Integrand size = 26, antiderivative size = 84 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40 \sqrt {10}} \]
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Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {100, 148, 56, 222} \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {1593 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{40 \sqrt {10}}+\frac {7 \sqrt {5 x+3} (3 x+2)^2}{33 (1-2 x)^{3/2}}-\frac {(95621-33462 x) \sqrt {5 x+3}}{14520 \sqrt {1-2 x}} \]
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Rule 56
Rule 100
Rule 148
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x) \left (155+\frac {507 x}{2}\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx \\ & = -\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593}{80} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{40 \sqrt {5}} \\ & = -\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40 \sqrt {10}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {-10 \sqrt {3+5 x} \left (83301-261664 x+39204 x^2\right )+578259 \sqrt {10-20 x} (-1+2 x) \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{145200 (1-2 x)^{3/2}} \]
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Time = 1.24 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.43
method | result | size |
default | \(\frac {\left (2313036 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-2313036 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -784080 x^{2} \sqrt {-10 x^{2}-x +3}+578259 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+5233280 x \sqrt {-10 x^{2}-x +3}-1666020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{290400 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(120\) |
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Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.08 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {578259 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (39204 \, x^{2} - 261664 \, x + 83301\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{290400 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
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\[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{3}}{\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {1593}{800} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {27}{40} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {343 \, \sqrt {-10 \, x^{2} - x + 3}}{132 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {11123 \, \sqrt {-10 \, x^{2} - x + 3}}{1452 \, {\left (2 \, x - 1\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.85 \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {1593}{400} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9801 \, \sqrt {5} {\left (5 \, x + 3\right )} - 385886 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 6360321 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1815000 \, {\left (2 \, x - 1\right )}^{2}} \]
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Timed out. \[ \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^3}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]
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