Optimal. Leaf size=57 \[ -\frac{2 a^2 \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.023212, antiderivative size = 57, 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} \[ -\frac{2 a^2 \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x}} x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt{a+b x}}-\frac{2 a \sqrt{a+b x}}{b^2}+\frac{(a+b x)^{3/2}}{b^2}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a^2 \sqrt{a+\frac{b}{x}}}{b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^3}\\ \end{align*}
Mathematica [A] time = 0.0220955, size = 40, normalized size = 0.7 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (8 a^2 x^2-4 a b x+3 b^2\right )}{15 b^3 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 44, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}-4\,xab+3\,{b}^{2} \right ) }{15\,{b}^{3}{x}^{3}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14922, size = 63, normalized size = 1.11 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}}{5 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a}{3 \, b^{3}} - \frac{2 \, \sqrt{a + \frac{b}{x}} a^{2}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4491, size = 88, normalized size = 1.54 \begin{align*} -\frac{2 \,{\left (8 \, a^{2} x^{2} - 4 \, a b x + 3 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{15 \, b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.78961, size = 813, normalized size = 14.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15703, size = 90, normalized size = 1.58 \begin{align*} -\frac{2 \,{\left (15 \, a^{2} \sqrt{\frac{a x + b}{x}} - \frac{10 \,{\left (a x + b\right )} a \sqrt{\frac{a x + b}{x}}}{x} + \frac{3 \,{\left (a x + b\right )}^{2} \sqrt{\frac{a x + b}{x}}}{x^{2}}\right )}}{15 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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