Integrand size = 24, antiderivative size = 133 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=-\frac {172 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {64 \sqrt {1-2 x} (2+3 x)^3}{2625}+\frac {11}{75} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac {4 \sqrt {1-2 x} (10998+3625 x)}{15625}-\frac {328 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}} \]
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Time = 0.04 (sec) , antiderivative size = 133, 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 = {99, 158, 152, 65, 212} \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=-\frac {328 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}}-\frac {\sqrt {1-2 x} (3 x+2)^5}{5 (5 x+3)}+\frac {11}{75} \sqrt {1-2 x} (3 x+2)^4+\frac {64 \sqrt {1-2 x} (3 x+2)^3}{2625}-\frac {172 \sqrt {1-2 x} (3 x+2)^2}{3125}-\frac {4 \sqrt {1-2 x} (3625 x+10998)}{15625} \]
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Rule 65
Rule 99
Rule 152
Rule 158
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-2 x} (2+3 x)^5}{5 (3+5 x)}+\frac {1}{5} \int \frac {(13-33 x) (2+3 x)^4}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {11}{75} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac {1}{225} \int \frac {(2+3 x)^3 (-180+192 x)}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {64 \sqrt {1-2 x} (2+3 x)^3}{2625}+\frac {11}{75} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{5 (3+5 x)}+\frac {\int \frac {(2+3 x)^2 (8568+10836 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{7875} \\ & = -\frac {172 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {64 \sqrt {1-2 x} (2+3 x)^3}{2625}+\frac {11}{75} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac {\int \frac {(-558432-913500 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{196875} \\ & = -\frac {172 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {64 \sqrt {1-2 x} (2+3 x)^3}{2625}+\frac {11}{75} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac {4 \sqrt {1-2 x} (10998+3625 x)}{15625}+\frac {164 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{15625} \\ & = -\frac {172 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {64 \sqrt {1-2 x} (2+3 x)^3}{2625}+\frac {11}{75} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac {4 \sqrt {1-2 x} (10998+3625 x)}{15625}-\frac {164 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15625} \\ & = -\frac {172 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {64 \sqrt {1-2 x} (2+3 x)^3}{2625}+\frac {11}{75} \sqrt {1-2 x} (2+3 x)^4-\frac {\sqrt {1-2 x} (2+3 x)^5}{5 (3+5 x)}-\frac {4 \sqrt {1-2 x} (10998+3625 x)}{15625}-\frac {328 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {\frac {55 \sqrt {1-2 x} \left (-862072-1133340 x+2225760 x^2+4760100 x^3+3864375 x^4+1181250 x^5\right )}{3+5 x}-2296 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6015625} \]
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Time = 0.98 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.50
method | result | size |
risch | \(-\frac {2362500 x^{6}+6547500 x^{5}+5655825 x^{4}-308580 x^{3}-4492440 x^{2}-590804 x +862072}{109375 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {328 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\) | \(66\) |
pseudoelliptic | \(\frac {-2296 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}+55 \sqrt {1-2 x}\, \left (1181250 x^{5}+3864375 x^{4}+4760100 x^{3}+2225760 x^{2}-1133340 x -862072\right )}{18046875+30078125 x}\) | \(67\) |
derivativedivides | \(\frac {27 \left (1-2 x \right )^{\frac {9}{2}}}{200}-\frac {8829 \left (1-2 x \right )^{\frac {7}{2}}}{7000}+\frac {107109 \left (1-2 x \right )^{\frac {5}{2}}}{25000}-\frac {144681 \left (1-2 x \right )^{\frac {3}{2}}}{25000}+\frac {6 \sqrt {1-2 x}}{3125}+\frac {2 \sqrt {1-2 x}}{78125 \left (-\frac {6}{5}-2 x \right )}-\frac {328 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\) | \(81\) |
default | \(\frac {27 \left (1-2 x \right )^{\frac {9}{2}}}{200}-\frac {8829 \left (1-2 x \right )^{\frac {7}{2}}}{7000}+\frac {107109 \left (1-2 x \right )^{\frac {5}{2}}}{25000}-\frac {144681 \left (1-2 x \right )^{\frac {3}{2}}}{25000}+\frac {6 \sqrt {1-2 x}}{3125}+\frac {2 \sqrt {1-2 x}}{78125 \left (-\frac {6}{5}-2 x \right )}-\frac {328 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{859375}\) | \(81\) |
trager | \(\frac {\sqrt {1-2 x}\, \left (1181250 x^{5}+3864375 x^{4}+4760100 x^{3}+2225760 x^{2}-1133340 x -862072\right )}{328125+546875 x}+\frac {164 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{859375}\) | \(87\) |
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Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {1148 \, \sqrt {55} {\left (5 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (1181250 \, x^{5} + 3864375 \, x^{4} + 4760100 \, x^{3} + 2225760 \, x^{2} - 1133340 \, x - 862072\right )} \sqrt {-2 \, x + 1}}{6015625 \, {\left (5 \, x + 3\right )}} \]
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Time = 48.91 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.66 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {27 \left (1 - 2 x\right )^{\frac {9}{2}}}{200} - \frac {8829 \left (1 - 2 x\right )^{\frac {7}{2}}}{7000} + \frac {107109 \left (1 - 2 x\right )^{\frac {5}{2}}}{25000} - \frac {144681 \left (1 - 2 x\right )^{\frac {3}{2}}}{25000} + \frac {6 \sqrt {1 - 2 x}}{3125} + \frac {163 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{859375} - \frac {44 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15625} \]
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Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {27}{200} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {8829}{7000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {107109}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {144681}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {164}{859375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {6}{3125} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {27}{200} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {8829}{7000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {107109}{25000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {144681}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {164}{859375} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {6}{3125} \, \sqrt {-2 \, x + 1} - \frac {\sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {6\,\sqrt {1-2\,x}}{3125}-\frac {2\,\sqrt {1-2\,x}}{78125\,\left (2\,x+\frac {6}{5}\right )}-\frac {144681\,{\left (1-2\,x\right )}^{3/2}}{25000}+\frac {107109\,{\left (1-2\,x\right )}^{5/2}}{25000}-\frac {8829\,{\left (1-2\,x\right )}^{7/2}}{7000}+\frac {27\,{\left (1-2\,x\right )}^{9/2}}{200}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,328{}\mathrm {i}}{859375} \]
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