Optimal. Leaf size=30 \[ -\frac{4 x^{3/2}}{3}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]
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Rubi [A] time = 0.0824455, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {6689, 43} \[ -\frac{4 x^{3/2}}{3}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]
Antiderivative was successfully verified.
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Rule 6689
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (\sqrt{-1+x}+\sqrt{x}\right )^2 \sqrt{-1+x}} \, dx &=\int \left (-\frac{1}{\sqrt{-1+x}}-2 \sqrt{x}+\frac{2 x}{\sqrt{-1+x}}\right ) \, dx\\ &=-2 \sqrt{-1+x}-\frac{4 x^{3/2}}{3}+2 \int \frac{x}{\sqrt{-1+x}} \, dx\\ &=-2 \sqrt{-1+x}-\frac{4 x^{3/2}}{3}+2 \int \left (\frac{1}{\sqrt{-1+x}}+\sqrt{-1+x}\right ) \, dx\\ &=2 \sqrt{-1+x}+\frac{4}{3} (-1+x)^{3/2}-\frac{4 x^{3/2}}{3}\\ \end{align*}
Mathematica [A] time = 0.0413616, size = 30, normalized size = 1. \[ -\frac{4 x^{3/2}}{3}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.7 \begin{align*}{\frac{4}{3} \left ( -1+x \right ) ^{{\frac{3}{2}}}}-{\frac{4}{3}{x}^{{\frac{3}{2}}}}+2\,\sqrt{-1+x} \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{1}{\sqrt{x - 1}{\left (\sqrt{x - 1} + \sqrt{x}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00777, size = 55, normalized size = 1.83 \begin{align*} \frac{2}{3} \,{\left (2 \, x + 1\right )} \sqrt{x - 1} - \frac{4}{3} \, x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.818293, size = 53, normalized size = 1.77 \begin{align*} - \frac{4 \sqrt{x}}{6 \sqrt{x} \sqrt{x - 1} + 6 x - 3} - \frac{2 \sqrt{x - 1}}{6 \sqrt{x} \sqrt{x - 1} + 6 x - 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07121, size = 27, normalized size = 0.9 \begin{align*} \frac{4}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} - \frac{4}{3} \, x^{\frac{3}{2}} + 2 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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