Integrand size = 26, antiderivative size = 113 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{5 \sqrt {3+5 x}}-\frac {7 (73-60 x) \sqrt {1-2 x} \sqrt {3+5 x}}{4000}+\frac {7}{25} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {10409 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{4000 \sqrt {10}} \]
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Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {99, 158, 152, 56, 222} \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {10409 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{4000 \sqrt {10}}-\frac {2 \sqrt {1-2 x} (3 x+2)^3}{5 \sqrt {5 x+3}}+\frac {7}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {7 (73-60 x) \sqrt {1-2 x} \sqrt {5 x+3}}{4000} \]
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Rule 56
Rule 99
Rule 152
Rule 158
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-2 x} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {(7-21 x) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {7}{25} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{75} \int \frac {(2+3 x) \left (-63+\frac {105 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^3}{5 \sqrt {3+5 x}}-\frac {7 (73-60 x) \sqrt {1-2 x} \sqrt {3+5 x}}{4000}+\frac {7}{25} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {10409 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{8000} \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^3}{5 \sqrt {3+5 x}}-\frac {7 (73-60 x) \sqrt {1-2 x} \sqrt {3+5 x}}{4000}+\frac {7}{25} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {10409 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{4000 \sqrt {5}} \\ & = -\frac {2 \sqrt {1-2 x} (2+3 x)^3}{5 \sqrt {3+5 x}}-\frac {7 (73-60 x) \sqrt {1-2 x} \sqrt {3+5 x}}{4000}+\frac {7}{25} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {10409 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{4000 \sqrt {10}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {10 \sqrt {1-2 x} \left (-893+3825 x+13140 x^2+7200 x^3\right )-10409 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{40000 \sqrt {3+5 x}} \]
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Time = 3.68 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\left (144000 x^{3} \sqrt {-10 x^{2}-x +3}+52045 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +262800 x^{2} \sqrt {-10 x^{2}-x +3}+31227 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+76500 x \sqrt {-10 x^{2}-x +3}-17860 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{80000 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(116\) |
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Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=-\frac {10409 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (7200 \, x^{3} + 13140 \, x^{2} + 3825 \, x - 893\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{80000 \, {\left (5 \, x + 3\right )}} \]
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\[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {1 - 2 x} \left (3 x + 2\right )^{3}}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {10409}{80000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {9}{250} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {81}{200} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {693}{20000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{625 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.39 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {9}{100000} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 463 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {10409}{40000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{6250 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{3125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^3}{{\left (5\,x+3\right )}^{3/2}} \,d x \]
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