Optimal. Leaf size=32 \[ \frac{2}{3} a^2 x^{3/2}+4 a b \sqrt{x}-\frac{2 b^2}{\sqrt{x}} \]
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Rubi [A] time = 0.0092571, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {263, 43} \[ \frac{2}{3} a^2 x^{3/2}+4 a b \sqrt{x}-\frac{2 b^2}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^2 \sqrt{x} \, dx &=\int \frac{(b+a x)^2}{x^{3/2}} \, dx\\ &=\int \left (\frac{b^2}{x^{3/2}}+\frac{2 a b}{\sqrt{x}}+a^2 \sqrt{x}\right ) \, dx\\ &=-\frac{2 b^2}{\sqrt{x}}+4 a b \sqrt{x}+\frac{2}{3} a^2 x^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0075991, size = 27, normalized size = 0.84 \[ \frac{2 \left (a^2 x^2+6 a b x-3 b^2\right )}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 24, normalized size = 0.8 \begin{align*}{\frac{2\,{a}^{2}{x}^{2}+12\,xab-6\,{b}^{2}}{3}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971106, size = 34, normalized size = 1.06 \begin{align*} \frac{2}{3} \,{\left (a^{2} + \frac{6 \, a b}{x}\right )} x^{\frac{3}{2}} - \frac{2 \, b^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94281, size = 55, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} + 6 \, a b x - 3 \, b^{2}\right )}}{3 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.489724, size = 31, normalized size = 0.97 \begin{align*} \frac{2 a^{2} x^{\frac{3}{2}}}{3} + 4 a b \sqrt{x} - \frac{2 b^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11774, size = 32, normalized size = 1. \begin{align*} \frac{2}{3} \, a^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, b^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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