Integrand size = 26, antiderivative size = 106 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx=-\frac {41}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {793 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{216 \sqrt {10}}-\frac {2}{27} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {103, 159, 163, 56, 222, 95, 210} \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx=\frac {793 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{216 \sqrt {10}}-\frac {2}{27} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {41}{72} \sqrt {1-2 x} \sqrt {5 x+3} \]
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Rule 56
Rule 95
Rule 103
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{6} \int \frac {\left (-9-\frac {41 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx \\ & = -\frac {41}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {1}{36} \int \frac {\frac {283}{2}+\frac {793 x}{4}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {41}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {7}{27} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {793}{432} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {41}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {14}{27} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {793 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{216 \sqrt {5}} \\ & = -\frac {41}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {793 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{216 \sqrt {10}}-\frac {2}{27} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx=\frac {150 \sqrt {1-2 x} \left (-3+31 x+60 x^2\right )-793 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-160 \sqrt {7} \sqrt {3+5 x} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2160 \sqrt {3+5 x}} \]
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Time = 3.62 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (793 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+160 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3600 x \sqrt {-10 x^{2}-x +3}-300 \sqrt {-10 x^{2}-x +3}\right )}{4320 \sqrt {-10 x^{2}-x +3}}\) | \(98\) |
risch | \(-\frac {5 \left (-1+12 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{72 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {793 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{4320}-\frac {\sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{27}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(126\) |
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Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx=\frac {5}{72} \, {\left (12 \, x - 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1}{27} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {793}{4320} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx=\int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}{3 x + 2}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx=\frac {5}{6} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {793}{4320} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {1}{27} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {5}{72} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (76) = 152\).
Time = 0.35 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.63 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx=\frac {1}{270} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{360} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 41 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {793}{4320} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}}{3\,x+2} \,d x \]
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