Optimal. Leaf size=88 \[ \frac {3 b x^{2/3} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {a x \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}} \]
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Rubi [A] time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1341, 1355, 14} \[ \frac {3 b x^{2/3} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {a x \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1341
Rule 1355
Rubi steps
\begin {align*} \int \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \, dx &=3 \operatorname {Subst}\left (\int \sqrt {a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}} 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 ) x^2 \, dx,x,\sqrt [3]{x}\right )}{a b+\frac {b^2}{\sqrt [3]{x}}}\\ &=\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 (b^2 x+a b x^2\right ) \, dx,x,\sqrt [3]{x}\right )}{a b+\frac {b^2}{\sqrt [3]{x}}}\\ &=\frac {3 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 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 0.56 \[ \frac {\sqrt {\frac {\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}} \left (2 a x^{4/3}+3 b x\right )}{2 \left (a \sqrt [3]{x}+b\right )} \]
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 = 0.40, size = 34, normalized size = 0.39 \[ a x \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\relax (x) + \frac {3}{2} \, b x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 50, normalized size = 0.57 \[ \frac {\sqrt {\frac {a^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}}\, \left (2 a x +3 b \,x^{\frac {2}{3}}\right ) x^{\frac {1}{3}}}{2 a \,x^{\frac {1}{3}}+2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 10, normalized size = 0.11 \[ a x + \frac {3}{2} \, b x^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 39, normalized size = 0.44 \[ \frac {x\,\left (a+\frac {3\,b}{2\,x^{1/3}}\right )\,\sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}}}{a+\frac {b}{x^{1/3}}} \]
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 \[ \int \sqrt {a^{2} + \frac {2 a b}{\sqrt [3]{x}} + \frac {b^{2}}{x^{\frac {2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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