Optimal. Leaf size=25 \[ \frac{4 \sqrt{x}}{b \sqrt{a x+b \sqrt{x}}} \]
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Rubi [A] time = 0.0051318, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2000} \[ \frac{4 \sqrt{x}}{b \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Rule 2000
Rubi steps
\begin{align*} \int \frac{1}{\left (b \sqrt{x}+a x\right )^{3/2}} \, dx &=\frac{4 \sqrt{x}}{b \sqrt{b \sqrt{x}+a x}}\\ \end{align*}
Mathematica [A] time = 0.0164931, size = 25, normalized size = 1. \[ \frac{4 \sqrt{x}}{b \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.008, size = 404, normalized size = 16.2 \begin{align*}{\frac{1}{{b}^{2}}\sqrt{b\sqrt{x}+ax} \left ( 2\,\sqrt{b\sqrt{x}+ax}{a}^{5/2}x+\ln \left ({\frac{1}{2} \left ( 2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b \right ){\frac{1}{\sqrt{a}}}} \right ) x{a}^{2}b+2\,{a}^{5/2}x\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b \right ){\frac{1}{\sqrt{a}}}} \right ) x{a}^{2}b+4\,\sqrt{b\sqrt{x}+ax}{a}^{3/2}\sqrt{x}b+2\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ) \sqrt{x}a{b}^{2}+4\,{a}^{3/2}\sqrt{x}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }b-4\,{a}^{3/2} \left ( \sqrt{x} \left ( b+a\sqrt{x} \right ) \right ) ^{3/2}-2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ) \sqrt{x}a{b}^{2}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}{b}^{2}+\ln \left ({\frac{1}{2} \left ( 2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b \right ){\frac{1}{\sqrt{a}}}} \right ){b}^{3}+2\,\sqrt{a}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{2}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b \right ){\frac{1}{\sqrt{a}}}} \right ){b}^{3} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}} \left ( b+a\sqrt{x} \right ) ^{-2}{\frac{1}{\sqrt{a}}}} \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*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28203, size = 77, normalized size = 3.08 \begin{align*} \frac{4 \, \sqrt{a x + b \sqrt{x}}{\left (a \sqrt{x} - b\right )}}{a^{2} b x - b^{3}} \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 \frac{1}{\left (a x + b \sqrt{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 = 1.14704, size = 46, normalized size = 1.84 \begin{align*} \frac{4}{{\left (\sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} + b\right )} \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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