Integrand size = 26, antiderivative size = 75 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3+5 x}}-14 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {96, 95, 210} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx=-14 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {5 x+3}} \]
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Rule 95
Rule 96
Rule 210
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}-7 \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3+5 x}}+49 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3+5 x}}+98 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right ) \\ & = -\frac {2 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3+5 x}}-14 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx=\frac {2 \sqrt {1-2 x} (62+107 x)}{3 (3+5 x)^{3/2}}-14 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(146\) vs. \(2(58)=116\).
Time = 1.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.96
method | result | size |
default | \(\frac {\left (525 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+630 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +189 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+214 x \sqrt {-10 x^{2}-x +3}+124 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{3 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(147\) |
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Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.15 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx=-\frac {21 \, \sqrt {7} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 2 \, {\left (107 \, x + 62\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.39 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx=7 \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {428 \, x}{15 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {8 \, x^{2}}{15 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1118}{75 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {488 \, x}{75 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {254}{75 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (58) = 116\).
Time = 0.34 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.53 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx=\frac {7}{10} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{1200} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {840 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {3360 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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