Optimal. Leaf size=77 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{b \sqrt{a+b \sqrt{x}}}{2 a \sqrt{x}}-\frac{\sqrt{a+b \sqrt{x}}}{x} \]
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Rubi [A] time = 0.0336536, antiderivative size = 77, 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 = {266, 47, 51, 63, 208} \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{b \sqrt{a+b \sqrt{x}}}{2 a \sqrt{x}}-\frac{\sqrt{a+b \sqrt{x}}}{x} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sqrt{x}}}{x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{x}+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{x}-\frac{b \sqrt{a+b \sqrt{x}}}{2 a \sqrt{x}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{4 a}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{x}-\frac{b \sqrt{a+b \sqrt{x}}}{2 a \sqrt{x}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sqrt{x}}\right )}{2 a}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{x}-\frac{b \sqrt{a+b \sqrt{x}}}{2 a \sqrt{x}}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0086862, size = 43, normalized size = 0.56 \[ -\frac{4 b^2 \left (a+b \sqrt{x}\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{\sqrt{x} b}{a}+1\right )}{3 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 59, normalized size = 0.8 \begin{align*} 4\,{b}^{2} \left ({\frac{1}{{b}^{2}x} \left ( -1/8\,{\frac{ \left ( a+b\sqrt{x} \right ) ^{3/2}}{a}}-1/8\,\sqrt{a+b\sqrt{x}} \right ) }+1/8\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) \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 = 1.32063, size = 351, normalized size = 4.56 \begin{align*} \left [\frac{\sqrt{a} b^{2} x \log \left (\frac{b x + 2 \, \sqrt{b \sqrt{x} + a} \sqrt{a} \sqrt{x} + 2 \, a \sqrt{x}}{x}\right ) - 2 \,{\left (a b \sqrt{x} + 2 \, a^{2}\right )} \sqrt{b \sqrt{x} + a}}{4 \, a^{2} x}, -\frac{\sqrt{-a} b^{2} x \arctan \left (\frac{\sqrt{b \sqrt{x} + a} \sqrt{-a}}{a}\right ) +{\left (a b \sqrt{x} + 2 \, a^{2}\right )} \sqrt{b \sqrt{x} + a}}{2 \, a^{2} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.48357, size = 105, normalized size = 1.36 \begin{align*} - \frac{a}{\sqrt{b} x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{3 \sqrt{b}}{2 x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{b^{\frac{3}{2}}}{2 a \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{2 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10865, size = 84, normalized size = 1.09 \begin{align*} -\frac{1}{2} \, b^{2}{\left (\frac{\arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} + \sqrt{b \sqrt{x} + a} a}{a b^{2} x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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