Optimal. Leaf size=189 \[ \frac {9 a^2 b x^{2/3} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {9 a b^2 \sqrt [3]{x} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {3 b^3 \log \left (\sqrt [3]{x}\right ) \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {a^3 x \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}} \]
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Rubi [A] time = 0.09, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1341, 1355, 263, 43} \[ \frac {a^3 x \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {9 a^2 b x^{2/3} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {9 a b^2 \sqrt [3]{x} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {3 b^3 \log \left (\sqrt [3]{x}\right ) \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rule 1341
Rule 1355
Rubi steps
\begin {align*} \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{3/2} \, dx &=3 \operatorname {Subst}\left (\int \left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right )^{3/2} x^2 \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \operatorname {Subst}\left (\int \left (a b+\frac {b^2}{x}\right )^3 x^2 \, dx,x,\sqrt [3]{x}\right )}{b^2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )}\\ &=\frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \operatorname {Subst}\left (\int \frac {\left (b^2+a b x\right )^3}{x} \, dx,x,\sqrt [3]{x}\right )}{b^2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )}\\ &=\frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \operatorname {Subst}\left (\int \left (3 a b^5+\frac {b^6}{x}+3 a^2 b^4 x+a^3 b^3 x^2\right ) \, dx,x,\sqrt [3]{x}\right )}{b^2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )}\\ &=\frac {9 a b^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \sqrt [3]{x}}{a b+\frac {b^2}{\sqrt [3]{x}}}+\frac {9 a^2 b^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x^{2/3}}{2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )}+\frac {a^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}}+\frac {b^4 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \log (x)}{a b+\frac {b^2}{\sqrt [3]{x}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 77, normalized size = 0.41 \[ \frac {\sqrt {\frac {\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}} \left (2 a^3 x^{4/3}+9 a^2 b x+18 a b^2 x^{2/3}+2 b^3 \sqrt [3]{x} \log (x)\right )}{2 \left (a \sqrt [3]{x}+b\right )} \]
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 [A] time = 0.41, size = 79, normalized size = 0.42 \[ a^{3} x \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\relax (x) + b^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\relax (x) + \frac {9}{2} \, a^{2} b x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\relax (x) + 9 \, a b^{2} x^{\frac {1}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 69, normalized size = 0.37 \[ \frac {\left (\frac {a^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {3}{2}} \left (2 a^{3} x +2 b^{3} \ln \relax (x )+9 a^{2} b \,x^{\frac {2}{3}}+18 a \,b^{2} x^{\frac {1}{3}}\right ) x}{2 \left (a \,x^{\frac {1}{3}}+b \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 30, normalized size = 0.16 \[ a^{3} x + b^{3} \log \relax (x) + \frac {9}{2} \, a^{2} b x^{\frac {2}{3}} + 9 \, a b^{2} x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}\right )}^{3/2} \,d x \]
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 \[ \int \left (a^{2} + \frac {2 a b}{\sqrt [3]{x}} + \frac {b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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