Optimal. Leaf size=72 \[ \frac{20}{9} a^2 b^3 x^{9/2}+\frac{5}{2} a^3 b^2 x^4+\frac{10}{7} a^4 b x^{7/2}+\frac{a^5 x^3}{3}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2} \]
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Rubi [A] time = 0.0402159, antiderivative size = 72, 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} \[ \frac{20}{9} a^2 b^3 x^{9/2}+\frac{5}{2} a^3 b^2 x^4+\frac{10}{7} a^4 b x^{7/2}+\frac{a^5 x^3}{3}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{x}\right )^5 x^2 \, dx &=2 \operatorname{Subst}\left (\int x^5 (a+b x)^5 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^5 x^5+5 a^4 b x^6+10 a^3 b^2 x^7+10 a^2 b^3 x^8+5 a b^4 x^9+b^5 x^{10}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^5 x^3}{3}+\frac{10}{7} a^4 b x^{7/2}+\frac{5}{2} a^3 b^2 x^4+\frac{20}{9} a^2 b^3 x^{9/2}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0251665, size = 72, normalized size = 1. \[ \frac{20}{9} a^2 b^3 x^{9/2}+\frac{5}{2} a^3 b^2 x^4+\frac{10}{7} a^4 b x^{7/2}+\frac{a^5 x^3}{3}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 57, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{3}}{3}}+{\frac{10\,{a}^{4}b}{7}{x}^{{\frac{7}{2}}}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{4}}{2}}+{\frac{20\,{a}^{2}{b}^{3}}{9}{x}^{{\frac{9}{2}}}}+a{b}^{4}{x}^{5}+{\frac{2\,{b}^{5}}{11}{x}^{{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949297, size = 132, normalized size = 1.83 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11}}{11 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{10} a}{b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )}^{9} a^{2}}{9 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{8} a^{3}}{2 \, b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{7} a^{4}}{7 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{6} a^{5}}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47659, size = 143, normalized size = 1.99 \begin{align*} a b^{4} x^{5} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{1}{3} \, a^{5} x^{3} + \frac{2}{693} \,{\left (63 \, b^{5} x^{5} + 770 \, a^{2} b^{3} x^{4} + 495 \, a^{4} b x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.90424, size = 70, normalized size = 0.97 \begin{align*} \frac{a^{5} x^{3}}{3} + \frac{10 a^{4} b x^{\frac{7}{2}}}{7} + \frac{5 a^{3} b^{2} x^{4}}{2} + \frac{20 a^{2} b^{3} x^{\frac{9}{2}}}{9} + a b^{4} x^{5} + \frac{2 b^{5} x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12196, size = 76, normalized size = 1.06 \begin{align*} \frac{2}{11} \, b^{5} x^{\frac{11}{2}} + a b^{4} x^{5} + \frac{20}{9} \, a^{2} b^{3} x^{\frac{9}{2}} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{10}{7} \, a^{4} b x^{\frac{7}{2}} + \frac{1}{3} \, a^{5} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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