Optimal. Leaf size=61 \[ a^2 x+\frac{6}{5} a b x^{5/3}+\frac{4}{3} a c x^{3/2}+\frac{3}{7} b^2 x^{7/3}+\frac{12}{13} b c x^{13/6}+\frac{c^2 x^2}{2} \]
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Rubi [A] time = 0.165853, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6741, 6742} \[ a^2 x+\frac{6}{5} a b x^{5/3}+\frac{4}{3} a c x^{3/2}+\frac{3}{7} b^2 x^{7/3}+\frac{12}{13} b c x^{13/6}+\frac{c^2 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 6741
Rule 6742
Rubi steps
\begin{align*} \int \left (a+c \sqrt{x}+b x^{2/3}\right )^2 \, dx &=6 \operatorname{Subst}\left (\int x^5 \left (a+x^3 (c+b x)\right )^2 \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname{Subst}\left (\int x^5 \left (a+c x^3+b x^4\right )^2 \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname{Subst}\left (\int \left (a^2 x^5+2 a c x^8+2 a b x^9+c^2 x^{11}+2 b c x^{12}+b^2 x^{13}\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=a^2 x+\frac{4}{3} a c x^{3/2}+\frac{6}{5} a b x^{5/3}+\frac{c^2 x^2}{2}+\frac{12}{13} b c x^{13/6}+\frac{3}{7} b^2 x^{7/3}\\ \end{align*}
Mathematica [A] time = 0.0460942, size = 61, normalized size = 1. \[ a^2 x+\frac{6}{5} a b x^{5/3}+\frac{4}{3} a c x^{3/2}+\frac{3}{7} b^2 x^{7/3}+\frac{12}{13} b c x^{13/6}+\frac{c^2 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 46, normalized size = 0.8 \begin{align*}{\frac{{c}^{2}{x}^{2}}{2}}+2\,c \left ({\frac{6\,b}{13}{x}^{{\frac{13}{6}}}}+2/3\,a{x}^{3/2} \right ) +x{a}^{2}+{\frac{3\,{b}^{2}}{7}{x}^{{\frac{7}{3}}}}+{\frac{6\,ab}{5}{x}^{{\frac{5}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11709, size = 61, normalized size = 1. \begin{align*} \frac{3}{7} \, b^{2} x^{\frac{7}{3}} + \frac{12}{13} \, b c x^{\frac{13}{6}} + \frac{1}{2} \, c^{2} x^{2} + a^{2} x + \frac{2}{15} \,{\left (9 \, b x^{\frac{5}{3}} + 10 \, c x^{\frac{3}{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70177, size = 130, normalized size = 2.13 \begin{align*} \frac{3}{7} \, b^{2} x^{\frac{7}{3}} + \frac{12}{13} \, b c x^{\frac{13}{6}} + \frac{1}{2} \, c^{2} x^{2} + \frac{6}{5} \, a b x^{\frac{5}{3}} + \frac{4}{3} \, a c 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 = 1.90677, size = 60, normalized size = 0.98 \begin{align*} a^{2} x + \frac{6 a b x^{\frac{5}{3}}}{5} + \frac{4 a c x^{\frac{3}{2}}}{3} + \frac{3 b^{2} x^{\frac{7}{3}}}{7} + \frac{12 b c x^{\frac{13}{6}}}{13} + \frac{c^{2} x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14186, size = 58, normalized size = 0.95 \begin{align*} \frac{3}{7} \, b^{2} x^{\frac{7}{3}} + \frac{12}{13} \, b c x^{\frac{13}{6}} + \frac{1}{2} \, c^{2} x^{2} + \frac{6}{5} \, a b x^{\frac{5}{3}} + \frac{4}{3} \, a c x^{\frac{3}{2}} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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