Optimal. Leaf size=27 \[ a^2 x+\frac{4}{3} a b x^{3/2}+\frac{b^2 x^2}{2} \]
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Rubi [A] time = 0.0160736, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ a^2 x+\frac{4}{3} a b x^{3/2}+\frac{b^2 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{x}\right )^2 \, dx &=2 \operatorname{Subst}\left (\int x (a+b x)^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,\sqrt{x}\right )\\ &=a^2 x+\frac{4}{3} a b x^{3/2}+\frac{b^2 x^2}{2}\\ \end{align*}
Mathematica [A] time = 0.0117133, size = 27, normalized size = 1. \[ a^2 x+\frac{4}{3} a b x^{3/2}+\frac{b^2 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 22, normalized size = 0.8 \begin{align*}{a}^{2}x+{\frac{4\,ab}{3}{x}^{{\frac{3}{2}}}}+{\frac{{b}^{2}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.944884, size = 28, normalized size = 1.04 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + \frac{4}{3} \, a b x^{\frac{3}{2}} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44104, size = 53, normalized size = 1.96 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + \frac{4}{3} \, a b x^{\frac{3}{2}} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.143745, size = 24, normalized size = 0.89 \begin{align*} a^{2} x + \frac{4 a b x^{\frac{3}{2}}}{3} + \frac{b^{2} x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09529, size = 28, normalized size = 1.04 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + \frac{4}{3} \, a b x^{\frac{3}{2}} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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