Optimal. Leaf size=34 \[ -\frac{2 a^2}{\sqrt{x}}-\frac{4 a b}{3 x^{3/2}}-\frac{2 b^2}{5 x^{5/2}} \]
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Rubi [A] time = 0.0092866, 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} \[ -\frac{2 a^2}{\sqrt{x}}-\frac{4 a b}{3 x^{3/2}}-\frac{2 b^2}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^2}{x^{3/2}} \, dx &=\int \frac{(b+a x)^2}{x^{7/2}} \, dx\\ &=\int \left (\frac{b^2}{x^{7/2}}+\frac{2 a b}{x^{5/2}}+\frac{a^2}{x^{3/2}}\right ) \, dx\\ &=-\frac{2 b^2}{5 x^{5/2}}-\frac{4 a b}{3 x^{3/2}}-\frac{2 a^2}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0076236, size = 28, normalized size = 0.82 \[ -\frac{2 \left (15 a^2 x^2+10 a b x+3 b^2\right )}{15 x^{5/2}} \]
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}+20\,xab+6\,{b}^{2}}{15}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987338, size = 32, normalized size = 0.94 \begin{align*} -\frac{2 \, a^{2}}{\sqrt{x}} - \frac{4 \, a b}{3 \, x^{\frac{3}{2}}} - \frac{2 \, b^{2}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87253, size = 63, normalized size = 1.85 \begin{align*} -\frac{2 \,{\left (15 \, a^{2} x^{2} + 10 \, a b x + 3 \, b^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.752224, size = 34, normalized size = 1. \begin{align*} - \frac{2 a^{2}}{\sqrt{x}} - \frac{4 a b}{3 x^{\frac{3}{2}}} - \frac{2 b^{2}}{5 x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11911, size = 32, normalized size = 0.94 \begin{align*} -\frac{2 \,{\left (15 \, a^{2} x^{2} + 10 \, a b x + 3 \, b^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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