Optimal. Leaf size=30 \[ -\frac{a^2}{2 x^2}-\frac{4 a b}{3 x^{3/2}}-\frac{b^2}{x} \]
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Rubi [A] time = 0.0137169, antiderivative size = 30, 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 = {266, 43} \[ -\frac{a^2}{2 x^2}-\frac{4 a b}{3 x^{3/2}}-\frac{b^2}{x} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^2}{x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^5} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^2}{x^5}+\frac{2 a b}{x^4}+\frac{b^2}{x^3}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^2}{2 x^2}-\frac{4 a b}{3 x^{3/2}}-\frac{b^2}{x}\\ \end{align*}
Mathematica [A] time = 0.0112022, size = 30, normalized size = 1. \[ -\frac{a^2}{2 x^2}-\frac{4 a b}{3 x^{3/2}}-\frac{b^2}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}-{\frac{4\,ab}{3}{x}^{-{\frac{3}{2}}}}-{\frac{{b}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960975, size = 32, normalized size = 1.07 \begin{align*} -\frac{6 \, b^{2} x + 8 \, a b \sqrt{x} + 3 \, a^{2}}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46438, size = 59, normalized size = 1.97 \begin{align*} -\frac{6 \, b^{2} x + 8 \, a b \sqrt{x} + 3 \, a^{2}}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.61202, size = 26, normalized size = 0.87 \begin{align*} - \frac{a^{2}}{2 x^{2}} - \frac{4 a b}{3 x^{\frac{3}{2}}} - \frac{b^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10237, size = 32, normalized size = 1.07 \begin{align*} -\frac{6 \, b^{2} x + 8 \, a b \sqrt{x} + 3 \, a^{2}}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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