Integrand size = 24, antiderivative size = 116 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {91}{250} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {91}{825} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1001 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{250 \sqrt {10}} \]
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Time = 0.02 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {79, 49, 52, 56, 222} \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx=-\frac {1001 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{250 \sqrt {10}}-\frac {2 (1-2 x)^{7/2}}{165 (5 x+3)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {5 x+3}}-\frac {91}{825} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {91}{250} \sqrt {5 x+3} \sqrt {1-2 x} \]
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Rule 49
Rule 52
Rule 56
Rule 79
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}+\frac {91}{165} \int \frac {(1-2 x)^{5/2}}{(3+5 x)^{3/2}} \, dx \\ & = -\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {182}{165} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {91}{825} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {91}{50} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {91}{250} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {91}{825} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1001}{500} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {91}{250} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {91}{825} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1001 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{250 \sqrt {5}} \\ & = -\frac {2 (1-2 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {182 (1-2 x)^{5/2}}{825 \sqrt {3+5 x}}-\frac {91}{250} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {91}{825} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{250 \sqrt {10}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx=\frac {\frac {5 \sqrt {1-2 x} \left (-3707-7970 x-2715 x^2+900 x^3\right )}{(3+5 x)^{3/2}}+3003 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{3750} \]
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Time = 1.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.12
method | result | size |
default | \(-\frac {\left (75075 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-18000 x^{3} \sqrt {-10 x^{2}-x +3}+90090 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +54300 x^{2} \sqrt {-10 x^{2}-x +3}+27027 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+159400 x \sqrt {-10 x^{2}-x +3}+74140 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{15000 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(130\) |
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Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx=\frac {3003 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (900 \, x^{3} - 2715 \, x^{2} - 7970 \, x - 3707\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{15000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
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\[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \cdot \left (3 x + 2\right )}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (83) = 166\).
Time = 0.30 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.60 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx=-\frac {1001}{5000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{25 \, {\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{50 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} - \frac {11 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{150 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{100 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {121 \, \sqrt {-10 \, x^{2} - x + 3}}{750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {2959 \, \sqrt {-10 \, x^{2} - x + 3}}{1500 \, {\left (5 \, x + 3\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (83) = 166\).
Time = 0.38 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.48 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx=\frac {1}{6250} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 289 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{150000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} - \frac {1001}{2500} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {627 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{12500 \, \sqrt {5 \, x + 3}} + \frac {11 \, \sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {171 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{9375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )}{{\left (5\,x+3\right )}^{5/2}} \,d x \]
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