Integrand size = 24, antiderivative size = 134 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=-\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac {13892 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \]
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Time = 0.03 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {99, 154, 150, 65, 212} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=-\frac {13892 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}}+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^4}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{15 (3 x+2)^5}-\frac {67 \sqrt {1-2 x} (5 x+3)^2}{315 (3 x+2)^3}-\frac {2 \sqrt {1-2 x} (15074 x+9529)}{9261 (3 x+2)^2} \]
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Rule 65
Rule 99
Rule 150
Rule 154
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx \\ & = -\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {1}{180} \int \frac {(3+5 x)^2 (-684+180 x)}{\sqrt {1-2 x} (2+3 x)^4} \, dx \\ & = -\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {\int \frac {(3+5 x) (-48960+6840 x)}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{11340} \\ & = -\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}+\frac {6946 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{9261} \\ & = -\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac {6946 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{9261} \\ & = -\frac {67 \sqrt {1-2 x} (3+5 x)^2}{315 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{15 (2+3 x)^5}+\frac {3 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^4}-\frac {2 \sqrt {1-2 x} (9529+15074 x)}{9261 (2+3 x)^2}-\frac {13892 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {4 \left (\frac {21 \sqrt {1-2 x} \left (300049+2619854 x+7992771 x^2+10375830 x^3+4904370 x^4\right )}{4 (2+3 x)^5}-17365 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{972405} \]
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Time = 1.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.46
method | result | size |
risch | \(-\frac {9808740 x^{5}+15847290 x^{4}+5609712 x^{3}-2753063 x^{2}-2019756 x -300049}{46305 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {13892 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(61\) |
pseudoelliptic | \(\frac {-69460 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}+21 \sqrt {1-2 x}\, \left (4904370 x^{4}+10375830 x^{3}+7992771 x^{2}+2619854 x +300049\right )}{972405 \left (2+3 x \right )^{5}}\) | \(65\) |
derivativedivides | \(\frac {-\frac {217972 \left (1-2 x \right )^{\frac {9}{2}}}{1029}+\frac {36616 \left (1-2 x \right )^{\frac {7}{2}}}{21}-\frac {1682344 \left (1-2 x \right )^{\frac {5}{2}}}{315}+\frac {194488 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {97244 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{5}}-\frac {13892 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(75\) |
default | \(\frac {-\frac {217972 \left (1-2 x \right )^{\frac {9}{2}}}{1029}+\frac {36616 \left (1-2 x \right )^{\frac {7}{2}}}{21}-\frac {1682344 \left (1-2 x \right )^{\frac {5}{2}}}{315}+\frac {194488 \left (1-2 x \right )^{\frac {3}{2}}}{27}-\frac {97244 \sqrt {1-2 x}}{27}}{\left (-4-6 x \right )^{5}}-\frac {13892 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{194481}\) | \(75\) |
trager | \(\frac {\left (4904370 x^{4}+10375830 x^{3}+7992771 x^{2}+2619854 x +300049\right ) \sqrt {1-2 x}}{46305 \left (2+3 x \right )^{5}}-\frac {6946 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{194481}\) | \(82\) |
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Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {34730 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (4904370 \, x^{4} + 10375830 \, x^{3} + 7992771 \, x^{2} + 2619854 \, x + 300049\right )} \sqrt {-2 \, x + 1}}{972405 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {6946}{194481} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (2452185 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 20184570 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 61826142 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 83386730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 41693365 \, \sqrt {-2 \, x + 1}\right )}}{46305 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {6946}{194481} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2452185 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 20184570 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 61826142 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 83386730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 41693365 \, \sqrt {-2 \, x + 1}}{370440 \, {\left (3 \, x + 2\right )}^{5}} \]
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Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {\frac {97244\,\sqrt {1-2\,x}}{6561}-\frac {194488\,{\left (1-2\,x\right )}^{3/2}}{6561}+\frac {1682344\,{\left (1-2\,x\right )}^{5/2}}{76545}-\frac {36616\,{\left (1-2\,x\right )}^{7/2}}{5103}+\frac {217972\,{\left (1-2\,x\right )}^{9/2}}{250047}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}}-\frac {13892\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{194481} \]
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