Optimal. Leaf size=487 \[ \frac{6 \sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} a^4 \left (1-2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}-\sqrt{3}+1\right )^2}} \left (-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{5 b^3 \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}-\sqrt{3}+1\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}-\frac{18 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{9 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \]
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Rubi [A] time = 0.973875, antiderivative size = 487, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {2011, 341, 50, 61, 622, 619, 236, 219} \[ \frac{6 \sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} a^4 \left (1-2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}-\sqrt{3}+1\right )^2}} \left (-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{5 b^3 \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}-\sqrt{3}+1\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}-\frac{18 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{9 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2011
Rule 341
Rule 50
Rule 61
Rule 622
Rule 619
Rule 236
Rule 219
Rubi steps
\begin{align*} \int \frac{1}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \, dx &=\frac{\left (\left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}\right ) \int \frac{1}{\left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}} \, dx}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\\ &=\frac{\left (3 \left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}\right ) \operatorname{Subst}\left (\int \frac{x^{4/3}}{(a+b x)^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\\ &=\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}-\frac{\left (12 a \left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{x}}{(a+b x)^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\\ &=-\frac{18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{\left (6 a^2 \left (a+b \sqrt [3]{x}\right )^{2/3} x^{2/9}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{2/3} (a+b x)^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\\ &=-\frac{18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{\left (6 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a x+b x^2\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{5 b^2}\\ &=-\frac{18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{\left (6 a^2 \left (-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{b x}{a}-\frac{b^2 x^2}{a^2}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\\ &=-\frac{18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}-\frac{\left (6 \sqrt [3]{2} a^4 \left (-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{a^2 x^2}{b^2}\right )^{2/3}} \, dx,x,-\frac{b \left (a+2 b \sqrt [3]{x}\right )}{a^2}\right )}{5 b^4 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\\ &=-\frac{18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}-\frac{\left (9 \sqrt [3]{2} a^4 \sqrt{-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}} \left (-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right )}{5 b^3 \left (a+2 b \sqrt [3]{x}\right ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\\ &=-\frac{18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac{6 \sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} a^4 \left (1-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right ) \sqrt{\frac{1+\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}+\left (1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right )^2}} \left (-\frac{b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}}{1-\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}}\right )|-7+4 \sqrt{3}\right )}{5 b^3 \sqrt{-\frac{1-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}}{\left (1-\sqrt{3}-\sqrt [3]{1-\frac{\left (a+2 b \sqrt [3]{x}\right )^2}{a^2}}\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0159076, size = 61, normalized size = 0.13 \[ \frac{9 x \left (\frac{b \sqrt [3]{x}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{7}{3};\frac{10}{3};-\frac{b \sqrt [3]{x}}{a}\right )}{7 \left (\sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.202, size = 0, normalized size = 0. \begin{align*} \int \left ( a\sqrt [3]{x}+b{x}^{{\frac{2}{3}}} \right ) ^{-{\frac{2}{3}}}\, dx \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 (b x^{\frac{2}{3}} + a x^{\frac{1}{3}}\right )}^{\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}{\left (a \sqrt [3]{x} + b x^{\frac{2}{3}}\right )^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{\frac{2}{3}} + a x^{\frac{1}{3}}\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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