Optimal. Leaf size=37 \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{5} b B x^{5/2} \]
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Rubi [A] time = 0.016267, 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} \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{5} b B x^{5/2} \]
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 )}{x^{5/2}} \, dx &=\int \left (\frac{a A}{x^{5/2}}+\frac{A b+a B}{\sqrt{x}}+b B x^{3/2}\right ) \, dx\\ &=-\frac{2 a A}{3 x^{3/2}}+2 (A b+a B) \sqrt{x}+\frac{2}{5} b B x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.012476, size = 36, normalized size = 0.97 \[ \frac{2 \left (3 b x^2 \left (5 A+B x^2\right )-5 a \left (A-3 B x^2\right )\right )}{15 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.9 \begin{align*} -{\frac{-6\,bB{x}^{4}-30\,A{x}^{2}b-30\,B{x}^{2}a+10\,Aa}{15}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03156, size = 36, normalized size = 0.97 \begin{align*} \frac{2}{5} \, B b x^{\frac{5}{2}} + 2 \,{\left (B a + A b\right )} \sqrt{x} - \frac{2 \, A a}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.663719, size = 74, normalized size = 2. \begin{align*} \frac{2 \,{\left (3 \, B b x^{4} + 15 \,{\left (B a + A b\right )} x^{2} - 5 \, A a\right )}}{15 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.25066, size = 42, normalized size = 1.14 \begin{align*} - \frac{2 A a}{3 x^{\frac{3}{2}}} + 2 A b \sqrt{x} + 2 B a \sqrt{x} + \frac{2 B b x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16129, size = 39, normalized size = 1.05 \begin{align*} \frac{2}{5} \, B b x^{\frac{5}{2}} + 2 \, B a \sqrt{x} + 2 \, A b \sqrt{x} - \frac{2 \, A a}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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