Optimal. Leaf size=44 \[ -\frac{54 x^{5/2}}{5}+72 x^{3/2}-\frac{125 \sqrt{x}}{x+1}-450 \sqrt{x}+575 \tan ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0118427, antiderivative size = 58, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {97, 153, 147, 63, 203} \[ -\frac{\sqrt{x} (2-3 x)^3}{x+1}-\frac{21}{5} \sqrt{x} (2-3 x)^2-\frac{3}{5} (917-171 x) \sqrt{x}+575 \tan ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 153
Rule 147
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{(2-3 x)^3 \sqrt{x}}{(1+x)^2} \, dx &=-\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 \operatorname{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*}
Mathematica [A] time = 0.017361, size = 38, normalized size = 0.86 \[ 575 \tan ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \left (54 x^3-306 x^2+1890 x+2875\right )}{5 (x+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 33, normalized size = 0.8 \begin{align*} 72\,{x}^{3/2}-{\frac{54}{5}{x}^{{\frac{5}{2}}}}+575\,\arctan \left ( \sqrt{x} \right ) -450\,\sqrt{x}-125\,{\frac{\sqrt{x}}{1+x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46261, size = 43, normalized size = 0.98 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32782, size = 120, normalized size = 2.73 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.39918, size = 75, normalized size = 1.7 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16128, size = 43, normalized size = 0.98 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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