Integrand size = 24, antiderivative size = 147 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}-\frac {33873 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375 \sqrt {55}} \]
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Time = 0.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 154, 158, 152, 65, 212} \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=-\frac {33873 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375 \sqrt {55}}+\frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)^2}-\frac {71 \sqrt {1-2 x} (3 x+2)^4}{1210 (5 x+3)^2}-\frac {2721 \sqrt {1-2 x} (3 x+2)^3}{66550 (5 x+3)}+\frac {377748 \sqrt {1-2 x} (3 x+2)^2}{831875}+\frac {63 \sqrt {1-2 x} (831375 x+2492512)}{8318750} \]
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Rule 65
Rule 100
Rule 152
Rule 154
Rule 158
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {1}{11} \int \frac {(2+3 x)^4 (173+312 x)}{\sqrt {1-2 x} (3+5 x)^3} \, dx \\ & = -\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\int \frac {(2+3 x)^3 (12450+21657 x)}{\sqrt {1-2 x} (3+5 x)^2} \, dx}{1210} \\ & = -\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}-\frac {\int \frac {(2+3 x)^2 (446523+755496 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{66550} \\ & = \frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {\int \frac {(-31392102-52376625 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{1663750} \\ & = \frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}+\frac {33873 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{8318750} \\ & = \frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}-\frac {33873 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{8318750} \\ & = \frac {377748 \sqrt {1-2 x} (2+3 x)^2}{831875}-\frac {71 \sqrt {1-2 x} (2+3 x)^4}{1210 (3+5 x)^2}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {2721 \sqrt {1-2 x} (2+3 x)^3}{66550 (3+5 x)}+\frac {63 \sqrt {1-2 x} (2492512+831375 x)}{8318750}-\frac {33873 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{4159375 \sqrt {55}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.50 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {-\frac {55 \left (-1702670584-4150263077 x+762244410 x^2+5682717810 x^3+1423105200 x^4+242574750 x^5\right )}{\sqrt {1-2 x} (3+5 x)^2}-67746 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{457531250} \]
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Time = 1.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.41
method | result | size |
risch | \(-\frac {242574750 x^{5}+1423105200 x^{4}+5682717810 x^{3}+762244410 x^{2}-4150263077 x -1702670584}{8318750 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {33873 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{228765625}\) | \(61\) |
pseudoelliptic | \(-\frac {729 \left (\frac {11291 \sqrt {55}\, \left (x +\frac {3}{5}\right )^{2} \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{88944075}+x^{5}+\frac {88 x^{4}}{15}+\frac {1757 x^{3}}{75}+\frac {25408147 x^{2}}{8085825}-\frac {4150263077 x}{242574750}-\frac {851335292}{121287375}\right )}{25 \sqrt {1-2 x}\, \left (3+5 x \right )^{2}}\) | \(70\) |
derivativedivides | \(\frac {729 \left (1-2 x \right )^{\frac {5}{2}}}{5000}-\frac {8991 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {333639 \sqrt {1-2 x}}{25000}+\frac {\frac {403 \left (1-2 x \right )^{\frac {3}{2}}}{831875}-\frac {81 \sqrt {1-2 x}}{75625}}{\left (-6-10 x \right )^{2}}-\frac {33873 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{228765625}+\frac {117649}{10648 \sqrt {1-2 x}}\) | \(84\) |
default | \(\frac {729 \left (1-2 x \right )^{\frac {5}{2}}}{5000}-\frac {8991 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {333639 \sqrt {1-2 x}}{25000}+\frac {\frac {403 \left (1-2 x \right )^{\frac {3}{2}}}{831875}-\frac {81 \sqrt {1-2 x}}{75625}}{\left (-6-10 x \right )^{2}}-\frac {33873 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{228765625}+\frac {117649}{10648 \sqrt {1-2 x}}\) | \(84\) |
trager | \(\frac {\left (242574750 x^{5}+1423105200 x^{4}+5682717810 x^{3}+762244410 x^{2}-4150263077 x -1702670584\right ) \sqrt {1-2 x}}{8318750 \left (3+5 x \right )^{2} \left (-1+2 x \right )}-\frac {33873 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{457531250}\) | \(94\) |
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Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {33873 \, \sqrt {55} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (242574750 \, x^{5} + 1423105200 \, x^{4} + 5682717810 \, x^{3} + 762244410 \, x^{2} - 4150263077 \, x - 1702670584\right )} \sqrt {-2 \, x + 1}}{457531250 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
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Timed out. \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {729}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {8991}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {33873}{457531250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {333639}{25000} \, \sqrt {-2 \, x + 1} + \frac {1838268849 \, {\left (2 \, x - 1\right )}^{2} + 16176751756 \, x + 808829747}{6655000 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {729}{5000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {8991}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {33873}{457531250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {333639}{25000} \, \sqrt {-2 \, x + 1} + \frac {117649}{10648 \, \sqrt {-2 \, x + 1}} + \frac {403 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 891 \, \sqrt {-2 \, x + 1}}{3327500 \, {\left (5 \, x + 3\right )}^{2}} \]
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Time = 1.43 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {\frac {367653449\,x}{3781250}+\frac {1838268849\,{\left (2\,x-1\right )}^2}{166375000}+\frac {73529977}{15125000}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}}+\frac {333639\,\sqrt {1-2\,x}}{25000}-\frac {8991\,{\left (1-2\,x\right )}^{3/2}}{5000}+\frac {729\,{\left (1-2\,x\right )}^{5/2}}{5000}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,33873{}\mathrm {i}}{228765625} \]
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