Optimal. Leaf size=32 \[ \frac{2 a}{3 b^2 (a+b x)^{3/2}}-\frac{2}{b^2 \sqrt{a+b x}} \]
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Rubi [A] time = 0.0084376, antiderivative size = 32, normalized size of antiderivative = 1., 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} \[ \frac{2 a}{3 b^2 (a+b x)^{3/2}}-\frac{2}{b^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x}{(a+b x)^{5/2}} \, dx &=\int \left (-\frac{a}{b (a+b x)^{5/2}}+\frac{1}{b (a+b x)^{3/2}}\right ) \, dx\\ &=\frac{2 a}{3 b^2 (a+b x)^{3/2}}-\frac{2}{b^2 \sqrt{a+b x}}\\ \end{align*}
Mathematica [A] time = 0.0117096, size = 24, normalized size = 0.75 \[ -\frac{2 (2 a+3 b x)}{3 b^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.7 \begin{align*} -{\frac{6\,bx+4\,a}{3\,{b}^{2}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04362, size = 35, normalized size = 1.09 \begin{align*} -\frac{2}{\sqrt{b x + a} b^{2}} + \frac{2 \, a}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60219, size = 89, normalized size = 2.78 \begin{align*} -\frac{2 \,{\left (3 \, b x + 2 \, a\right )} \sqrt{b x + a}}{3 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.40403, size = 80, normalized size = 2.5 \begin{align*} \begin{cases} - \frac{4 a}{3 a b^{2} \sqrt{a + b x} + 3 b^{3} x \sqrt{a + b x}} - \frac{6 b x}{3 a b^{2} \sqrt{a + b x} + 3 b^{3} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12735, size = 27, normalized size = 0.84 \begin{align*} -\frac{2 \,{\left (3 \, b x + 2 \, a\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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