Integrand size = 24, antiderivative size = 127 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {11237 \sqrt {1-2 x}}{111132 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac {\sqrt {1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}+\frac {11237 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{55566 \sqrt {21}} \]
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Time = 0.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {99, 154, 150, 44, 65, 212} \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {11237 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{55566 \sqrt {21}}-\frac {\sqrt {1-2 x} (5 x+3)^3}{15 (3 x+2)^5}-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{630 (3 x+2)^4}-\frac {\sqrt {1-2 x} (59665 x+37224)}{79380 (3 x+2)^3}+\frac {11237 \sqrt {1-2 x}}{111132 (3 x+2)} \]
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Rule 44
Rule 65
Rule 99
Rule 150
Rule 154
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {(12-35 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx \\ & = -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}+\frac {\int \frac {(346-3310 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx}{1260} \\ & = -\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac {\sqrt {1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}-\frac {11237 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{15876} \\ & = \frac {11237 \sqrt {1-2 x}}{111132 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac {\sqrt {1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}-\frac {11237 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{111132} \\ & = \frac {11237 \sqrt {1-2 x}}{111132 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac {\sqrt {1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}+\frac {11237 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{111132} \\ & = \frac {11237 \sqrt {1-2 x}}{111132 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac {\sqrt {1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}+\frac {11237 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{55566 \sqrt {21}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-1984928-8471518 x-10100352 x^2+240615 x^3+4550985 x^4\right )}{2 (2+3 x)^5}+56185 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5834430} \]
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Time = 2.99 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48
method | result | size |
risch | \(-\frac {9101970 x^{5}-4069755 x^{4}-20441319 x^{3}-6842684 x^{2}+4501662 x +1984928}{555660 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {11237 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1166886}\) | \(61\) |
pseudoelliptic | \(\frac {112370 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}+21 \sqrt {1-2 x}\, \left (4550985 x^{4}+240615 x^{3}-10100352 x^{2}-8471518 x -1984928\right )}{11668860 \left (2+3 x \right )^{5}}\) | \(65\) |
derivativedivides | \(\frac {-\frac {11237 \left (1-2 x \right )^{\frac {9}{2}}}{686}+\frac {4237 \left (1-2 x \right )^{\frac {7}{2}}}{63}+\frac {39632 \left (1-2 x \right )^{\frac {5}{2}}}{945}-\frac {263117 \left (1-2 x \right )^{\frac {3}{2}}}{567}+\frac {78659 \sqrt {1-2 x}}{162}}{\left (-4-6 x \right )^{5}}+\frac {11237 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1166886}\) | \(75\) |
default | \(\frac {-\frac {11237 \left (1-2 x \right )^{\frac {9}{2}}}{686}+\frac {4237 \left (1-2 x \right )^{\frac {7}{2}}}{63}+\frac {39632 \left (1-2 x \right )^{\frac {5}{2}}}{945}-\frac {263117 \left (1-2 x \right )^{\frac {3}{2}}}{567}+\frac {78659 \sqrt {1-2 x}}{162}}{\left (-4-6 x \right )^{5}}+\frac {11237 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1166886}\) | \(75\) |
trager | \(\frac {\left (4550985 x^{4}+240615 x^{3}-10100352 x^{2}-8471518 x -1984928\right ) \sqrt {1-2 x}}{555660 \left (2+3 x \right )^{5}}+\frac {11237 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2333772}\) | \(82\) |
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Time = 0.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {56185 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (4550985 \, x^{4} + 240615 \, x^{3} - 10100352 \, x^{2} - 8471518 \, x - 1984928\right )} \sqrt {-2 \, x + 1}}{11668860 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=-\frac {11237}{2333772} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4550985 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 18685170 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 11651808 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 128927330 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 134900185 \, \sqrt {-2 \, x + 1}}{277830 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=-\frac {11237}{2333772} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4550985 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 18685170 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 11651808 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 128927330 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 134900185 \, \sqrt {-2 \, x + 1}}{8890560 \, {\left (3 \, x + 2\right )}^{5}} \]
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Time = 1.56 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {11237\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1166886}-\frac {\frac {78659\,\sqrt {1-2\,x}}{39366}-\frac {263117\,{\left (1-2\,x\right )}^{3/2}}{137781}+\frac {39632\,{\left (1-2\,x\right )}^{5/2}}{229635}+\frac {4237\,{\left (1-2\,x\right )}^{7/2}}{15309}-\frac {11237\,{\left (1-2\,x\right )}^{9/2}}{166698}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
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