Optimal. Leaf size=487 \[ -\frac {18 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{5 b^2 \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {9 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{5 b \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}}+\frac {6 \sqrt [3]{2} 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \left (1-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right ) \sqrt {\frac {1+2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}+2 \sqrt [3]{2} \left (-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right )^2}} \left (-\frac {b \left (a \sqrt [3]{x}+b x^{2/3}\right )}{a^2}\right )^{2/3} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}{1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}\right )|-7+4 \sqrt {3}\right )}{5 b^3 \sqrt {-\frac {1-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}}{\left (1-\sqrt {3}-2^{2/3} \sqrt [3]{-\frac {b \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{a^2}}\right )^2}} \left (a+2 b \sqrt [3]{x}\right ) \left (a \sqrt [3]{x}+b x^{2/3}\right )^{2/3}} \]
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Rubi [A]
time = 0.71, 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 = {2036, 348, 52,
63, 636, 633, 242, 225} \begin {gather*} \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 (\text {ArcSin}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 63
Rule 225
Rule 242
Rule 348
Rule 633
Rule 636
Rule 2036
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 61, normalized size = 0.13 \begin {gather*} \frac {9 \left (1+\frac {b \sqrt [3]{x}}{a}\right )^{2/3} x \, _2F_1\left (\frac {2}{3},\frac {7}{3};\frac {10}{3};-\frac {b \sqrt [3]{x}}{a}\right )}{7 \left (\left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{\frac {1}{3}}+b \,x^{\frac {2}{3}}\right )^{\frac {2}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (a \sqrt [3]{x} + b x^{\frac {2}{3}}\right )^{\frac {2}{3}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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