Optimal. Leaf size=34 \[ \frac{2}{5} a^2 x^{5/2}+\frac{4}{3} a b x^{3/2}+2 b^2 \sqrt{x} \]
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Rubi [A] time = 0.0092085, antiderivative size = 34, 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} \[ \frac{2}{5} a^2 x^{5/2}+\frac{4}{3} a b x^{3/2}+2 b^2 \sqrt{x} \]
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^{3/2} \, dx &=\int \frac{(b+a x)^2}{\sqrt{x}} \, dx\\ &=\int \left (\frac{b^2}{\sqrt{x}}+2 a b \sqrt{x}+a^2 x^{3/2}\right ) \, dx\\ &=2 b^2 \sqrt{x}+\frac{4}{3} a b x^{3/2}+\frac{2}{5} a^2 x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0069384, size = 28, normalized size = 0.82 \[ \frac{2}{15} \sqrt{x} \left (3 a^2 x^2+10 a b x+15 b^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.7 \begin{align*}{\frac{6\,{a}^{2}{x}^{2}+20\,xab+30\,{b}^{2}}{15}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983316, size = 35, normalized size = 1.03 \begin{align*} \frac{2}{15} \,{\left (3 \, a^{2} + \frac{10 \, a b}{x} + \frac{15 \, b^{2}}{x^{2}}\right )} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68799, size = 62, normalized size = 1.82 \begin{align*} \frac{2}{15} \,{\left (3 \, a^{2} x^{2} + 10 \, a b x + 15 \, b^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00118, size = 32, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{5}{2}}}{5} + \frac{4 a b x^{\frac{3}{2}}}{3} + 2 b^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16179, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{5} \, a^{2} x^{\frac{5}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} + 2 \, b^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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