Optimal. Leaf size=39 \[ \frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{9} b B x^{9/2} \]
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Rubi [A] time = 0.0150596, 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} \[ \frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{9} b B x^{9/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right ) \left (A+B x^3\right )}{x^{5/2}} \, dx &=\int \left (\frac{a A}{x^{5/2}}+(A b+a B) \sqrt{x}+b B x^{7/2}\right ) \, dx\\ &=-\frac{2 a A}{3 x^{3/2}}+\frac{2}{3} (A b+a B) x^{3/2}+\frac{2}{9} b B x^{9/2}\\ \end{align*}
Mathematica [A] time = 0.0111206, size = 34, normalized size = 0.87 \[ \frac{2 \left (-3 a A+3 a B x^3+3 A b x^3+b B x^6\right )}{9 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 32, normalized size = 0.8 \begin{align*} -{\frac{-2\,bB{x}^{6}-6\,A{x}^{3}b-6\,B{x}^{3}a+6\,Aa}{9}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.936978, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{3} \,{\left (B a + A b\right )} x^{\frac{3}{2}} - \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 = 1.71192, size = 69, normalized size = 1.77 \begin{align*} \frac{2 \,{\left (B b x^{6} + 3 \,{\left (B a + A b\right )} x^{3} - 3 \, A a\right )}}{9 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.10627, size = 46, normalized size = 1.18 \begin{align*} - \frac{2 A a}{3 x^{\frac{3}{2}}} + \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 B a x^{\frac{3}{2}}}{3} + \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 = 1.12935, size = 39, normalized size = 1. \begin{align*} \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} - \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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