Optimal. Leaf size=179 \[ \frac{a^3 x \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}+\frac{6 a^2 b \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}-\frac{2 b^3 \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{\sqrt{x} \left (a+\frac{b}{\sqrt{x}}\right )}+\frac{6 a b^2 \log \left (\sqrt{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}} \]
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Rubi [A] time = 0.0922638, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1341, 1355, 263, 43} \[ \frac{a^3 x \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}+\frac{6 a^2 b \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}}-\frac{2 b^3 \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{\sqrt{x} \left (a+\frac{b}{\sqrt{x}}\right )}+\frac{6 a b^2 \log \left (\sqrt{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt{x}}+\frac{b^2}{x}}}{a+\frac{b}{\sqrt{x}}} \]
Antiderivative was successfully verified.
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Rule 1341
Rule 1355
Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a^2+\frac{b^2}{x}+\frac{2 a b}{\sqrt{x}}\right )^{3/2} \, dx &=2 \operatorname{Subst}\left (\int \left (a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}\right )^{3/2} x \, dx,x,\sqrt{x}\right )\\ &=\frac{\left (2 \sqrt{a^2+\frac{b^2}{x}+\frac{2 a b}{\sqrt{x}}}\right ) \operatorname{Subst}\left (\int \left (a b+\frac{b^2}{x}\right )^3 x \, dx,x,\sqrt{x}\right )}{b^2 \left (a b+\frac{b^2}{\sqrt{x}}\right )}\\ &=\frac{\left (2 \sqrt{a^2+\frac{b^2}{x}+\frac{2 a b}{\sqrt{x}}}\right ) \operatorname{Subst}\left (\int \frac{\left (b^2+a b x\right )^3}{x^2} \, dx,x,\sqrt{x}\right )}{b^2 \left (a b+\frac{b^2}{\sqrt{x}}\right )}\\ &=\frac{\left (2 \sqrt{a^2+\frac{b^2}{x}+\frac{2 a b}{\sqrt{x}}}\right ) \operatorname{Subst}\left (\int \left (3 a^2 b^4+\frac{b^6}{x^2}+\frac{3 a b^5}{x}+a^3 b^3 x\right ) \, dx,x,\sqrt{x}\right )}{b^2 \left (a b+\frac{b^2}{\sqrt{x}}\right )}\\ &=-\frac{2 b^4 \sqrt{a^2+\frac{b^2}{x}+\frac{2 a b}{\sqrt{x}}}}{\left (a b+\frac{b^2}{\sqrt{x}}\right ) \sqrt{x}}+\frac{6 a^2 b^2 \sqrt{a^2+\frac{b^2}{x}+\frac{2 a b}{\sqrt{x}}} \sqrt{x}}{a b+\frac{b^2}{\sqrt{x}}}+\frac{a^3 \sqrt{a^2+\frac{b^2}{x}+\frac{2 a b}{\sqrt{x}}} x}{a+\frac{b}{\sqrt{x}}}+\frac{3 a b^3 \sqrt{a^2+\frac{b^2}{x}+\frac{2 a b}{\sqrt{x}}} \log (x)}{a b+\frac{b^2}{\sqrt{x}}}\\ \end{align*}
Mathematica [A] time = 0.0335123, size = 66, normalized size = 0.37 \[ \frac{\sqrt{\frac{\left (a \sqrt{x}+b\right )^2}{x}} \left (6 a^2 b x+a^3 x^{3/2}+3 a b^2 \sqrt{x} \log (x)-2 b^3\right )}{a \sqrt{x}+b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 68, normalized size = 0.4 \begin{align*}{\sqrt{{ \left ({a}^{2}{x}^{{\frac{3}{2}}}+{b}^{2}\sqrt{x}+2\,abx \right ){x}^{-{\frac{3}{2}}}}} \left ({x}^{{\frac{3}{2}}}{a}^{3}+6\,x{a}^{2}b+3\,\sqrt{x}\ln \left ( x \right ) a{b}^{2}-2\,{b}^{3} \right ) \left ( a\sqrt{x}+b \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} x + 3 \, a b^{2} \int \frac{1}{x}\,{d x} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, b^{3}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a^{2} + \frac{2 a b}{\sqrt{x}} + \frac{b^{2}}{x}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18833, size = 108, normalized size = 0.6 \begin{align*} a^{3} x \mathrm{sgn}\left (a x + b \sqrt{x}\right ) \mathrm{sgn}\left (x\right ) + 3 \, a b^{2} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (a x + b \sqrt{x}\right ) \mathrm{sgn}\left (x\right ) + 6 \, a^{2} b \sqrt{x} \mathrm{sgn}\left (a x + b \sqrt{x}\right ) \mathrm{sgn}\left (x\right ) - \frac{2 \, b^{3} \mathrm{sgn}\left (a x + b \sqrt{x}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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