Optimal. Leaf size=487 \[ \frac {6 \sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \left (1-2^{2/3} \sqrt [3]{-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}-\sqrt {3}+1\right )^2}} \left (-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3} F\left (\sin ^{-1}\left (\frac {-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}+\sqrt {3}+1}{-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{5 b^3 \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac {b \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{a^2}}-\sqrt {3}+1\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}-\frac {18 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \]
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Rubi [A] time = 0.97, antiderivative size = 487, normalized size of antiderivative = 1.00, 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 50
Rule 61
Rule 219
Rule 236
Rule 341
Rule 619
Rule 622
Rule 2011
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*}
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Mathematica [C] time = 0.02, 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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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{\frac {2}{3}} + a x^{\frac {1}{3}}\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{\frac {2}{3}}+a \,x^{\frac {1}{3}}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{\frac {2}{3}} + a x^{\frac {1}{3}}\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.25, size = 42, normalized size = 0.09 \[ \frac {9\,x\,{\left (\frac {b\,x^{1/3}}{a}+1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {2}{3},\frac {7}{3};\ \frac {10}{3};\ -\frac {b\,x^{1/3}}{a}\right )}{7\,{\left (a\,x^{1/3}+b\,x^{2/3}\right )}^{2/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sqrt [3]{x} + b x^{\frac {2}{3}}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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