Optimal. Leaf size=39 \[ \frac{2}{7} x^{7/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{11} b B x^{11/2} \]
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Rubi [A] time = 0.0153497, antiderivative size = 39, 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 = {448} \[ \frac{2}{7} x^{7/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{11} b B x^{11/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \sqrt{x} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx &=\int \left (a A \sqrt{x}+(A b+a B) x^{5/2}+b B x^{9/2}\right ) \, dx\\ &=\frac{2}{3} a A x^{3/2}+\frac{2}{7} (A b+a B) x^{7/2}+\frac{2}{11} b B x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0146051, size = 33, normalized size = 0.85 \[ \frac{2}{231} x^{3/2} \left (33 x^2 (a B+A b)+77 a A+21 b B x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 32, normalized size = 0.8 \begin{align*}{\frac{42\,bB{x}^{4}+66\,A{x}^{2}b+66\,B{x}^{2}a+154\,Aa}{231}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02638, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{11} \, B b x^{\frac{11}{2}} + \frac{2}{7} \,{\left (B a + A b\right )} x^{\frac{7}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.889685, size = 81, normalized size = 2.08 \begin{align*} \frac{2}{231} \,{\left (21 \, B b x^{5} + 33 \,{\left (B a + A b\right )} x^{3} + 77 \, A a x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.67964, size = 37, normalized size = 0.95 \begin{align*} \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{11}{2}}}{11} + \frac{2 x^{\frac{7}{2}} \left (A b + B a\right )}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1176, size = 39, normalized size = 1. \begin{align*} \frac{2}{11} \, B b x^{\frac{11}{2}} + \frac{2}{7} \, B a x^{\frac{7}{2}} + \frac{2}{7} \, A b x^{\frac{7}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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