Optimal. Leaf size=32 \[ \frac{a^2 x^3}{3}+\frac{4}{7} a b x^{7/2}+\frac{b^2 x^4}{4} \]
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Rubi [A] time = 0.0214081, 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 = {266, 43} \[ \frac{a^2 x^3}{3}+\frac{4}{7} a b x^{7/2}+\frac{b^2 x^4}{4} \]
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^2 \, dx &=2 \operatorname{Subst}\left (\int x^5 (a+b x)^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^2 x^5+2 a b x^6+b^2 x^7\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^3}{3}+\frac{4}{7} a b x^{7/2}+\frac{b^2 x^4}{4}\\ \end{align*}
Mathematica [A] time = 0.0157046, size = 28, normalized size = 0.88 \[ \frac{1}{84} x^3 \left (28 a^2+48 a b \sqrt{x}+21 b^2 x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 25, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{4\,ab}{7}{x}^{{\frac{7}{2}}}}+{\frac{{b}^{2}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.965583, size = 132, normalized size = 4.12 \begin{align*} \frac{{\left (b \sqrt{x} + a\right )}^{8}}{4 \, b^{6}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{7} a}{7 \, b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{6} a^{2}}{3 \, b^{6}} - \frac{4 \,{\left (b \sqrt{x} + a\right )}^{5} a^{3}}{b^{6}} + \frac{5 \,{\left (b \sqrt{x} + a\right )}^{4} a^{4}}{2 \, b^{6}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3} a^{5}}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46816, size = 61, normalized size = 1.91 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + \frac{4}{7} \, a b x^{\frac{7}{2}} + \frac{1}{3} \, a^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.411889, size = 27, normalized size = 0.84 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{4 a b x^{\frac{7}{2}}}{7} + \frac{b^{2} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12686, size = 32, normalized size = 1. \begin{align*} \frac{1}{4} \, b^{2} x^{4} + \frac{4}{7} \, a b x^{\frac{7}{2}} + \frac{1}{3} \, a^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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