Optimal. Leaf size=32 \[ \frac{a^2 x^2}{2}+\frac{4}{5} a b x^{5/2}+\frac{b^2 x^3}{3} \]
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Rubi [A] time = 0.0178599, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^2 x^2}{2}+\frac{4}{5} a b x^{5/2}+\frac{b^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{x}\right )^2 x \, dx &=2 \operatorname{Subst}\left (\int x^3 (a+b x)^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^2 x^3+2 a b x^4+b^2 x^5\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^2}{2}+\frac{4}{5} a b x^{5/2}+\frac{b^2 x^3}{3}\\ \end{align*}
Mathematica [A] time = 0.0133287, size = 28, normalized size = 0.88 \[ \frac{1}{30} x^2 \left (15 a^2+24 a b \sqrt{x}+10 b^2 x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{4\,ab}{5}{x}^{{\frac{5}{2}}}}+{\frac{{b}^{2}{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.97089, size = 86, normalized size = 2.69 \begin{align*} \frac{{\left (b \sqrt{x} + a\right )}^{6}}{3 \, b^{4}} - \frac{6 \,{\left (b \sqrt{x} + a\right )}^{5} a}{5 \, b^{4}} + \frac{3 \,{\left (b \sqrt{x} + a\right )}^{4} a^{2}}{2 \, b^{4}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3} a^{3}}{3 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46985, size = 61, normalized size = 1.91 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + \frac{4}{5} \, a b x^{\frac{5}{2}} + \frac{1}{2} \, a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.225381, size = 27, normalized size = 0.84 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{4 a b x^{\frac{5}{2}}}{5} + \frac{b^{2} x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10795, size = 32, normalized size = 1. \begin{align*} \frac{1}{3} \, b^{2} x^{3} + \frac{4}{5} \, a b x^{\frac{5}{2}} + \frac{1}{2} \, a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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