Integrand size = 22, antiderivative size = 148 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}+\frac {205 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \]
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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 = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 44, 65, 212} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\frac {205 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}}+\frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}-\frac {41 (1-2 x)^{5/2}}{378 (3 x+2)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac {205 \sqrt {1-2 x}}{444528 (3 x+2)}+\frac {205 \sqrt {1-2 x}}{190512 (3 x+2)^2}-\frac {205 \sqrt {1-2 x}}{13608 (3 x+2)^3} \]
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Rule 43
Rule 44
Rule 65
Rule 79
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}+\frac {205}{126} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6} \, dx \\ & = \frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}-\frac {205}{378} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5} \, dx \\ & = \frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac {205 \int \frac {\sqrt {1-2 x}}{(2+3 x)^4} \, dx}{1512} \\ & = \frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}-\frac {205 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{13608} \\ & = \frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}-\frac {205 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{63504} \\ & = \frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}-\frac {205 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{444528} \\ & = \frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}+\frac {205 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{444528} \\ & = \frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}+\frac {205 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-51904+154312 x-176850 x^2-824526 x^3+204795 x^4+49815 x^5\right )}{2 (2+3 x)^6}+205 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4667544} \]
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Time = 3.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45
method | result | size |
risch | \(-\frac {99630 x^{6}+359775 x^{5}-1853847 x^{4}+470826 x^{3}+485474 x^{2}-258120 x +51904}{444528 \left (2+3 x \right )^{6} \sqrt {1-2 x}}+\frac {205 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\) | \(66\) |
pseudoelliptic | \(\frac {410 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{6} \sqrt {21}+21 \sqrt {1-2 x}\, \left (49815 x^{5}+204795 x^{4}-824526 x^{3}-176850 x^{2}+154312 x -51904\right )}{9335088 \left (2+3 x \right )^{6}}\) | \(70\) |
derivativedivides | \(-\frac {46656 \left (\frac {205 \left (1-2 x \right )^{\frac {11}{2}}}{42674688}-\frac {3485 \left (1-2 x \right )^{\frac {9}{2}}}{54867456}-\frac {439 \left (1-2 x \right )^{\frac {7}{2}}}{3919104}+\frac {451 \left (1-2 x \right )^{\frac {5}{2}}}{559872}-\frac {24395 \left (1-2 x \right )^{\frac {3}{2}}}{30233088}+\frac {10045 \sqrt {1-2 x}}{30233088}\right )}{\left (-4-6 x \right )^{6}}+\frac {205 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\) | \(84\) |
default | \(-\frac {46656 \left (\frac {205 \left (1-2 x \right )^{\frac {11}{2}}}{42674688}-\frac {3485 \left (1-2 x \right )^{\frac {9}{2}}}{54867456}-\frac {439 \left (1-2 x \right )^{\frac {7}{2}}}{3919104}+\frac {451 \left (1-2 x \right )^{\frac {5}{2}}}{559872}-\frac {24395 \left (1-2 x \right )^{\frac {3}{2}}}{30233088}+\frac {10045 \sqrt {1-2 x}}{30233088}\right )}{\left (-4-6 x \right )^{6}}+\frac {205 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\) | \(84\) |
trager | \(\frac {\left (49815 x^{5}+204795 x^{4}-824526 x^{3}-176850 x^{2}+154312 x -51904\right ) \sqrt {1-2 x}}{444528 \left (2+3 x \right )^{6}}-\frac {205 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{9335088}\) | \(87\) |
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Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\frac {205 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (49815 \, x^{5} + 204795 \, x^{4} - 824526 \, x^{3} - 176850 \, x^{2} + 154312 \, x - 51904\right )} \sqrt {-2 \, x + 1}}{9335088 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=-\frac {205}{9335088} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {49815 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 658665 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 1161594 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 8353422 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 8367485 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 3445435 \, \sqrt {-2 \, x + 1}}{222264 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]
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Time = 0.35 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=-\frac {205}{9335088} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {49815 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 658665 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 1161594 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 8353422 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 8367485 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3445435 \, \sqrt {-2 \, x + 1}}{14224896 \, {\left (3 \, x + 2\right )}^{6}} \]
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Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\frac {205\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4667544}-\frac {\frac {10045\,\sqrt {1-2\,x}}{472392}-\frac {24395\,{\left (1-2\,x\right )}^{3/2}}{472392}+\frac {451\,{\left (1-2\,x\right )}^{5/2}}{8748}-\frac {439\,{\left (1-2\,x\right )}^{7/2}}{61236}-\frac {3485\,{\left (1-2\,x\right )}^{9/2}}{857304}+\frac {205\,{\left (1-2\,x\right )}^{11/2}}{666792}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]
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