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 = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.44, size = 24, normalized size = 0.75 \[ \frac {2 \, {\left (3 \, b^{2} x^{2} - 6 \, a b x - a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 23, normalized size = 0.72 \[ 2 \, b^{2} \sqrt {x} - \frac {2 \, {\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.72 \[ -\frac {2 \left (-3 b^{2} x^{2}+6 a b x +a^{2}\right )}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 23, normalized size = 0.72 \[ 2 \, b^{2} \sqrt {x} - \frac {2 \, {\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 0.75 \[ -\frac {2\,a^2+12\,a\,b\,x-6\,b^2\,x^2}{3\,x^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 31, normalized size = 0.97 \[ - \frac {2 a^{2}}{3 x^{\frac {3}{2}}} - \frac {4 a b}{\sqrt {x}} + 2 b^{2} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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