Integrand size = 24, antiderivative size = 94 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{20 \sqrt {10}} \]
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Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {79, 52, 56, 222} \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=\frac {33 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{20 \sqrt {10}}-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {5 x+3}}+\frac {1}{22} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {3}{20} \sqrt {5 x+3} \sqrt {1-2 x} \]
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Rule 52
Rule 56
Rule 79
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {5}{11} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {3}{4} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33}{40} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{20 \sqrt {5}} \\ & = -\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{20 \sqrt {10}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=\frac {5 \sqrt {1-2 x} \left (11+17 x-12 x^2\right )-33 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{100 \sqrt {3+5 x}} \]
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Time = 1.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\left (165 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -240 x^{2} \sqrt {-10 x^{2}-x +3}+99 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+340 x \sqrt {-10 x^{2}-x +3}+220 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{400 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(99\) |
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Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=-\frac {33 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (12 \, x^{2} - 17 \, x - 11\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{400 \, {\left (5 \, x + 3\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \cdot \left (3 x + 2\right )}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.03 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=\frac {33}{400} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {99}{500} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{25 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{50 \, {\left (5 \, x + 3\right )}} - \frac {33 \, \sqrt {-10 \, x^{2} - x + 3}}{125 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.47 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.18 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=-\frac {1}{2500} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 157 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {33}{200} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{1250 \, \sqrt {5 \, x + 3}} + \frac {22 \, \sqrt {10} \sqrt {5 \, x + 3}}{625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,\left (3\,x+2\right )}{{\left (5\,x+3\right )}^{3/2}} \,d x \]
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