Optimal. Leaf size=55 \[ \frac{2 a^2}{b^3 \sqrt{a+\frac{b}{x}}}+\frac{4 a \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.0249515, 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} \[ \frac{2 a^2}{b^3 \sqrt{a+\frac{b}{x}}}+\frac{4 a \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{3/2}}-\frac{2 a}{b^2 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^2}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^2}{b^3 \sqrt{a+\frac{b}{x}}}+\frac{4 a \sqrt{a+\frac{b}{x}}}{b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^3}\\ \end{align*}
Mathematica [A] time = 0.022299, size = 40, normalized size = 0.73 \[ \frac{16 a^2 x^2+8 a b x-2 b^2}{3 b^3 x^2 \sqrt{a+\frac{b}{x}}} \]
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-{b}^{2} \right ) }{3\,{b}^{3}{x}^{3}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00635, size = 63, normalized size = 1.15 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b^{3}} + \frac{4 \, \sqrt{a + \frac{b}{x}} a}{b^{3}} + \frac{2 \, a^{2}}{\sqrt{a + \frac{b}{x}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48166, size = 96, normalized size = 1.75 \begin{align*} \frac{2 \,{\left (8 \, a^{2} x^{2} + 4 \, a b x - b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a b^{3} x^{2} + b^{4} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.70181, size = 457, normalized size = 8.31 \begin{align*} \frac{16 a^{\frac{9}{2}} b^{\frac{7}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} + \frac{24 a^{\frac{7}{2}} b^{\frac{9}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} + \frac{6 a^{\frac{5}{2}} b^{\frac{11}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{16 a^{5} b^{3} x^{\frac{7}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{32 a^{4} b^{4} x^{\frac{5}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} - \frac{16 a^{3} b^{5} x^{\frac{3}{2}}}{3 a^{\frac{7}{2}} b^{6} x^{\frac{7}{2}} + 6 a^{\frac{5}{2}} b^{7} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{8} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27384, size = 93, normalized size = 1.69 \begin{align*} \frac{2}{3} \, b{\left (\frac{3 \, a^{2}}{b^{4} \sqrt{\frac{a x + b}{x}}} + \frac{6 \, a b^{8} \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} b^{8} \sqrt{\frac{a x + b}{x}}}{x}}{b^{12}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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