Integrand size = 24, antiderivative size = 102 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {350}{81} \sqrt {1-2 x}-\frac {50}{81} (1-2 x)^{3/2}-\frac {10}{63} (1-2 x)^{5/2}-\frac {25}{63} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{63 (2+3 x)}+\frac {350}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 81, 52, 65, 212} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^2} \, dx=\frac {350}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {(1-2 x)^{7/2}}{63 (3 x+2)}-\frac {25}{63} (1-2 x)^{7/2}-\frac {10}{63} (1-2 x)^{5/2}-\frac {50}{81} (1-2 x)^{3/2}-\frac {350}{81} \sqrt {1-2 x} \]
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{7/2}}{63 (2+3 x)}+\frac {1}{63} \int \frac {(1-2 x)^{5/2} (275+525 x)}{2+3 x} \, dx \\ & = -\frac {25}{63} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{63 (2+3 x)}-\frac {25}{21} \int \frac {(1-2 x)^{5/2}}{2+3 x} \, dx \\ & = -\frac {10}{63} (1-2 x)^{5/2}-\frac {25}{63} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{63 (2+3 x)}-\frac {25}{9} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx \\ & = -\frac {50}{81} (1-2 x)^{3/2}-\frac {10}{63} (1-2 x)^{5/2}-\frac {25}{63} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{63 (2+3 x)}-\frac {175}{27} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx \\ & = -\frac {350}{81} \sqrt {1-2 x}-\frac {50}{81} (1-2 x)^{3/2}-\frac {10}{63} (1-2 x)^{5/2}-\frac {25}{63} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{63 (2+3 x)}-\frac {1225}{81} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx \\ & = -\frac {350}{81} \sqrt {1-2 x}-\frac {50}{81} (1-2 x)^{3/2}-\frac {10}{63} (1-2 x)^{5/2}-\frac {25}{63} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{63 (2+3 x)}+\frac {1225}{81} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {350}{81} \sqrt {1-2 x}-\frac {50}{81} (1-2 x)^{3/2}-\frac {10}{63} (1-2 x)^{5/2}-\frac {25}{63} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{63 (2+3 x)}+\frac {350}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^2} \, dx=\frac {\sqrt {1-2 x} \left (-6239-4471 x+1002 x^2-5508 x^3+5400 x^4\right )}{567 (2+3 x)}+\frac {350}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Time = 1.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.60
method | result | size |
risch | \(-\frac {10800 x^{5}-16416 x^{4}+7512 x^{3}-9944 x^{2}-8007 x +6239}{567 \left (2+3 x \right ) \sqrt {1-2 x}}+\frac {350 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(61\) |
pseudoelliptic | \(\frac {2450 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}+3 \sqrt {1-2 x}\, \left (5400 x^{4}-5508 x^{3}+1002 x^{2}-4471 x -6239\right )}{3402+5103 x}\) | \(62\) |
derivativedivides | \(-\frac {25 \left (1-2 x \right )^{\frac {7}{2}}}{63}-\frac {4 \left (1-2 x \right )^{\frac {5}{2}}}{27}-\frac {16 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {1036 \sqrt {1-2 x}}{243}+\frac {98 \sqrt {1-2 x}}{729 \left (-\frac {4}{3}-2 x \right )}+\frac {350 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(72\) |
default | \(-\frac {25 \left (1-2 x \right )^{\frac {7}{2}}}{63}-\frac {4 \left (1-2 x \right )^{\frac {5}{2}}}{27}-\frac {16 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {1036 \sqrt {1-2 x}}{243}+\frac {98 \sqrt {1-2 x}}{729 \left (-\frac {4}{3}-2 x \right )}+\frac {350 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(72\) |
trager | \(\frac {\sqrt {1-2 x}\, \left (5400 x^{4}-5508 x^{3}+1002 x^{2}-4471 x -6239\right )}{1134+1701 x}+\frac {175 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{243}\) | \(82\) |
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Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^2} \, dx=\frac {1225 \, \sqrt {7} \sqrt {3} {\left (3 \, x + 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 3 \, {\left (5400 \, x^{4} - 5508 \, x^{3} + 1002 \, x^{2} - 4471 \, x - 6239\right )} \sqrt {-2 \, x + 1}}{1701 \, {\left (3 \, x + 2\right )}} \]
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Time = 31.61 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.07 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^2} \, dx=- \frac {25 \left (1 - 2 x\right )^{\frac {7}{2}}}{63} - \frac {4 \left (1 - 2 x\right )^{\frac {5}{2}}}{27} - \frac {16 \left (1 - 2 x\right )^{\frac {3}{2}}}{27} - \frac {1036 \sqrt {1 - 2 x}}{243} - \frac {532 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{729} - \frac {1372 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{243} \]
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Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {25}{63} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {4}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {16}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {175}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1036}{243} \, \sqrt {-2 \, x + 1} - \frac {49 \, \sqrt {-2 \, x + 1}}{243 \, {\left (3 \, x + 2\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^2} \, dx=\frac {25}{63} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {4}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {16}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {175}{243} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {1036}{243} \, \sqrt {-2 \, x + 1} - \frac {49 \, \sqrt {-2 \, x + 1}}{243 \, {\left (3 \, x + 2\right )}} \]
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Time = 1.60 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^2} \, dx=-\frac {98\,\sqrt {1-2\,x}}{729\,\left (2\,x+\frac {4}{3}\right )}-\frac {1036\,\sqrt {1-2\,x}}{243}-\frac {16\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {4\,{\left (1-2\,x\right )}^{5/2}}{27}-\frac {25\,{\left (1-2\,x\right )}^{7/2}}{63}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,350{}\mathrm {i}}{243} \]
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