Integrand size = 24, antiderivative size = 133 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {4313 \sqrt {1-2 x}}{72 (2+3 x)^2}+\frac {100145 \sqrt {1-2 x}}{168 (2+3 x)}+\frac {3454265 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}-1210 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 154, 156, 162, 65, 212} \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {3454265 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}-1210 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}+\frac {100145 \sqrt {1-2 x}}{168 (3 x+2)}+\frac {4313 \sqrt {1-2 x}}{72 (3 x+2)^2}+\frac {301 \sqrt {1-2 x}}{36 (3 x+2)^3} \]
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Rule 65
Rule 100
Rule 154
Rule 156
Rule 162
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {(195-159 x) \sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx \\ & = \frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^3}-\frac {1}{108} \int \frac {-15777+21621 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx \\ & = \frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {4313 \sqrt {1-2 x}}{72 (2+3 x)^2}-\frac {\int \frac {-1197315+1358595 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{1512} \\ & = \frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {4313 \sqrt {1-2 x}}{72 (2+3 x)^2}+\frac {100145 \sqrt {1-2 x}}{168 (2+3 x)}-\frac {\int \frac {-51509115+31545675 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{10584} \\ & = \frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {4313 \sqrt {1-2 x}}{72 (2+3 x)^2}+\frac {100145 \sqrt {1-2 x}}{168 (2+3 x)}-\frac {3454265}{168} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+33275 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {4313 \sqrt {1-2 x}}{72 (2+3 x)^2}+\frac {100145 \sqrt {1-2 x}}{168 (2+3 x)}+\frac {3454265}{168} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-33275 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = \frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^3}+\frac {4313 \sqrt {1-2 x}}{72 (2+3 x)^2}+\frac {100145 \sqrt {1-2 x}}{168 (2+3 x)}+\frac {3454265 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}-1210 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (844322+3730002 x+5498403 x^2+2703915 x^3\right )}{168 (2+3 x)^4}+\frac {3454265 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{84 \sqrt {21}}-1210 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 1.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.56
method | result | size |
risch | \(-\frac {5407830 x^{4}+8292891 x^{3}+1961601 x^{2}-2041358 x -844322}{168 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {3454265 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1764}-1210 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(74\) |
derivativedivides | \(-1210 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {162 \left (\frac {100145 \left (1-2 x \right )^{\frac {7}{2}}}{504}-\frac {909931 \left (1-2 x \right )^{\frac {5}{2}}}{648}+\frac {2144065 \left (1-2 x \right )^{\frac {3}{2}}}{648}-\frac {5053615 \sqrt {1-2 x}}{1944}\right )}{\left (-4-6 x \right )^{4}}+\frac {3454265 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1764}\) | \(84\) |
default | \(-1210 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {162 \left (\frac {100145 \left (1-2 x \right )^{\frac {7}{2}}}{504}-\frac {909931 \left (1-2 x \right )^{\frac {5}{2}}}{648}+\frac {2144065 \left (1-2 x \right )^{\frac {3}{2}}}{648}-\frac {5053615 \sqrt {1-2 x}}{1944}\right )}{\left (-4-6 x \right )^{4}}+\frac {3454265 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1764}\) | \(84\) |
pseudoelliptic | \(\frac {6908530 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-4268880 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{4} \sqrt {55}+21 \sqrt {1-2 x}\, \left (2703915 x^{3}+5498403 x^{2}+3730002 x +844322\right )}{3528 \left (2+3 x \right )^{4}}\) | \(85\) |
trager | \(\frac {\left (2703915 x^{3}+5498403 x^{2}+3730002 x +844322\right ) \sqrt {1-2 x}}{168 \left (2+3 x \right )^{4}}+605 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )+\frac {3454265 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{3528}\) | \(121\) |
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Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {2134440 \, \sqrt {55} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 3454265 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (2703915 \, x^{3} + 5498403 \, x^{2} + 3730002 \, x + 844322\right )} \sqrt {-2 \, x + 1}}{3528 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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Time = 125.25 (sec) , antiderivative size = 838, normalized size of antiderivative = 6.30 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\text {Too large to display} \]
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Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=605 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3454265}{3528} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2703915 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 19108551 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 45025365 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 35375305 \, \sqrt {-2 \, x + 1}}{84 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=605 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3454265}{3528} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2703915 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 19108551 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 45025365 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 35375305 \, \sqrt {-2 \, x + 1}}{1344 \, {\left (3 \, x + 2\right )}^{4}} \]
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Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {3454265\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1764}-1210\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {5053615\,\sqrt {1-2\,x}}{972}-\frac {2144065\,{\left (1-2\,x\right )}^{3/2}}{324}+\frac {909931\,{\left (1-2\,x\right )}^{5/2}}{324}-\frac {100145\,{\left (1-2\,x\right )}^{7/2}}{252}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]
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