Optimal. Leaf size=42 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{a+\frac{b}{x}}} \]
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Rubi [A] time = 0.0207117, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2}{a \sqrt{a+\frac{b}{x}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{2}{a \sqrt{a+\frac{b}{x}}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{a b}\\ &=-\frac{2}{a \sqrt{a+\frac{b}{x}}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0127945, size = 34, normalized size = 0.81 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b}{a x}+1\right )}{a \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 198, normalized size = 4.7 \begin{align*}{\frac{x}{b \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( -2\,{a}^{5/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{a}^{2}b+2\,{a}^{3/2} \left ( \left ( ax+b \right ) x \right ) ^{3/2}-4\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}xb+2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) xa{b}^{2}-2\,\sqrt{a}\sqrt{ \left ( ax+b \right ) x}{b}^{2}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){b}^{3} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}} \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 = 1.51984, size = 293, normalized size = 6.98 \begin{align*} \left [-\frac{2 \, a x \sqrt{\frac{a x + b}{x}} -{\left (a x + b\right )} \sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right )}{a^{3} x + a^{2} b}, -\frac{2 \,{\left (a x \sqrt{\frac{a x + b}{x}} +{\left (a x + b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right )\right )}}{a^{3} x + a^{2} b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.0464, size = 148, normalized size = 3.52 \begin{align*} - \frac{2 a^{3} x \sqrt{1 + \frac{b}{a x}}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} - \frac{a^{3} x \log{\left (\frac{b}{a x} \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} + \frac{2 a^{3} x \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x} \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15898, size = 70, normalized size = 1.67 \begin{align*} -2 \, b{\left (\frac{\arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b} + \frac{1}{a b \sqrt{\frac{a x + b}{x}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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