Optimal. Leaf size=34 \[ 2 a^2 \sqrt{x}+\frac{4}{3} a b x^{3/2}+\frac{2}{5} b^2 x^{5/2} \]
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Rubi [A] time = 0.0069611, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ 2 a^2 \sqrt{x}+\frac{4}{3} a b x^{3/2}+\frac{2}{5} b^2 x^{5/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{\sqrt{x}} \, dx &=\int \left (\frac{a^2}{\sqrt{x}}+2 a b \sqrt{x}+b^2 x^{3/2}\right ) \, dx\\ &=2 a^2 \sqrt{x}+\frac{4}{3} a b x^{3/2}+\frac{2}{5} b^2 x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0073289, size = 28, normalized size = 0.82 \[ \frac{2}{15} \sqrt{x} \left (15 a^2+10 a b x+3 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.7 \begin{align*}{\frac{6\,{b}^{2}{x}^{2}+20\,abx+30\,{a}^{2}}{15}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04053, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{5} \, b^{2} x^{\frac{5}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} + 2 \, a^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50465, size = 62, normalized size = 1.82 \begin{align*} \frac{2}{15} \,{\left (3 \, b^{2} x^{2} + 10 \, a b x + 15 \, a^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.359422, size = 32, normalized size = 0.94 \begin{align*} 2 a^{2} \sqrt{x} + \frac{4 a b x^{\frac{3}{2}}}{3} + \frac{2 b^{2} x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23031, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{5} \, b^{2} x^{\frac{5}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} + 2 \, a^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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