Optimal. Leaf size=32 \[ \frac{2}{5} (x-1)^{5/2}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]
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Rubi [A] time = 0.0054953, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2}{5} (x-1)^{5/2}+\frac{4}{3} (x-1)^{3/2}+2 \sqrt{x-1} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{-1+x}} \, dx &=\int \left (\frac{1}{\sqrt{-1+x}}+2 \sqrt{-1+x}+(-1+x)^{3/2}\right ) \, dx\\ &=2 \sqrt{-1+x}+\frac{4}{3} (-1+x)^{3/2}+\frac{2}{5} (-1+x)^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0055076, size = 21, normalized size = 0.66 \[ \frac{2}{15} \sqrt{x-1} \left (3 x^2+4 x+8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 18, normalized size = 0.6 \begin{align*}{\frac{6\,{x}^{2}+8\,x+16}{15}\sqrt{-1+x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.925678, size = 30, normalized size = 0.94 \begin{align*} \frac{2}{5} \,{\left (x - 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} + 2 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79603, size = 49, normalized size = 1.53 \begin{align*} \frac{2}{15} \,{\left (3 \, x^{2} + 4 \, x + 8\right )} \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.07552, size = 76, normalized size = 2.38 \begin{align*} \begin{cases} \frac{2 x^{2} \sqrt{x - 1}}{5} + \frac{8 x \sqrt{x - 1}}{15} + \frac{16 \sqrt{x - 1}}{15} & \text{for}\: \left |{x}\right | > 1 \\\frac{2 i x^{2} \sqrt{1 - x}}{5} + \frac{8 i x \sqrt{1 - x}}{15} + \frac{16 i \sqrt{1 - x}}{15} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05024, size = 30, normalized size = 0.94 \begin{align*} \frac{2}{5} \,{\left (x - 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} + 2 \, \sqrt{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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