Optimal. Leaf size=36 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{5} a b x^{5/2}+\frac{2}{3} b^2 x^{3/2} \]
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Rubi [A] time = 0.0098707, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {263, 43} \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{5} a b x^{5/2}+\frac{2}{3} b^2 x^{3/2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^2 x^{5/2} \, dx &=\int \sqrt{x} (b+a x)^2 \, dx\\ &=\int \left (b^2 \sqrt{x}+2 a b x^{3/2}+a^2 x^{5/2}\right ) \, dx\\ &=\frac{2}{3} b^2 x^{3/2}+\frac{4}{5} a b x^{5/2}+\frac{2}{7} a^2 x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0074049, size = 28, normalized size = 0.78 \[ \frac{2}{105} x^{3/2} \left (15 a^2 x^2+42 a b x+35 b^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.7 \begin{align*}{\frac{30\,{a}^{2}{x}^{2}+84\,xab+70\,{b}^{2}}{105}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962952, size = 35, normalized size = 0.97 \begin{align*} \frac{2}{105} \,{\left (15 \, a^{2} + \frac{42 \, a b}{x} + \frac{35 \, b^{2}}{x^{2}}\right )} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5976, size = 70, normalized size = 1.94 \begin{align*} \frac{2}{105} \,{\left (15 \, a^{2} x^{3} + 42 \, a b x^{2} + 35 \, b^{2} x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.67022, size = 34, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{5}{2}}}{5} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09946, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{7} \, a^{2} x^{\frac{7}{2}} + \frac{4}{5} \, a b x^{\frac{5}{2}} + \frac{2}{3} \, b^{2} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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