Optimal. Leaf size=32 \[ \frac{a^2 x^4}{4}+\frac{4}{9} a b x^{9/2}+\frac{b^2 x^5}{5} \]
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Rubi [A] time = 0.023469, 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} \[ \frac{a^2 x^4}{4}+\frac{4}{9} a b x^{9/2}+\frac{b^2 x^5}{5} \]
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^3 \, dx &=2 \operatorname{Subst}\left (\int x^7 (a+b x)^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^2 x^7+2 a b x^8+b^2 x^9\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^4}{4}+\frac{4}{9} a b x^{9/2}+\frac{b^2 x^5}{5}\\ \end{align*}
Mathematica [A] time = 0.0174577, size = 32, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{4}{9} a b x^{9/2}+\frac{b^2 x^5}{5} \]
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}^{4}}{4}}+{\frac{4\,ab}{9}{x}^{{\frac{9}{2}}}}+{\frac{{b}^{2}{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.961564, size = 178, normalized size = 5.56 \begin{align*} \frac{{\left (b \sqrt{x} + a\right )}^{10}}{5 \, b^{8}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{9} a}{9 \, b^{8}} + \frac{21 \,{\left (b \sqrt{x} + a\right )}^{8} a^{2}}{4 \, b^{8}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{7} a^{3}}{b^{8}} + \frac{35 \,{\left (b \sqrt{x} + a\right )}^{6} a^{4}}{3 \, b^{8}} - \frac{42 \,{\left (b \sqrt{x} + a\right )}^{5} a^{5}}{5 \, b^{8}} + \frac{7 \,{\left (b \sqrt{x} + a\right )}^{4} a^{6}}{2 \, b^{8}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3} a^{7}}{3 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45151, size = 61, normalized size = 1.91 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.744018, size = 27, normalized size = 0.84 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{4 a b x^{\frac{9}{2}}}{9} + \frac{b^{2} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1019, size = 32, normalized size = 1. \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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