Integrand size = 22, antiderivative size = 61 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {64}{147 \sqrt {1-2 x}}+\frac {1}{21 \sqrt {1-2 x} (2+3 x)}-\frac {64 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \]
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Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 53, 65, 212} \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=-\frac {64 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}+\frac {64}{147 \sqrt {1-2 x}}+\frac {1}{21 \sqrt {1-2 x} (3 x+2)} \]
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Rule 53
Rule 65
Rule 79
Rule 212
Rubi steps \begin{align*} \text {integral}& = \frac {1}{21 \sqrt {1-2 x} (2+3 x)}+\frac {32}{21} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)} \, dx \\ & = \frac {64}{147 \sqrt {1-2 x}}+\frac {1}{21 \sqrt {1-2 x} (2+3 x)}+\frac {32}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx \\ & = \frac {64}{147 \sqrt {1-2 x}}+\frac {1}{21 \sqrt {1-2 x} (2+3 x)}-\frac {32}{49} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = \frac {64}{147 \sqrt {1-2 x}}+\frac {1}{21 \sqrt {1-2 x} (2+3 x)}-\frac {64 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {45+64 x}{49 \sqrt {1-2 x} (2+3 x)}-\frac {64 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \]
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Time = 1.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {64 x +45}{49 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {64 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\) | \(41\) |
derivativedivides | \(-\frac {2 \sqrt {1-2 x}}{147 \left (-\frac {4}{3}-2 x \right )}-\frac {64 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}+\frac {22}{49 \sqrt {1-2 x}}\) | \(45\) |
default | \(-\frac {2 \sqrt {1-2 x}}{147 \left (-\frac {4}{3}-2 x \right )}-\frac {64 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}+\frac {22}{49 \sqrt {1-2 x}}\) | \(45\) |
pseudoelliptic | \(-\frac {192 \left (\sqrt {1-2 x}\, \sqrt {21}\, \left (\frac {2}{3}+x \right ) \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )-7 x -\frac {315}{64}\right )}{\sqrt {1-2 x}\, \left (2058+3087 x \right )}\) | \(49\) |
trager | \(-\frac {\left (64 x +45\right ) \sqrt {1-2 x}}{49 \left (6 x^{2}+x -2\right )}-\frac {32 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1029}\) | \(70\) |
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Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {32 \, \sqrt {21} {\left (6 \, x^{2} + x - 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (64 \, x + 45\right )} \sqrt {-2 \, x + 1}}{1029 \, {\left (6 \, x^{2} + x - 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (53) = 106\).
Time = 32.90 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.84 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {11 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{343} + \frac {4 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{7} + \frac {22}{49 \sqrt {1 - 2 x}} \]
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Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {32}{1029} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (64 \, x + 45\right )}}{49 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.11 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {32}{1029} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2 \, {\left (64 \, x + 45\right )}}{49 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \]
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Time = 1.41 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {\frac {128\,x}{147}+\frac {30}{49}}{\frac {7\,\sqrt {1-2\,x}}{3}-{\left (1-2\,x\right )}^{3/2}}-\frac {64\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1029} \]
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