Optimal. Leaf size=37 \[ \frac{2}{5} x^{5/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{9} b B x^{9/2} \]
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Rubi [A] time = 0.0153744, antiderivative size = 37, 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} \[ \frac{2}{5} x^{5/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{9} b B x^{9/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (A+B x^2\right )}{\sqrt{x}} \, dx &=\int \left (\frac{a A}{\sqrt{x}}+(A b+a B) x^{3/2}+b B x^{7/2}\right ) \, dx\\ &=2 a A \sqrt{x}+\frac{2}{5} (A b+a B) x^{5/2}+\frac{2}{9} b B x^{9/2}\\ \end{align*}
Mathematica [A] time = 0.0139022, size = 33, normalized size = 0.89 \[ \frac{2}{45} \sqrt{x} \left (9 x^2 (a B+A b)+45 a A+5 b B x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.9 \begin{align*}{\frac{10\,bB{x}^{4}+18\,A{x}^{2}b+18\,B{x}^{2}a+90\,Aa}{45}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00321, size = 36, normalized size = 0.97 \begin{align*} \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{5} \,{\left (B a + A b\right )} x^{\frac{5}{2}} + 2 \, A a \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.828137, size = 74, normalized size = 2. \begin{align*} \frac{2}{45} \,{\left (5 \, B b x^{4} + 9 \,{\left (B a + A b\right )} x^{2} + 45 \, A a\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.76436, size = 44, normalized size = 1.19 \begin{align*} 2 A a \sqrt{x} + \frac{2 A b x^{\frac{5}{2}}}{5} + \frac{2 B a x^{\frac{5}{2}}}{5} + \frac{2 B b x^{\frac{9}{2}}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12298, size = 39, normalized size = 1.05 \begin{align*} \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{5} \, B a x^{\frac{5}{2}} + \frac{2}{5} \, A b x^{\frac{5}{2}} + 2 \, A a \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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