Integrand size = 24, antiderivative size = 153 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}+\frac {584179 \sqrt {1-2 x}}{196 (2+3 x)}+\frac {20149879 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 154, 156, 162, 65, 212} \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {20149879 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac {584179 \sqrt {1-2 x}}{196 (3 x+2)}+\frac {25159 \sqrt {1-2 x}}{84 (3 x+2)^2}+\frac {1201 \sqrt {1-2 x}}{30 (3 x+2)^3}+\frac {63 \sqrt {1-2 x}}{10 (3 x+2)^4} \]
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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}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {(228-225 x) \sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx \\ & = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}-\frac {1}{180} \int \frac {-25182+37890 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx \\ & = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}-\frac {\int \frac {-2761290+3783150 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx}{3780} \\ & = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}-\frac {\int \frac {-209531070+237752550 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{52920} \\ & = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}+\frac {584179 \sqrt {1-2 x}}{196 (2+3 x)}-\frac {\int \frac {-9014096070+5520491550 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{370440} \\ & = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}+\frac {584179 \sqrt {1-2 x}}{196 (2+3 x)}-\frac {20149879}{196} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+166375 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = \frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}+\frac {584179 \sqrt {1-2 x}}{196 (2+3 x)}+\frac {20149879}{196} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-166375 \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}}{15 (2+3 x)^5}+\frac {63 \sqrt {1-2 x}}{10 (2+3 x)^4}+\frac {1201 \sqrt {1-2 x}}{30 (2+3 x)^3}+\frac {25159 \sqrt {1-2 x}}{84 (2+3 x)^2}+\frac {584179 \sqrt {1-2 x}}{196 (2+3 x)}+\frac {20149879 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (147756688+874383298 x+1941349752 x^2+1916515215 x^3+709777485 x^4\right )}{2940 (2+3 x)^5}+\frac {20149879 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{98 \sqrt {21}}-6050 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Time = 1.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {1419554970 x^{5}+3123252945 x^{4}+1966184289 x^{3}-192583156 x^{2}-578869922 x -147756688}{2940 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {20149879 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2058}-6050 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(79\) |
pseudoelliptic | \(\frac {201498790 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}-124509000 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{5} \sqrt {55}+7 \sqrt {1-2 x}\, \left (709777485 x^{4}+1916515215 x^{3}+1941349752 x^{2}+874383298 x +147756688\right )}{20580 \left (2+3 x \right )^{5}}\) | \(90\) |
derivativedivides | \(-6050 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {486 \left (\frac {584179 \left (1-2 x \right )^{\frac {9}{2}}}{588}-\frac {504319 \left (1-2 x \right )^{\frac {7}{2}}}{54}+\frac {13335122 \left (1-2 x \right )^{\frac {5}{2}}}{405}-\frac {75232787 \left (1-2 x \right )^{\frac {3}{2}}}{1458}+\frac {29479429 \sqrt {1-2 x}}{972}\right )}{\left (-4-6 x \right )^{5}}+\frac {20149879 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2058}\) | \(93\) |
default | \(-6050 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {486 \left (\frac {584179 \left (1-2 x \right )^{\frac {9}{2}}}{588}-\frac {504319 \left (1-2 x \right )^{\frac {7}{2}}}{54}+\frac {13335122 \left (1-2 x \right )^{\frac {5}{2}}}{405}-\frac {75232787 \left (1-2 x \right )^{\frac {3}{2}}}{1458}+\frac {29479429 \sqrt {1-2 x}}{972}\right )}{\left (-4-6 x \right )^{5}}+\frac {20149879 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2058}\) | \(93\) |
trager | \(\frac {\left (709777485 x^{4}+1916515215 x^{3}+1941349752 x^{2}+874383298 x +147756688\right ) \sqrt {1-2 x}}{2940 \left (2+3 x \right )^{5}}+3025 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )+\frac {20149879 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{4116}\) | \(126\) |
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Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.11 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {62254500 \, \sqrt {55} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 100749395 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \, {\left (709777485 \, x^{4} + 1916515215 \, x^{3} + 1941349752 \, x^{2} + 874383298 \, x + 147756688\right )} \sqrt {-2 \, x + 1}}{20580 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\text {Timed out} \]
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Time = 0.37 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=3025 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {20149879}{4116} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {709777485 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 6672140370 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 23523155208 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 36864065630 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 21667380315 \, \sqrt {-2 \, x + 1}}{1470 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=3025 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {20149879}{4116} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {709777485 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 6672140370 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 23523155208 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 36864065630 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 21667380315 \, \sqrt {-2 \, x + 1}}{47040 \, {\left (3 \, x + 2\right )}^{5}} \]
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Time = 0.10 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx=\frac {20149879\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2058}-6050\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {29479429\,\sqrt {1-2\,x}}{486}-\frac {75232787\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {26670244\,{\left (1-2\,x\right )}^{5/2}}{405}-\frac {504319\,{\left (1-2\,x\right )}^{7/2}}{27}+\frac {584179\,{\left (1-2\,x\right )}^{9/2}}{294}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
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