Optimal. Leaf size=63 \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]
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Rubi [A] time = 0.0276435, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^{3/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int x^{3/2} (a+b x)^2 (A+B x) \, dx\\ &=\int \left (a^2 A x^{3/2}+a (2 A b+a B) x^{5/2}+b (A b+2 a B) x^{7/2}+b^2 B x^{9/2}\right ) \, dx\\ &=\frac{2}{5} a^2 A x^{5/2}+\frac{2}{7} a (2 A b+a B) x^{7/2}+\frac{2}{9} b (A b+2 a B) x^{9/2}+\frac{2}{11} b^2 B x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0169175, size = 52, normalized size = 0.83 \[ \frac{2 x^{5/2} \left (99 a^2 (7 A+5 B x)+110 a b x (9 A+7 B x)+35 b^2 x^2 (11 A+9 B x)\right )}{3465} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 52, normalized size = 0.8 \begin{align*}{\frac{630\,{b}^{2}B{x}^{3}+770\,A{b}^{2}{x}^{2}+1540\,B{x}^{2}ab+1980\,aAbx+990\,{a}^{2}Bx+1386\,A{a}^{2}}{3465}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00835, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53849, size = 140, normalized size = 2.22 \begin{align*} \frac{2}{3465} \,{\left (315 \, B b^{2} x^{5} + 693 \, A a^{2} x^{2} + 385 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 495 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.83519, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A a^{2} x^{\frac{5}{2}}}{5} + \frac{4 A a b x^{\frac{7}{2}}}{7} + \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{2 B a^{2} x^{\frac{7}{2}}}{7} + \frac{4 B a b x^{\frac{9}{2}}}{9} + \frac{2 B b^{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 = 1.18957, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{9} \, B a b x^{\frac{9}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{2}{7} \, B a^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A a b x^{\frac{7}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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