Integrand size = 24, antiderivative size = 107 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (18016+26075 x)}{3969 (2+3 x)}-\frac {92996 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}} \]
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Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {99, 154, 151, 65, 212} \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=-\frac {92996 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}}-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{189 (3 x+2)^2}+\frac {2 \sqrt {1-2 x} (26075 x+18016)}{3969 (3 x+2)} \]
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Rule 65
Rule 99
Rule 151
Rule 154
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {(12-35 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx \\ & = -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {1}{378} \int \frac {(544-2980 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^2} \, dx \\ & = -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (18016+26075 x)}{3969 (2+3 x)}+\frac {46498 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{3969} \\ & = -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (18016+26075 x)}{3969 (2+3 x)}-\frac {46498 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3969} \\ & = -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{189 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (18016+26075 x)}{3969 (2+3 x)}-\frac {92996 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (112187+484618 x+695043 x^2+330750 x^3\right )}{(2+3 x)^3}-92996 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{83349} \]
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Time = 0.98 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {661500 x^{4}+1059336 x^{3}+274193 x^{2}-260244 x -112187}{3969 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {92996 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\) | \(56\) |
pseudoelliptic | \(\frac {-92996 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}+21 \sqrt {1-2 x}\, \left (330750 x^{3}+695043 x^{2}+484618 x +112187\right )}{83349 \left (2+3 x \right )^{3}}\) | \(60\) |
derivativedivides | \(\frac {250 \sqrt {1-2 x}}{81}+\frac {-\frac {7454 \left (1-2 x \right )^{\frac {5}{2}}}{441}+\frac {44092 \left (1-2 x \right )^{\frac {3}{2}}}{567}-\frac {7246 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{3}}-\frac {92996 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\) | \(66\) |
default | \(\frac {250 \sqrt {1-2 x}}{81}+\frac {-\frac {7454 \left (1-2 x \right )^{\frac {5}{2}}}{441}+\frac {44092 \left (1-2 x \right )^{\frac {3}{2}}}{567}-\frac {7246 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{3}}-\frac {92996 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\) | \(66\) |
trager | \(\frac {\left (330750 x^{3}+695043 x^{2}+484618 x +112187\right ) \sqrt {1-2 x}}{3969 \left (2+3 x \right )^{3}}+\frac {46498 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{83349}\) | \(77\) |
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Time = 0.23 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {46498 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (330750 \, x^{3} + 695043 \, x^{2} + 484618 \, x + 112187\right )} \sqrt {-2 \, x + 1}}{83349 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {46498}{83349} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {250}{81} \, \sqrt {-2 \, x + 1} + \frac {2 \, {\left (33543 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 154322 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 177527 \, \sqrt {-2 \, x + 1}\right )}}{3969 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {46498}{83349} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {250}{81} \, \sqrt {-2 \, x + 1} + \frac {33543 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 154322 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 177527 \, \sqrt {-2 \, x + 1}}{15876 \, {\left (3 \, x + 2\right )}^{3}} \]
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Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {250\,\sqrt {1-2\,x}}{81}-\frac {92996\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{83349}+\frac {\frac {7246\,\sqrt {1-2\,x}}{2187}-\frac {44092\,{\left (1-2\,x\right )}^{3/2}}{15309}+\frac {7454\,{\left (1-2\,x\right )}^{5/2}}{11907}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]
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