Optimal. Leaf size=30 \[ \frac {2 a}{b^2 \sqrt {a+b x}}+\frac {2 \sqrt {a+b x}}{b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \[ \frac {2 a}{b^2 \sqrt {a+b x}}+\frac {2 \sqrt {a+b x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x}{(a+b x)^{3/2}} \, dx &=\int \left (-\frac {a}{b (a+b x)^{3/2}}+\frac {1}{b \sqrt {a+b x}}\right ) \, dx\\ &=\frac {2 a}{b^2 \sqrt {a+b x}}+\frac {2 \sqrt {a+b x}}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.70 \[ \frac {2 (2 a+b x)}{b^2 \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 29, normalized size = 0.97 \[ \frac {2 \, {\left (b x + 2 \, a\right )} \sqrt {b x + a}}{b^{3} x + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 29, normalized size = 0.97 \[ \frac {2 \, {\left (\frac {\sqrt {b x + a}}{b} + \frac {a}{\sqrt {b x + a} b}\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.67 \[ \frac {2 b x +4 a}{b^{2} \sqrt {b x +a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 26, normalized size = 0.87 \[ \frac {2 \, \sqrt {b x + a}}{b^{2}} + \frac {2 \, a}{\sqrt {b x + a} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 19, normalized size = 0.63 \[ \frac {4\,a+2\,b\,x}{b^2\,\sqrt {a+b\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 37, normalized size = 1.23 \[ \begin {cases} \frac {4 a}{b^{2} \sqrt {a + b x}} + \frac {2 x}{b \sqrt {a + b x}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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