Optimal. Leaf size=45 \[ \frac{1}{4} (x-1)^{3/2} x (x+1)^{3/2}+\frac{1}{8} \sqrt{x-1} x \sqrt{x+1}-\frac{1}{8} \cosh ^{-1}(x) \]
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Rubi [A] time = 0.0061312, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {90, 38, 52} \[ \frac{1}{4} (x-1)^{3/2} x (x+1)^{3/2}+\frac{1}{8} \sqrt{x-1} x \sqrt{x+1}-\frac{1}{8} \cosh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 90
Rule 38
Rule 52
Rubi steps
\begin{align*} \int \sqrt{-1+x} x^2 \sqrt{1+x} \, dx &=\frac{1}{4} (-1+x)^{3/2} x (1+x)^{3/2}+\frac{1}{4} \int \sqrt{-1+x} \sqrt{1+x} \, dx\\ &=\frac{1}{8} \sqrt{-1+x} x \sqrt{1+x}+\frac{1}{4} (-1+x)^{3/2} x (1+x)^{3/2}-\frac{1}{8} \int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx\\ &=\frac{1}{8} \sqrt{-1+x} x \sqrt{1+x}+\frac{1}{4} (-1+x)^{3/2} x (1+x)^{3/2}-\frac{1}{8} \cosh ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0205533, size = 63, normalized size = 1.4 \[ \frac{x \sqrt{x+1} \left (2 x^3-2 x^2-x+1\right )+2 \sqrt{1-x} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )}{8 \sqrt{x-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 52, normalized size = 1.2 \begin{align*} -{\frac{1}{8}\sqrt{-1+x}\sqrt{1+x} \left ( -2\,{x}^{3}\sqrt{{x}^{2}-1}+x\sqrt{{x}^{2}-1}+\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.37858, size = 50, normalized size = 1.11 \begin{align*} \frac{1}{4} \,{\left (x^{2} - 1\right )}^{\frac{3}{2}} x + \frac{1}{8} \, \sqrt{x^{2} - 1} x - \frac{1}{8} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62247, size = 108, normalized size = 2.4 \begin{align*} \frac{1}{8} \,{\left (2 \, x^{3} - x\right )} \sqrt{x + 1} \sqrt{x - 1} + \frac{1}{8} \, \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\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 x^{2} \sqrt{x - 1} \sqrt{x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40156, size = 62, normalized size = 1.38 \begin{align*} \frac{1}{8} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{x - 1} + \frac{1}{4} \, \log \left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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