Optimal. Leaf size=32 \[ -\frac{2 a^2}{3 x^{3/2}}-\frac{4 a b}{\sqrt{x}}+2 b^2 \sqrt{x} \]
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Rubi [A] time = 0.0074047, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ -\frac{2 a^2}{3 x^{3/2}}-\frac{4 a b}{\sqrt{x}}+2 b^2 \sqrt{x} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x^{5/2}} \, dx &=\int \left (\frac{a^2}{x^{5/2}}+\frac{2 a b}{x^{3/2}}+\frac{b^2}{\sqrt{x}}\right ) \, dx\\ &=-\frac{2 a^2}{3 x^{3/2}}-\frac{4 a b}{\sqrt{x}}+2 b^2 \sqrt{x}\\ \end{align*}
Mathematica [A] time = 0.0087331, size = 26, normalized size = 0.81 \[ -\frac{2 \left (a^2+6 a b x-3 b^2 x^2\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 0.7 \begin{align*} -{\frac{-6\,{b}^{2}{x}^{2}+12\,abx+2\,{a}^{2}}{3}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05362, size = 31, normalized size = 0.97 \begin{align*} 2 \, b^{2} \sqrt{x} - \frac{2 \,{\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40446, size = 55, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{2} - 6 \, a b x - a^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.883137, size = 31, normalized size = 0.97 \begin{align*} - \frac{2 a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 a b}{\sqrt{x}} + 2 b^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21539, size = 31, normalized size = 0.97 \begin{align*} 2 \, b^{2} \sqrt{x} - \frac{2 \,{\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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