Optimal. Leaf size=90 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{4 b^{3/2}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{4 b x^{2/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{2 x} \]
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Rubi [A] time = 0.138582, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2020, 2025, 2029, 206} \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{4 b^{3/2}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{4 b x^{2/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{2 x} \]
Antiderivative was successfully verified.
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Rule 2020
Rule 2025
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{b x^{2/3}+a x}}{x^2} \, dx &=-\frac{3 \sqrt{b x^{2/3}+a x}}{2 x}+\frac{1}{4} a \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{2 x}-\frac{3 a \sqrt{b x^{2/3}+a x}}{4 b x^{2/3}}-\frac{a^2 \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{8 b}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{2 x}-\frac{3 a \sqrt{b x^{2/3}+a x}}{4 b x^{2/3}}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{4 b}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{2 x}-\frac{3 a \sqrt{b x^{2/3}+a x}}{4 b x^{2/3}}+\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{4 b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0415882, size = 57, normalized size = 0.63 \[ -\frac{2 a^2 \left (a \sqrt [3]{x}+b\right ) \sqrt{a x+b x^{2/3}} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^3 \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 80, normalized size = 0.9 \begin{align*}{\frac{3}{4\,x}\sqrt{b{x}^{{\frac{2}{3}}}+ax} \left ({\it Artanh} \left ({\sqrt{b+a\sqrt [3]{x}}{\frac{1}{\sqrt{b}}}} \right ) b{a}^{2}{x}^{{\frac{2}{3}}}-{b}^{{\frac{3}{2}}} \left ( b+a\sqrt [3]{x} \right ) ^{{\frac{3}{2}}}-{b}^{{\frac{5}{2}}}\sqrt{b+a\sqrt [3]{x}} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{b+a\sqrt [3]{x}}}}} \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{\sqrt{a x + b x^{\frac{2}{3}}}}{x^{2}}\,{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{\sqrt{a x + b x^{\frac{2}{3}}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21738, size = 97, normalized size = 1.08 \begin{align*} -\frac{3 \,{\left (\frac{a^{3} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{3} + \sqrt{a x^{\frac{1}{3}} + b} a^{3} b}{a^{2} b x^{\frac{2}{3}}}\right )}}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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