Integrand size = 24, antiderivative size = 140 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {1404 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {9 \sqrt {1-2 x} (32+1375 x)}{31250}-\frac {12279 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}} \]
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Time = 0.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {99, 154, 158, 152, 65, 212} \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {12279 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}}-\frac {129 \sqrt {1-2 x} (3 x+2)^4}{50 (5 x+3)}-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac {2643 \sqrt {1-2 x} (3 x+2)^3}{1750}+\frac {1404 \sqrt {1-2 x} (3 x+2)^2}{3125}+\frac {9 \sqrt {1-2 x} (1375 x+32)}{31250} \]
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Rule 65
Rule 99
Rule 152
Rule 154
Rule 158
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}+\frac {1}{10} \int \frac {(6-33 x) \sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^2} \, dx \\ & = -\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {1}{50} \int \frac {(870-2643 x) (2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}-\frac {\int \frac {(2+3 x)^2 (-5397+19656 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{1750} \\ & = \frac {1404 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {\int \frac {(33978-86625 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{43750} \\ & = \frac {1404 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {9 \sqrt {1-2 x} (32+1375 x)}{31250}+\frac {12279 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{31250} \\ & = \frac {1404 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {9 \sqrt {1-2 x} (32+1375 x)}{31250}-\frac {12279 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{31250} \\ & = \frac {1404 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {9 \sqrt {1-2 x} (32+1375 x)}{31250}-\frac {12279 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {-\frac {55 \sqrt {1-2 x} \left (96776-489445 x-2120880 x^2-496350 x^3+3267000 x^4+2025000 x^5\right )}{(3+5 x)^2}-171906 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{12031250} \]
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Time = 1.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.47
method | result | size |
risch | \(\frac {4050000 x^{6}+4509000 x^{5}-4259700 x^{4}-3745410 x^{3}+1141990 x^{2}+682997 x -96776}{218750 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {12279 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\) | \(66\) |
pseudoelliptic | \(\frac {-171906 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}-55 \sqrt {1-2 x}\, \left (2025000 x^{5}+3267000 x^{4}-496350 x^{3}-2120880 x^{2}-489445 x +96776\right )}{12031250 \left (3+5 x \right )^{2}}\) | \(70\) |
derivativedivides | \(\frac {81 \left (1-2 x \right )^{\frac {7}{2}}}{1750}-\frac {1107 \left (1-2 x \right )^{\frac {5}{2}}}{6250}+\frac {36 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {228 \sqrt {1-2 x}}{3125}+\frac {\frac {259 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {2871 \sqrt {1-2 x}}{15625}}{\left (-6-10 x \right )^{2}}-\frac {12279 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\) | \(84\) |
default | \(\frac {81 \left (1-2 x \right )^{\frac {7}{2}}}{1750}-\frac {1107 \left (1-2 x \right )^{\frac {5}{2}}}{6250}+\frac {36 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {228 \sqrt {1-2 x}}{3125}+\frac {\frac {259 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {2871 \sqrt {1-2 x}}{15625}}{\left (-6-10 x \right )^{2}}-\frac {12279 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\) | \(84\) |
trager | \(-\frac {\left (2025000 x^{5}+3267000 x^{4}-496350 x^{3}-2120880 x^{2}-489445 x +96776\right ) \sqrt {1-2 x}}{218750 \left (3+5 x \right )^{2}}+\frac {12279 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1718750}\) | \(87\) |
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Time = 0.23 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {85953 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (2025000 \, x^{5} + 3267000 \, x^{4} - 496350 \, x^{3} - 2120880 \, x^{2} - 489445 \, x + 96776\right )} \sqrt {-2 \, x + 1}}{12031250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
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Time = 166.13 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.69 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {81 \left (1 - 2 x\right )^{\frac {7}{2}}}{1750} - \frac {1107 \left (1 - 2 x\right )^{\frac {5}{2}}}{6250} + \frac {36 \left (1 - 2 x\right )^{\frac {3}{2}}}{3125} + \frac {228 \sqrt {1 - 2 x}}{3125} + \frac {1202 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{171875} - \frac {5632 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15625} + \frac {968 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15625} \]
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Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {81}{1750} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1107}{6250} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {36}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {12279}{1718750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {228}{3125} \, \sqrt {-2 \, x + 1} + \frac {1295 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2871 \, \sqrt {-2 \, x + 1}}{15625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=-\frac {81}{1750} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1107}{6250} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {36}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {12279}{1718750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {228}{3125} \, \sqrt {-2 \, x + 1} + \frac {1295 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2871 \, \sqrt {-2 \, x + 1}}{62500 \, {\left (5 \, x + 3\right )}^{2}} \]
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Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {228\,\sqrt {1-2\,x}}{3125}+\frac {36\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {1107\,{\left (1-2\,x\right )}^{5/2}}{6250}+\frac {81\,{\left (1-2\,x\right )}^{7/2}}{1750}-\frac {\frac {2871\,\sqrt {1-2\,x}}{390625}-\frac {259\,{\left (1-2\,x\right )}^{3/2}}{78125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,12279{}\mathrm {i}}{859375} \]
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