Optimal. Leaf size=75 \[ 2 a^2 A \sqrt{x}+\frac{2}{3} a^2 B x^{3/2}+\frac{4}{5} a A c x^{5/2}+\frac{4}{7} a B c x^{7/2}+\frac{2}{9} A c^2 x^{9/2}+\frac{2}{11} B c^2 x^{11/2} \]
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Rubi [A] time = 0.0248111, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {766} \[ 2 a^2 A \sqrt{x}+\frac{2}{3} a^2 B x^{3/2}+\frac{4}{5} a A c x^{5/2}+\frac{4}{7} a B c x^{7/2}+\frac{2}{9} A c^2 x^{9/2}+\frac{2}{11} B c^2 x^{11/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^2}{\sqrt{x}} \, dx &=\int \left (\frac{a^2 A}{\sqrt{x}}+a^2 B \sqrt{x}+2 a A c x^{3/2}+2 a B c x^{5/2}+A c^2 x^{7/2}+B c^2 x^{9/2}\right ) \, dx\\ &=2 a^2 A \sqrt{x}+\frac{2}{3} a^2 B x^{3/2}+\frac{4}{5} a A c x^{5/2}+\frac{4}{7} a B c x^{7/2}+\frac{2}{9} A c^2 x^{9/2}+\frac{2}{11} B c^2 x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0273959, size = 59, normalized size = 0.79 \[ \frac{2}{3} a^2 \sqrt{x} (3 A+B x)+\frac{4}{35} a c x^{5/2} (7 A+5 B x)+\frac{2}{99} c^2 x^{9/2} (11 A+9 B x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 54, normalized size = 0.7 \begin{align*}{\frac{630\,B{c}^{2}{x}^{5}+770\,A{c}^{2}{x}^{4}+1980\,aBc{x}^{3}+2772\,aAc{x}^{2}+2310\,{a}^{2}Bx+6930\,A{a}^{2}}{3465}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00578, size = 72, normalized size = 0.96 \begin{align*} \frac{2}{11} \, B c^{2} x^{\frac{11}{2}} + \frac{2}{9} \, A c^{2} x^{\frac{9}{2}} + \frac{4}{7} \, B a c x^{\frac{7}{2}} + \frac{4}{5} \, A a c x^{\frac{5}{2}} + \frac{2}{3} \, B a^{2} x^{\frac{3}{2}} + 2 \, A a^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20221, size = 147, normalized size = 1.96 \begin{align*} \frac{2}{3465} \,{\left (315 \, B c^{2} x^{5} + 385 \, A c^{2} x^{4} + 990 \, B a c x^{3} + 1386 \, A a c x^{2} + 1155 \, B a^{2} x + 3465 \, A a^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.55386, size = 78, normalized size = 1.04 \begin{align*} 2 A a^{2} \sqrt{x} + \frac{4 A a c x^{\frac{5}{2}}}{5} + \frac{2 A c^{2} x^{\frac{9}{2}}}{9} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3} + \frac{4 B a c x^{\frac{7}{2}}}{7} + \frac{2 B c^{2} x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2478, size = 72, normalized size = 0.96 \begin{align*} \frac{2}{11} \, B c^{2} x^{\frac{11}{2}} + \frac{2}{9} \, A c^{2} x^{\frac{9}{2}} + \frac{4}{7} \, B a c x^{\frac{7}{2}} + \frac{4}{5} \, A a c x^{\frac{5}{2}} + \frac{2}{3} \, B a^{2} x^{\frac{3}{2}} + 2 \, A a^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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