Optimal. Leaf size=122 \[ -\frac{4 (x-2) (x+1)}{3 \sqrt{(x-2) (x+1)^3}}-\frac{\sqrt{2} \sqrt{x-2} (x+1)^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{x+1}}{\sqrt{x-2}}\right )}{\sqrt{(x-2) (x+1)^3}}+\frac{2 \sqrt{x-2} (x+1)^{3/2} \sinh ^{-1}\left (\frac{\sqrt{x-2}}{\sqrt{3}}\right )}{\sqrt{(x-2) (x+1)^3}} \]
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Rubi [A] time = 0.345962, antiderivative size = 133, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {1593, 6719, 1614, 21, 105, 54, 215, 93, 204} \[ \frac{4 (2-x) (x+1)}{3 \sqrt{-(2-x) (x+1)^3}}-\frac{\sqrt{2} \sqrt{x-2} (x+1)^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{x+1}}{\sqrt{x-2}}\right )}{\sqrt{-(2-x) (x+1)^3}}+\frac{2 \sqrt{x-2} (x+1)^{3/2} \sinh ^{-1}\left (\frac{\sqrt{x-2}}{\sqrt{3}}\right )}{\sqrt{-(2-x) (x+1)^3}} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 6719
Rule 1614
Rule 21
Rule 105
Rule 54
Rule 215
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{\frac{1}{x}+x}{\sqrt{(-2+x) (1+x)^3}} \, dx &=\int \frac{1+x^2}{x \sqrt{(-2+x) (1+x)^3}} \, dx\\ &=\frac{\left (\sqrt{-2+x} (1+x)^{3/2}\right ) \int \frac{1+x^2}{\sqrt{-2+x} x (1+x)^{3/2}} \, dx}{\sqrt{(-2+x) (1+x)^3}}\\ &=\frac{4 (2-x) (1+x)}{3 \sqrt{-(2-x) (1+x)^3}}-\frac{\left (2 \sqrt{-2+x} (1+x)^{3/2}\right ) \int \frac{-\frac{3}{2}-\frac{3 x}{2}}{\sqrt{-2+x} x \sqrt{1+x}} \, dx}{3 \sqrt{(-2+x) (1+x)^3}}\\ &=\frac{4 (2-x) (1+x)}{3 \sqrt{-(2-x) (1+x)^3}}+\frac{\left (\sqrt{-2+x} (1+x)^{3/2}\right ) \int \frac{\sqrt{1+x}}{\sqrt{-2+x} x} \, dx}{\sqrt{(-2+x) (1+x)^3}}\\ &=\frac{4 (2-x) (1+x)}{3 \sqrt{-(2-x) (1+x)^3}}+\frac{\left (\sqrt{-2+x} (1+x)^{3/2}\right ) \int \frac{1}{\sqrt{-2+x} \sqrt{1+x}} \, dx}{\sqrt{(-2+x) (1+x)^3}}+\frac{\left (\sqrt{-2+x} (1+x)^{3/2}\right ) \int \frac{1}{\sqrt{-2+x} x \sqrt{1+x}} \, dx}{\sqrt{(-2+x) (1+x)^3}}\\ &=\frac{4 (2-x) (1+x)}{3 \sqrt{-(2-x) (1+x)^3}}+\frac{\left (2 \sqrt{-2+x} (1+x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-2 x^2} \, dx,x,\frac{\sqrt{1+x}}{\sqrt{-2+x}}\right )}{\sqrt{(-2+x) (1+x)^3}}+\frac{\left (2 \sqrt{-2+x} (1+x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+x^2}} \, dx,x,\sqrt{-2+x}\right )}{\sqrt{(-2+x) (1+x)^3}}\\ &=\frac{4 (2-x) (1+x)}{3 \sqrt{-(2-x) (1+x)^3}}+\frac{2 \sqrt{-2+x} (1+x)^{3/2} \sinh ^{-1}\left (\frac{\sqrt{-2+x}}{\sqrt{3}}\right )}{\sqrt{-(2-x) (1+x)^3}}-\frac{\sqrt{2} \sqrt{-2+x} (1+x)^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{1+x}}{\sqrt{-2+x}}\right )}{\sqrt{-(2-x) (1+x)^3}}\\ \end{align*}
