Optimal. Leaf size=55 \[ \frac{2 a^2}{3 b^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 a}{b^3 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^3} \]
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Rubi [A] time = 0.0248751, antiderivative size = 55, 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} \[ \frac{2 a^2}{3 b^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 a}{b^3 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{5/2}}-\frac{2 a}{b^2 (a+b x)^{3/2}}+\frac{1}{b^2 \sqrt{a+b x}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^2}{3 b^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 a}{b^3 \sqrt{a+\frac{b}{x}}}-\frac{2 \sqrt{a+\frac{b}{x}}}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0310525, size = 44, normalized size = 0.8 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (8 a^2 x^2+12 a b x+3 b^2\right )}{3 b^3 (a x+b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 44, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}+12\,xab+3\,{b}^{2} \right ) }{3\,{b}^{3}{x}^{3}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992823, size = 63, normalized size = 1.15 \begin{align*} -\frac{2 \, \sqrt{a + \frac{b}{x}}}{b^{3}} - \frac{4 \, a}{\sqrt{a + \frac{b}{x}} b^{3}} + \frac{2 \, a^{2}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72509, size = 117, normalized size = 2.13 \begin{align*} -\frac{2 \,{\left (8 \, a^{2} x^{2} + 12 \, a b x + 3 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.6538, size = 136, normalized size = 2.47 \begin{align*} \begin{cases} - \frac{16 a^{2} x^{2}}{3 a b^{3} x^{2} \sqrt{a + \frac{b}{x}} + 3 b^{4} x \sqrt{a + \frac{b}{x}}} - \frac{24 a b x}{3 a b^{3} x^{2} \sqrt{a + \frac{b}{x}} + 3 b^{4} x \sqrt{a + \frac{b}{x}}} - \frac{6 b^{2}}{3 a b^{3} x^{2} \sqrt{a + \frac{b}{x}} + 3 b^{4} x \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{3 a^{\frac{5}{2}} x^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30114, size = 78, normalized size = 1.42 \begin{align*} \frac{2}{3} \, b{\left (\frac{{\left (a^{2} - \frac{6 \,{\left (a x + b\right )} a}{x}\right )} x}{{\left (a x + b\right )} b^{4} \sqrt{\frac{a x + b}{x}}} - \frac{3 \, \sqrt{\frac{a x + b}{x}}}{b^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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