Optimal. Leaf size=74 \[ -\frac{3 \sqrt{a+\frac{b}{x}}}{b^2 \sqrt{x}}+\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}+\frac{2}{b x^{3/2} \sqrt{a+\frac{b}{x}}} \]
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Rubi [A] time = 0.0399742, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {337, 288, 321, 217, 206} \[ -\frac{3 \sqrt{a+\frac{b}{x}}}{b^2 \sqrt{x}}+\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}+\frac{2}{b x^{3/2} \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Rule 337
Rule 288
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{7/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} x^{3/2}}-\frac{6 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{b}\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} x^{3/2}}-\frac{3 \sqrt{a+\frac{b}{x}}}{b^2 \sqrt{x}}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{b^2}\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} x^{3/2}}-\frac{3 \sqrt{a+\frac{b}{x}}}{b^2 \sqrt{x}}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{b^2}\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} x^{3/2}}-\frac{3 \sqrt{a+\frac{b}{x}}}{b^2 \sqrt{x}}+\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0164479, size = 56, normalized size = 0.76 \[ -\frac{2 \sqrt{\frac{b}{a x}+1} \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};-\frac{b}{a x}\right )}{5 a x^{5/2} \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 61, normalized size = 0.8 \begin{align*} -{\frac{1}{ax+b}\sqrt{{\frac{ax+b}{x}}} \left ( -3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}xa+3\,ax\sqrt{b}+{b}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{x}}}{b}^{-{\frac{5}{2}}}} \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 = 1.56506, size = 410, normalized size = 5.54 \begin{align*} \left [\frac{3 \,{\left (a^{2} x^{2} + a b x\right )} \sqrt{b} \log \left (\frac{a x + 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) - 2 \,{\left (3 \, a b x + b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{2 \,{\left (a b^{3} x^{2} + b^{4} x\right )}}, -\frac{3 \,{\left (a^{2} x^{2} + a b x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) +{\left (3 \, a b x + b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{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 [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 = 1.3323, size = 78, normalized size = 1.05 \begin{align*} -a{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \, a x + b}{{\left ({\left (a x + b\right )}^{\frac{3}{2}} - \sqrt{a x + b} b\right )} b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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