Integrand size = 26, antiderivative size = 72 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {49 \sqrt {3+5 x}}{66 (1-2 x)^{3/2}}-\frac {448 \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {9 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2 \sqrt {10}} \]
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Time = 0.02 (sec) , antiderivative size = 72, 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 = {91, 79, 56, 222} \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {9 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {448 \sqrt {5 x+3}}{363 \sqrt {1-2 x}}+\frac {49 \sqrt {5 x+3}}{66 (1-2 x)^{3/2}} \]
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Rule 56
Rule 79
Rule 91
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {49 \sqrt {3+5 x}}{66 (1-2 x)^{3/2}}-\frac {1}{66} \int \frac {\frac {599}{2}+297 x}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx \\ & = \frac {49 \sqrt {3+5 x}}{66 (1-2 x)^{3/2}}-\frac {448 \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {9}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = \frac {49 \sqrt {3+5 x}}{66 (1-2 x)^{3/2}}-\frac {448 \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {9 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2 \sqrt {5}} \\ & = \frac {49 \sqrt {3+5 x}}{66 (1-2 x)^{3/2}}-\frac {448 \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {9 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2 \sqrt {10}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {70 \sqrt {3+5 x} (-51+256 x)+3267 \sqrt {10-20 x} (-1+2 x) \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{7260 (1-2 x)^{3/2}} \]
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Time = 4.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.43
method | result | size |
default | \(\frac {\left (13068 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-13068 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +3267 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+35840 x \sqrt {-10 x^{2}-x +3}-7140 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{14520 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(103\) |
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Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.19 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {3267 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 140 \, {\left (256 \, x - 51\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14520 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
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\[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{2}}{\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {9}{40} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{66 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {448 \, \sqrt {-10 \, x^{2} - x + 3}}{363 \, {\left (2 \, x - 1\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81 \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {9}{20} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {7 \, {\left (256 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1023 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{18150 \, {\left (2 \, x - 1\right )}^{2}} \]
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Timed out. \[ \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^2}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]
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