3)}-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac {1117}{750} (1-2 x)^{3/2} (3 x+2)^3+\frac {1903 (1-2 x)^{3/2} (3 x+2)^2}{4375}+\frac {(1-2 x)^{3/2} (24939 x+734)}{93750}+\frac {11763 \sqrt {1-2 x}}{78125} \]
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Rule 52
Rule 65
Rule 99
Rule 152
Rule 154
Rule 158
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}+\frac {1}{10} \int \frac {(2-39 x) (1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^2} \, dx \\ & = -\frac {(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac {1}{50} \int \frac {(342-3351 x) \sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx \\ & = \frac {1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}-\frac {\int \frac {\sqrt {1-2 x} (2+3 x)^2 (-621+34254 x)}{3+5 x} \, dx}{2250} \\ & = \frac {1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac {1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac {\int \frac {(43470-174573 x) \sqrt {1-2 x} (2+3 x)}{3+5 x} \, dx}{78750} \\ & = \frac {1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac {1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac {(1-2 x)^{3/2} (734+24939 x)}{93750}+\frac {11763 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{31250} \\ & = \frac {11763 \sqrt {1-2 x}}{78125}+\frac {1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac {1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac {(1-2 x)^{3/2} (734+24939 x)}{93750}+\frac {129393 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{156250} \\ & = \frac {11763 \sqrt {1-2 x}}{78125}+\frac {1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac {1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac {(1-2 x)^{3/2} (734+24939 x)}{93750}-\frac {129393 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{156250} \\ & = \frac {11763 \sqrt {1-2 x}}{78125}+\frac {1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac {1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac {(1-2 x)^{3/2} (734+24939 x)}{93750}-\frac {11763 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.50 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {\frac {5 \sqrt {1-2 x} \left (871208+6891315 x+9372960 x^2-11139550 x^3-16051500 x^4+15075000 x^5+15750000 x^6\right )}{(3+5 x)^2}-164682 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5468750} \]
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Time = 1.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.46
method | result | size |
risch | \(-\frac {31500000 x^{7}+14400000 x^{6}-47178000 x^{5}-6227600 x^{4}+29885470 x^{3}+4409670 x^{2}-5148899 x -871208}{1093750 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {11763 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(71\) |
pseudoelliptic | \(\frac {-164682 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}+5 \sqrt {1-2 x}\, \left (15750000 x^{6}+15075000 x^{5}-16051500 x^{4}-11139550 x^{3}+9372960 x^{2}+6891315 x +871208\right )}{5468750 \left (3+5 x \right )^{2}}\) | \(75\) |
trager | \(\frac {\left (15750000 x^{6}+15075000 x^{5}-16051500 x^{4}-11139550 x^{3}+9372960 x^{2}+6891315 x +871208\right ) \sqrt {1-2 x}}{1093750 \left (3+5 x \right )^{2}}+\frac {11763 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{781250}\) | \(92\) |
derivativedivides | \(\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{250}-\frac {1107 \left (1-2 x \right )^{\frac {7}{2}}}{8750}+\frac {108 \left (1-2 x \right )^{\frac {5}{2}}}{15625}+\frac {76 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {2404 \sqrt {1-2 x}}{15625}+\frac {\frac {561 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {31097 \sqrt {1-2 x}}{78125}}{\left (-6-10 x \right )^{2}}-\frac {11763 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(93\) |
default | \(\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{250}-\frac {1107 \left (1-2 x \right )^{\frac {7}{2}}}{8750}+\frac {108 \left (1-2 x \right )^{\frac {5}{2}}}{15625}+\frac {76 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {2404 \sqrt {1-2 x}}{15625}+\frac {\frac {561 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {31097 \sqrt {1-2 x}}{78125}}{\left (-6-10 x \right )^{2}}-\frac {11763 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(93\) |
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Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {82341 \, \sqrt {11} \sqrt {5} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (15750000 \, x^{6} + 15075000 \, x^{5} - 16051500 \, x^{4} - 11139550 \, x^{3} + 9372960 \, x^{2} + 6891315 \, x + 871208\right )} \sqrt {-2 \, x + 1}}{5468750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
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Time = 164.62 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.51 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {9 \left (1 - 2 x\right )^{\frac {9}{2}}}{250} - \frac {1107 \left (1 - 2 x\right )^{\frac {7}{2}}}{8750} + \frac {108 \left (1 - 2 x\right )^{\frac {5}{2}}}{15625} + \frac {76 \left (1 - 2 x\right )^{\frac {3}{2}}}{3125} + \frac {2404 \sqrt {1 - 2 x}}{15625} + \frac {5754 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{390625} - \frac {60984 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{78125} + \frac {10648 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{78125} \]
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Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {9}{250} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {1107}{8750} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {108}{15625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {76}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11763}{781250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2404}{15625} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (1275 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2827 \, \sqrt {-2 \, x + 1}\right )}}{78125 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {9}{250} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {1107}{8750} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {108}{15625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {76}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11763}{781250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2404}{15625} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (1275 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2827 \, \sqrt {-2 \, x + 1}\right )}}{312500 \, {\left (5 \, x + 3\right )}^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {2404\,\sqrt {1-2\,x}}{15625}+\frac {76\,{\left (1-2\,x\right )}^{3/2}}{3125}+\frac {108\,{\left (1-2\,x\right )}^{5/2}}{15625}-\frac {1107\,{\left (1-2\,x\right )}^{7/2}}{8750}+\frac {9\,{\left (1-2\,x\right )}^{9/2}}{250}-\frac {\frac {31097\,\sqrt {1-2\,x}}{1953125}-\frac {561\,{\left (1-2\,x\right )}^{3/2}}{78125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,11763{}\mathrm {i}}{390625} \]
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