Mathematica [A] time = 0.130333, size = 114, normalized size = 0.93 \[ -\frac{(x+1) \left (-4 (2-x)^{3/2}-6 (x-2) \sqrt{x+1} \sin ^{-1}\left (\frac{\sqrt{2-x}}{\sqrt{3}}\right )-3 \sqrt{2} \sqrt{-(x-2)^2} \sqrt{x+1} \tan ^{-1}\left (\frac{\sqrt{\frac{x-2}{x+1}}}{\sqrt{2}}\right )\right )}{3 \sqrt{2-x} \sqrt{(x-2) (x+1)^3}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.018, size = 118, normalized size = 1. \begin{align*}{\frac{1}{6} \left ( -3\,\sqrt{2}\arctan \left ( 1/4\,{\frac{ \left ( 4+x \right ) \sqrt{2}}{\sqrt{{x}^{2}-x-2}}} \right ) x+6\,\ln \left ( x-1/2+\sqrt{{x}^{2}-x-2} \right ) x-3\,\sqrt{2}\arctan \left ( 1/4\,{\frac{ \left ( 4+x \right ) \sqrt{2}}{\sqrt{{x}^{2}-x-2}}} \right ) +6\,\ln \left ( x-1/2+\sqrt{{x}^{2}-x-2} \right ) -8\,\sqrt{{x}^{2}-x-2} \right ) \sqrt{ \left ( 1+x \right ) \left ( -2+x \right ) }{\frac{1}{\sqrt{ \left ( -2+x \right ) \left ( 1+x \right ) ^{3}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + \frac{1}{x}}{\sqrt{{\left (x + 1\right )}^{3}{\left (x - 2\right )}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7969, size = 375, normalized size = 3.07 \begin{align*} \frac{3 \, \sqrt{2}{\left (x^{2} + 2 \, x + 1\right )} \arctan \left (-\frac{\sqrt{2}{\left (x^{2} + x\right )} - \sqrt{2} \sqrt{x^{4} + x^{3} - 3 \, x^{2} - 5 \, x - 2}}{2 \,{\left (x + 1\right )}}\right ) - 4 \, x^{2} - 3 \,{\left (x^{2} + 2 \, x + 1\right )} \log \left (-\frac{2 \, x^{2} + x - 2 \, \sqrt{x^{4} + x^{3} - 3 \, x^{2} - 5 \, x - 2} - 1}{x + 1}\right ) - 8 \, x - 4 \, \sqrt{x^{4} + x^{3} - 3 \, x^{2} - 5 \, x - 2} - 4}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} + 1}{x \sqrt{\left (x - 2\right ) \left (x + 1\right )^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19029, size = 230, normalized size = 1.89 \begin{align*} -\frac{\sqrt{2} \arcsin \left (\frac{4}{3 \, x} + \frac{1}{3}\right )}{2 \, \mathrm{sgn}\left (\frac{1}{x^{2}} + \frac{1}{x^{3}}\right )} - \frac{\log \left (\frac{{\left | -4 \, \sqrt{2} + \frac{2 \,{\left (2 \, \sqrt{2} \sqrt{-\frac{1}{x} - \frac{2}{x^{2}} + 1} - 3\right )}}{\frac{4}{x} + 1} + 6 \right |}}{{\left | 4 \, \sqrt{2} + \frac{2 \,{\left (2 \, \sqrt{2} \sqrt{-\frac{1}{x} - \frac{2}{x^{2}} + 1} - 3\right )}}{\frac{4}{x} + 1} + 6 \right |}}\right )}{\mathrm{sgn}\left (\frac{1}{x^{2}} + \frac{1}{x^{3}}\right )} + \frac{8 \, \sqrt{2}}{3 \,{\left (\frac{2 \, \sqrt{2} \sqrt{-\frac{1}{x} - \frac{2}{x^{2}} + 1} - 3}{\frac{4}{x} + 1} - 1\right )} \mathrm{sgn}\left (\frac{1}{x^{2}} + \frac{1}{x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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