Optimal. Leaf size=73 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\sqrt{x}}{4 a b (a x+b)}-\frac{\sqrt{x}}{2 a (a x+b)^2} \]
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Rubi [A] time = 0.0222098, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {263, 47, 51, 63, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\sqrt{x}}{4 a b (a x+b)}-\frac{\sqrt{x}}{2 a (a x+b)^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 47
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^{5/2}} \, dx &=\int \frac{\sqrt{x}}{(b+a x)^3} \, dx\\ &=-\frac{\sqrt{x}}{2 a (b+a x)^2}+\frac{\int \frac{1}{\sqrt{x} (b+a x)^2} \, dx}{4 a}\\ &=-\frac{\sqrt{x}}{2 a (b+a x)^2}+\frac{\sqrt{x}}{4 a b (b+a x)}+\frac{\int \frac{1}{\sqrt{x} (b+a x)} \, dx}{8 a b}\\ &=-\frac{\sqrt{x}}{2 a (b+a x)^2}+\frac{\sqrt{x}}{4 a b (b+a x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{4 a b}\\ &=-\frac{\sqrt{x}}{2 a (b+a x)^2}+\frac{\sqrt{x}}{4 a b (b+a x)}+\frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{3/2} b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0047959, size = 27, normalized size = 0.37 \[ \frac{2 x^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};-\frac{a x}{b}\right )}{3 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 52, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{ \left ( ax+b \right ) ^{2}} \left ( 1/8\,{\frac{{x}^{3/2}}{b}}-1/8\,{\frac{\sqrt{x}}{a}} \right ) }+{\frac{1}{4\,ab}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74412, size = 412, normalized size = 5.64 \begin{align*} \left [-\frac{{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{-a b} \log \left (\frac{a x - b - 2 \, \sqrt{-a b} \sqrt{x}}{a x + b}\right ) - 2 \,{\left (a^{2} b x - a b^{2}\right )} \sqrt{x}}{8 \,{\left (a^{4} b^{2} x^{2} + 2 \, a^{3} b^{3} x + a^{2} b^{4}\right )}}, -\frac{{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{a \sqrt{x}}\right ) -{\left (a^{2} b x - a b^{2}\right )} \sqrt{x}}{4 \,{\left (a^{4} b^{2} x^{2} + 2 \, a^{3} b^{3} x + a^{2} b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09483, size = 70, normalized size = 0.96 \begin{align*} \frac{\arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a b} + \frac{a x^{\frac{3}{2}} - b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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