Optimal. Leaf size=47 \[ \frac{2 \sqrt{a x+b x^{2/3}}}{a}-\frac{4 b \sqrt{a x+b x^{2/3}}}{a^2 \sqrt [3]{x}} \]
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Rubi [A] time = 0.0495725, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2002, 2014} \[ \frac{2 \sqrt{a x+b x^{2/3}}}{a}-\frac{4 b \sqrt{a x+b x^{2/3}}}{a^2 \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b x^{2/3}+a x}} \, dx &=\frac{2 \sqrt{b x^{2/3}+a x}}{a}-\frac{(2 b) \int \frac{1}{\sqrt [3]{x} \sqrt{b x^{2/3}+a x}} \, dx}{3 a}\\ &=\frac{2 \sqrt{b x^{2/3}+a x}}{a}-\frac{4 b \sqrt{b x^{2/3}+a x}}{a^2 \sqrt [3]{x}}\\ \end{align*}
Mathematica [A] time = 0.0274036, size = 36, normalized size = 0.77 \[ \frac{2 \left (a \sqrt [3]{x}-2 b\right ) \sqrt{a x+b x^{2/3}}}{a^2 \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt [3]{x} \left ( b+a\sqrt [3]{x} \right ) \left ( a\sqrt [3]{x}-2\,b \right ) }{\sqrt{b{x}^{2/3}+ax}{a}^{2}}} \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}{\sqrt{a x + b x^{\frac{2}{3}}}}\,{d 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 \frac{1}{\sqrt{a x + b x^{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1424, size = 49, normalized size = 1.04 \begin{align*} \frac{4 \, b^{\frac{3}{2}}}{a^{2}} + \frac{2 \,{\left ({\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x^{\frac{1}{3}} + b} b\right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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