Optimal. Leaf size=63 \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{9} b^2 B x^{9/2} \]
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Rubi [A] time = 0.0259567, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {76} \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{9} b^2 B x^{9/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \sqrt{x} (a+b x)^2 (A+B x) \, dx &=\int \left (a^2 A \sqrt{x}+a (2 A b+a B) x^{3/2}+b (A b+2 a B) x^{5/2}+b^2 B x^{7/2}\right ) \, dx\\ &=\frac{2}{3} a^2 A x^{3/2}+\frac{2}{5} a (2 A b+a B) x^{5/2}+\frac{2}{7} b (A b+2 a B) x^{7/2}+\frac{2}{9} b^2 B x^{9/2}\\ \end{align*}
Mathematica [A] time = 0.0157279, size = 52, normalized size = 0.83 \[ \frac{2}{315} x^{3/2} \left (21 a^2 (5 A+3 B x)+18 a b x (7 A+5 B x)+5 b^2 x^2 (9 A+7 B x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 52, normalized size = 0.8 \begin{align*}{\frac{70\,B{b}^{2}{x}^{3}+90\,A{b}^{2}{x}^{2}+180\,B{x}^{2}ab+252\,aAbx+126\,{a}^{2}Bx+210\,{a}^{2}A}{315}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08824, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} + \frac{2}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43284, size = 132, normalized size = 2.1 \begin{align*} \frac{2}{315} \,{\left (35 \, B b^{2} x^{4} + 105 \, A a^{2} x + 45 \,{\left (2 \, B a b + A b^{2}\right )} x^{3} + 63 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.03942, size = 66, normalized size = 1.05 \begin{align*} \frac{2 A a^{2} x^{\frac{3}{2}}}{3} + \frac{2 B b^{2} x^{\frac{9}{2}}}{9} + \frac{2 x^{\frac{7}{2}} \left (A b^{2} + 2 B a b\right )}{7} + \frac{2 x^{\frac{5}{2}} \left (2 A a b + B a^{2}\right )}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21914, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{4}{7} \, B a b x^{\frac{7}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B a^{2} x^{\frac{5}{2}} + \frac{4}{5} \, A a b x^{\frac{5}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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