Integrand size = 24, antiderivative size = 100 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {487+720 x}{294 \sqrt {1-2 x} (2+3 x)^2}+\frac {905 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 149, 44, 65, 212} \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {905 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {720 x+487}{294 \sqrt {1-2 x} (3 x+2)^2}+\frac {905 \sqrt {1-2 x}}{2058 (3 x+2)} \]
[In]
[Out]
Rule 44
Rule 65
Rule 100
Rule 149
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {1}{21} \int \frac {(3+5 x) (31+15 x)}{(1-2 x)^{3/2} (2+3 x)^3} \, dx \\ & = \frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {487+720 x}{294 \sqrt {1-2 x} (2+3 x)^2}-\frac {905}{294} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx \\ & = \frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {487+720 x}{294 \sqrt {1-2 x} (2+3 x)^2}-\frac {905 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2058} \\ & = \frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {487+720 x}{294 \sqrt {1-2 x} (2+3 x)^2}+\frac {905 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2058} \\ & = \frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {487+720 x}{294 \sqrt {1-2 x} (2+3 x)^2}+\frac {905 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.65 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {\frac {21 \left (8103+29593 x+33410 x^2+10860 x^3\right )}{2 (1-2 x)^{3/2} (2+3 x)^2}+905 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609} \]
[In]
[Out]
Time = 3.49 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.58
method | result | size |
risch | \(-\frac {10860 x^{3}+33410 x^{2}+29593 x +8103}{2058 \left (-1+2 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{2}}+\frac {905 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(58\) |
derivativedivides | \(-\frac {18 \left (-\frac {199 \left (1-2 x \right )^{\frac {3}{2}}}{18}+\frac {1379 \sqrt {1-2 x}}{54}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {905 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}+\frac {1331}{1029 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {726}{2401 \sqrt {1-2 x}}\) | \(66\) |
default | \(-\frac {18 \left (-\frac {199 \left (1-2 x \right )^{\frac {3}{2}}}{18}+\frac {1379 \sqrt {1-2 x}}{54}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {905 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}+\frac {1331}{1029 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {726}{2401 \sqrt {1-2 x}}\) | \(66\) |
pseudoelliptic | \(-\frac {1810 \left (\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{2} \sqrt {21}}{18}-7 x^{3}-\frac {23387 x^{2}}{1086}-\frac {207151 x}{10860}-\frac {18907}{3620}\right )}{2401 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{2}}\) | \(69\) |
trager | \(\frac {\left (10860 x^{3}+33410 x^{2}+29593 x +8103\right ) \sqrt {1-2 x}}{2058 \left (6 x^{2}+x -2\right )^{2}}-\frac {905 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{43218}\) | \(80\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {905 \, \sqrt {21} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (10860 \, x^{3} + 33410 \, x^{2} + 29593 \, x + 8103\right )} \sqrt {-2 \, x + 1}}{43218 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
[In]
[Out]
Time = 146.01 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.54 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=- \frac {363 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{16807} - \frac {404 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{1029} - \frac {8 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{147} - \frac {726}{2401 \sqrt {1 - 2 x}} + \frac {1331}{1029 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.92 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {905}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2715 \, {\left (2 \, x - 1\right )}^{3} + 24850 \, {\left (2 \, x - 1\right )}^{2} + 142296 \, x - 5929}{1029 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 49 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.89 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {905}{43218} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {121 \, {\left (36 \, x + 59\right )}}{7203 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {597 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1379 \, \sqrt {-2 \, x + 1}}{28812 \, {\left (3 \, x + 2\right )}^{2}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.71 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {905\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}+\frac {\frac {968\,x}{63}+\frac {3550\,{\left (2\,x-1\right )}^2}{1323}+\frac {905\,{\left (2\,x-1\right )}^3}{3087}-\frac {121}{189}}{\frac {49\,{\left (1-2\,x\right )}^{3/2}}{9}-\frac {14\,{\left (1-2\,x\right )}^{5/2}}{3}+{\left (1-2\,x\right )}^{7/2}} \]
[In]
[Out]