Integrand size = 24, antiderivative size = 68 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}-\frac {257 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {91, 79, 65, 212} \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {257 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}+\frac {137 \sqrt {1-2 x}}{882 (3 x+2)}-\frac {\sqrt {1-2 x}}{126 (3 x+2)^2} \]
[In]
[Out]
Rule 65
Rule 79
Rule 91
Rule 212
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {1}{126} \int \frac {563+1050 x}{\sqrt {1-2 x} (2+3 x)^2} \, dx \\ & = -\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}+\frac {257}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx \\ & = -\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}-\frac {257}{98} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {\sqrt {1-2 x}}{126 (2+3 x)^2}+\frac {137 \sqrt {1-2 x}}{882 (2+3 x)}-\frac {257 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {\frac {7 \sqrt {1-2 x} (89+137 x)}{(2+3 x)^2}-514 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2058} \]
[In]
[Out]
Time = 1.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {274 x^{2}+41 x -89}{294 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {257 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\) | \(46\) |
derivativedivides | \(\frac {-\frac {137 \left (1-2 x \right )^{\frac {3}{2}}}{147}+\frac {15 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{2}}-\frac {257 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\) | \(48\) |
default | \(\frac {-\frac {137 \left (1-2 x \right )^{\frac {3}{2}}}{147}+\frac {15 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{2}}-\frac {257 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\) | \(48\) |
pseudoelliptic | \(\frac {-514 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}+7 \sqrt {1-2 x}\, \left (137 x +89\right )}{2058 \left (2+3 x \right )^{2}}\) | \(50\) |
trager | \(\frac {\left (137 x +89\right ) \sqrt {1-2 x}}{294 \left (2+3 x \right )^{2}}-\frac {257 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2058}\) | \(67\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {257 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \, {\left (137 \, x + 89\right )} \sqrt {-2 \, x + 1}}{2058 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (56) = 112\).
Time = 95.64 (sec) , antiderivative size = 330, normalized size of antiderivative = 4.85 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {25 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{189} + \frac {40 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} + \frac {8 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{9} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {257}{2058} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 315 \, \sqrt {-2 \, x + 1}}{147 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {257}{2058} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 315 \, \sqrt {-2 \, x + 1}}{588 \, {\left (3 \, x + 2\right )}^{2}} \]
[In]
[Out]
Time = 1.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {\frac {5\,\sqrt {1-2\,x}}{21}-\frac {137\,{\left (1-2\,x\right )}^{3/2}}{1323}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {257\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1029} \]
[In]
[Out]