Optimal. Leaf size=63 \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2} \]
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Rubi [A] time = 0.0293406, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \sqrt{x} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx &=\int \left (a^2 A \sqrt{x}+a (2 A b+a B) x^{5/2}+b (A b+2 a B) x^{9/2}+b^2 B x^{13/2}\right ) \, dx\\ &=\frac{2}{3} a^2 A x^{3/2}+\frac{2}{7} a (2 A b+a B) x^{7/2}+\frac{2}{11} b (A b+2 a B) x^{11/2}+\frac{2}{15} b^2 B x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.0287531, size = 53, normalized size = 0.84 \[ \frac{2 x^{3/2} \left (385 a^2 A+105 b x^4 (2 a B+A b)+165 a x^2 (a B+2 A b)+77 b^2 B x^6\right )}{1155} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 56, normalized size = 0.9 \begin{align*}{\frac{154\,B{b}^{2}{x}^{6}+210\,A{b}^{2}{x}^{4}+420\,B{x}^{4}ab+660\,aAb{x}^{2}+330\,B{x}^{2}{a}^{2}+770\,{a}^{2}A}{1155}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04412, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{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 = 0.912158, size = 136, normalized size = 2.16 \begin{align*} \frac{2}{1155} \,{\left (77 \, B b^{2} x^{7} + 105 \,{\left (2 \, B a b + A b^{2}\right )} x^{5} + 385 \, A a^{2} x + 165 \,{\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 = 2.46814, 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{15}{2}}}{15} + \frac{2 x^{\frac{11}{2}} \left (A b^{2} + 2 B a b\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (2 A a b + B a^{2}\right )}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15207, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{4}{11} \, B a b x^{\frac{11}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} + \frac{2}{7} \, B a^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A a b x^{\frac{7}{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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