Integrand size = 24, antiderivative size = 97 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx=-\frac {10 \sqrt {1-2 x}}{3+5 x}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}-138 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+134 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 97, 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 = {101, 156, 162, 65, 212} \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx=-138 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+134 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-\frac {10 \sqrt {1-2 x}}{5 x+3}+\frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)} \]
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Rule 65
Rule 101
Rule 156
Rule 162
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}-\int \frac {-13+15 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx \\ & = -\frac {10 \sqrt {1-2 x}}{3+5 x}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}+\frac {1}{11} \int \frac {-539+330 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx \\ & = -\frac {10 \sqrt {1-2 x}}{3+5 x}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}+207 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-335 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {10 \sqrt {1-2 x}}{3+5 x}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}-207 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+335 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {10 \sqrt {1-2 x}}{3+5 x}+\frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)}-138 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+134 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} (19+30 x)}{6+19 x+15 x^2}-138 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+134 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 3.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70
method | result | size |
risch | \(\frac {\left (-1+2 x \right ) \left (19+30 x \right )}{\left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}-\frac {138 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}+\frac {134 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}\) | \(68\) |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+\frac {134 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {2 \sqrt {1-2 x}}{-\frac {4}{3}-2 x}-\frac {138 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(70\) |
default | \(\frac {2 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+\frac {134 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {2 \sqrt {1-2 x}}{-\frac {4}{3}-2 x}-\frac {138 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(70\) |
pseudoelliptic | \(\frac {-1518 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (15 x^{2}+19 x +6\right ) \sqrt {21}+938 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (15 x^{2}+19 x +6\right ) \sqrt {55}-77 \sqrt {1-2 x}\, \left (19+30 x \right )}{1155 x^{2}+1463 x +462}\) | \(85\) |
trager | \(-\frac {\left (19+30 x \right ) \sqrt {1-2 x}}{15 x^{2}+19 x +6}+\frac {67 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{11}-\frac {69 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{7}\) | \(116\) |
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Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx=\frac {469 \, \sqrt {11} \sqrt {5} {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 759 \, \sqrt {7} \sqrt {3} {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (30 \, x + 19\right )} \sqrt {-2 \, x + 1}}{77 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \]
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Time = 28.06 (sec) , antiderivative size = 321, normalized size of antiderivative = 3.31 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx=\frac {68 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{7} - \frac {68 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{11} - 84 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 220 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx=-\frac {67}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {69}{7} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (15 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 34 \, \sqrt {-2 \, x + 1}\right )}}{15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \]
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Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx=-\frac {67}{11} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {69}{7} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (15 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 34 \, \sqrt {-2 \, x + 1}\right )}}{15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9} \]
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Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 (3+5 x)^2} \, dx=\frac {134\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}-\frac {138\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {\frac {136\,\sqrt {1-2\,x}}{15}-4\,{\left (1-2\,x\right )}^{3/2}}{\frac {136\,x}{15}+{\left (2\,x-1\right )}^2+\frac {3}{5}} \]
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