Integrand size = 26, antiderivative size = 115 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {1111 \sqrt {1-2 x}}{15 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}-\frac {8}{45} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {665}{9} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 115, 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 = {100, 155, 163, 56, 222, 95, 210} \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {8}{45} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {665}{9} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}-\frac {1111 \sqrt {1-2 x}}{15 \sqrt {5 x+3}} \]
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Rule 56
Rule 95
Rule 100
Rule 155
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}+\frac {1}{3} \int \frac {\sqrt {1-2 x} \left (\frac {227}{2}+4 x\right )}{(2+3 x) (3+5 x)^{3/2}} \, dx \\ & = -\frac {1111 \sqrt {1-2 x}}{15 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}+\frac {2}{15} \int \frac {-\frac {7769}{4}-4 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {1111 \sqrt {1-2 x}}{15 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}-\frac {8}{45} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx-\frac {4655}{18} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {1111 \sqrt {1-2 x}}{15 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}-\frac {4655}{9} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {16 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{45 \sqrt {5}} \\ & = -\frac {1111 \sqrt {1-2 x}}{15 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}-\frac {8}{45} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {665}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=\frac {1}{225} \left (-\frac {15 \sqrt {1-2 x} (2187+3403 x)}{(2+3 x) \sqrt {3+5 x}}+8 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+16625 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(190\) vs. \(2(83)=166\).
Time = 3.84 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.66
method | result | size |
default | \(-\frac {\left (120 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+249375 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+152 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +315875 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +48 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+99750 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+102090 x \sqrt {-10 x^{2}-x +3}+65610 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{450 \left (2+3 x \right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(191\) |
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Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.23 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=\frac {8 \, \sqrt {5} \sqrt {2} {\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 16625 \, \sqrt {7} {\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 30 \, {\left (3403 \, x + 2187\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{450 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \]
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\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {4}{225} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {665}{18} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {6806 \, x}{45 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {10699}{135 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343}{27 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (83) = 166\).
Time = 0.39 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.77 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {133}{36} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {4}{225} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121}{50} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {1078 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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