Integrand size = 18, antiderivative size = 44 \[ \int \frac {(2-3 x)^3 \sqrt {x}}{(1+x)^2} \, dx=-450 \sqrt {x}+72 x^{3/2}-\frac {54 x^{5/2}}{5}-\frac {125 \sqrt {x}}{1+x}+575 \arctan \left (\sqrt {x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {99, 158, 152, 65, 209} \[ \int \frac {(2-3 x)^3 \sqrt {x}}{(1+x)^2} \, dx=575 \arctan \left (\sqrt {x}\right )-\frac {\sqrt {x} (2-3 x)^3}{x+1}-\frac {21}{5} \sqrt {x} (2-3 x)^2-\frac {3}{5} (917-171 x) \sqrt {x} \]
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Rule 65
Rule 99
Rule 152
Rule 158
Rule 209
Rubi steps \begin{align*} \text {integral}& = -\frac {(2-3 x)^3 \sqrt {x}}{1+x}+\int \frac {\left (1-\frac {21 x}{2}\right ) (2-3 x)^2}{\sqrt {x} (1+x)} \, dx \\ & = -\frac {21}{5} (2-3 x)^2 \sqrt {x}-\frac {(2-3 x)^3 \sqrt {x}}{1+x}+\frac {2}{5} \int \frac {\left (\frac {31}{2}-\frac {513 x}{4}\right ) (2-3 x)}{\sqrt {x} (1+x)} \, dx \\ & = -\frac {3}{5} (917-171 x) \sqrt {x}-\frac {21}{5} (2-3 x)^2 \sqrt {x}-\frac {(2-3 x)^3 \sqrt {x}}{1+x}+\frac {575}{2} \int \frac {1}{\sqrt {x} (1+x)} \, dx \\ & = -\frac {3}{5} (917-171 x) \sqrt {x}-\frac {21}{5} (2-3 x)^2 \sqrt {x}-\frac {(2-3 x)^3 \sqrt {x}}{1+x}+575 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {3}{5} (917-171 x) \sqrt {x}-\frac {21}{5} (2-3 x)^2 \sqrt {x}-\frac {(2-3 x)^3 \sqrt {x}}{1+x}+575 \tan ^{-1}\left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int \frac {(2-3 x)^3 \sqrt {x}}{(1+x)^2} \, dx=-\frac {\sqrt {x} \left (2875+1890 x-306 x^2+54 x^3\right )}{5 (1+x)}+575 \arctan \left (\sqrt {x}\right ) \]
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Time = 0.48 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(72 x^{\frac {3}{2}}-\frac {54 x^{\frac {5}{2}}}{5}+575 \arctan \left (\sqrt {x}\right )-450 \sqrt {x}-\frac {125 \sqrt {x}}{1+x}\) | \(33\) |
default | \(72 x^{\frac {3}{2}}-\frac {54 x^{\frac {5}{2}}}{5}+575 \arctan \left (\sqrt {x}\right )-450 \sqrt {x}-\frac {125 \sqrt {x}}{1+x}\) | \(33\) |
risch | \(-\frac {\left (54 x^{3}-306 x^{2}+1890 x +2875\right ) \sqrt {x}}{5 \left (1+x \right )}+575 \arctan \left (\sqrt {x}\right )\) | \(33\) |
trager | \(-\frac {\left (54 x^{3}-306 x^{2}+1890 x +2875\right ) \sqrt {x}}{5 \left (1+x \right )}-\frac {575 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x}-x +1}{1+x}\right )}{2}\) | \(59\) |
meijerg | \(-\frac {8 \sqrt {x}}{1+x}+575 \arctan \left (\sqrt {x}\right )-\frac {36 \sqrt {x}\, \left (10 x +15\right )}{5 \left (1+x \right )}-\frac {18 \sqrt {x}\, \left (-14 x^{2}+70 x +105\right )}{7 \left (1+x \right )}-\frac {3 \sqrt {x}\, \left (18 x^{3}-42 x^{2}+210 x +315\right )}{5 \left (1+x \right )}\) | \(78\) |
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Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {(2-3 x)^3 \sqrt {x}}{(1+x)^2} \, dx=\frac {2875 \, {\left (x + 1\right )} \arctan \left (\sqrt {x}\right ) - {\left (54 \, x^{3} - 306 \, x^{2} + 1890 \, x + 2875\right )} \sqrt {x}}{5 \, {\left (x + 1\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.70 \[ \int \frac {(2-3 x)^3 \sqrt {x}}{(1+x)^2} \, dx=- \frac {54 x^{\frac {7}{2}}}{5 x + 5} + \frac {306 x^{\frac {5}{2}}}{5 x + 5} - \frac {1890 x^{\frac {3}{2}}}{5 x + 5} - \frac {2875 \sqrt {x}}{5 x + 5} + \frac {2875 x \operatorname {atan}{\left (\sqrt {x} \right )}}{5 x + 5} + \frac {2875 \operatorname {atan}{\left (\sqrt {x} \right )}}{5 x + 5} \]
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Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73 \[ \int \frac {(2-3 x)^3 \sqrt {x}}{(1+x)^2} \, dx=-\frac {54}{5} \, x^{\frac {5}{2}} + 72 \, x^{\frac {3}{2}} - 450 \, \sqrt {x} - \frac {125 \, \sqrt {x}}{x + 1} + 575 \, \arctan \left (\sqrt {x}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73 \[ \int \frac {(2-3 x)^3 \sqrt {x}}{(1+x)^2} \, dx=-\frac {54}{5} \, x^{\frac {5}{2}} + 72 \, x^{\frac {3}{2}} - 450 \, \sqrt {x} - \frac {125 \, \sqrt {x}}{x + 1} + 575 \, \arctan \left (\sqrt {x}\right ) \]
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Time = 0.37 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73 \[ \int \frac {(2-3 x)^3 \sqrt {x}}{(1+x)^2} \, dx=575\,\mathrm {atan}\left (\sqrt {x}\right )-\frac {125\,\sqrt {x}}{x+1}-450\,\sqrt {x}+72\,x^{3/2}-\frac {54\,x^{5/2}}{5} \]
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