Integrand size = 24, antiderivative size = 148 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac {1315 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {1315 \sqrt {1-2 x}}{148176 (2+3 x)}+\frac {1315 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{74088 \sqrt {21}} \]
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Time = 0.03 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {91, 79, 43, 44, 65, 212} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {1315 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{74088 \sqrt {21}}+\frac {83 (1-2 x)^{7/2}}{2646 (3 x+2)^5}-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}-\frac {263 (1-2 x)^{5/2}}{1176 (3 x+2)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (3 x+2)^3}+\frac {1315 \sqrt {1-2 x}}{148176 (3 x+2)}-\frac {1315 \sqrt {1-2 x}}{21168 (3 x+2)^2} \]
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Rule 43
Rule 44
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {1}{378} \int \frac {(1-2 x)^{5/2} (1685+3150 x)}{(2+3 x)^6} \, dx \\ & = -\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}+\frac {263}{98} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5} \, dx \\ & = -\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}-\frac {1315 \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4} \, dx}{1176} \\ & = -\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}+\frac {1315 \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx}{3528} \\ & = -\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac {1315 \sqrt {1-2 x}}{21168 (2+3 x)^2}-\frac {1315 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{21168} \\ & = -\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac {1315 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {1315 \sqrt {1-2 x}}{148176 (2+3 x)}-\frac {1315 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{148176} \\ & = -\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac {1315 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {1315 \sqrt {1-2 x}}{148176 (2+3 x)}+\frac {1315 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{148176} \\ & = -\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac {1315 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {1315 \sqrt {1-2 x}}{148176 (2+3 x)}+\frac {1315 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{74088 \sqrt {21}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-81568-106808 x-587502 x^2-2360850 x^3-1979115 x^4+319545 x^5\right )}{2 (2+3 x)^6}+1315 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1555848} \]
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Time = 3.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45
method | result | size |
risch | \(-\frac {639090 x^{6}-4277775 x^{5}-2742585 x^{4}+1185846 x^{3}+373886 x^{2}-56328 x +81568}{148176 \left (2+3 x \right )^{6} \sqrt {1-2 x}}+\frac {1315 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1555848}\) | \(66\) |
pseudoelliptic | \(\frac {2630 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{6} \sqrt {21}+21 \sqrt {1-2 x}\, \left (319545 x^{5}-1979115 x^{4}-2360850 x^{3}-587502 x^{2}-106808 x -81568\right )}{3111696 \left (2+3 x \right )^{6}}\) | \(70\) |
derivativedivides | \(\frac {-\frac {11835 \left (1-2 x \right )^{\frac {11}{2}}}{2744}-\frac {112405 \left (1-2 x \right )^{\frac {9}{2}}}{3528}+\frac {8345 \left (1-2 x \right )^{\frac {7}{2}}}{28}-\frac {2893 \left (1-2 x \right )^{\frac {5}{2}}}{4}+\frac {156485 \left (1-2 x \right )^{\frac {3}{2}}}{216}-\frac {64435 \sqrt {1-2 x}}{216}}{\left (-4-6 x \right )^{6}}+\frac {1315 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1555848}\) | \(84\) |
default | \(\frac {-\frac {11835 \left (1-2 x \right )^{\frac {11}{2}}}{2744}-\frac {112405 \left (1-2 x \right )^{\frac {9}{2}}}{3528}+\frac {8345 \left (1-2 x \right )^{\frac {7}{2}}}{28}-\frac {2893 \left (1-2 x \right )^{\frac {5}{2}}}{4}+\frac {156485 \left (1-2 x \right )^{\frac {3}{2}}}{216}-\frac {64435 \sqrt {1-2 x}}{216}}{\left (-4-6 x \right )^{6}}+\frac {1315 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1555848}\) | \(84\) |
trager | \(\frac {\left (319545 x^{5}-1979115 x^{4}-2360850 x^{3}-587502 x^{2}-106808 x -81568\right ) \sqrt {1-2 x}}{148176 \left (2+3 x \right )^{6}}+\frac {1315 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{3111696}\) | \(87\) |
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Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {1315 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (319545 \, x^{5} - 1979115 \, x^{4} - 2360850 \, x^{3} - 587502 \, x^{2} - 106808 \, x - 81568\right )} \sqrt {-2 \, x + 1}}{3111696 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {1315}{3111696} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {319545 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 2360505 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 22080870 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 53584146 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 53674355 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 22101205 \, \sqrt {-2 \, x + 1}}{74088 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {1315}{3111696} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {319545 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - 2360505 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 22080870 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 53584146 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 53674355 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 22101205 \, \sqrt {-2 \, x + 1}}{4741632 \, {\left (3 \, x + 2\right )}^{6}} \]
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Time = 1.72 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {1315\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1555848}-\frac {\frac {64435\,\sqrt {1-2\,x}}{157464}-\frac {156485\,{\left (1-2\,x\right )}^{3/2}}{157464}+\frac {2893\,{\left (1-2\,x\right )}^{5/2}}{2916}-\frac {8345\,{\left (1-2\,x\right )}^{7/2}}{20412}+\frac {112405\,{\left (1-2\,x\right )}^{9/2}}{2571912}+\frac {1315\,{\left (1-2\,x\right )}^{11/2}}{222264}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]
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