Optimal. Leaf size=88 \[ \frac{3 b x^{4/3} \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{4 \left (a+b \sqrt [3]{x}\right )}+\frac{a x \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{a+b \sqrt [3]{x}} \]
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Rubi [A] time = 0.0398336, antiderivative size = 88, normalized size of antiderivative = 1., 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, 646, 43} \[ \frac{3 b x^{4/3} \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{4 \left (a+b \sqrt [3]{x}\right )}+\frac{a x \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{a+b \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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Rule 1341
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \, dx &=3 \operatorname{Subst}\left (\int x^2 \sqrt{a^2+2 a b x+b^2 x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{\left (3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}\right ) \operatorname{Subst}\left (\int x^2 \left (a b+b^2 x\right ) \, dx,x,\sqrt [3]{x}\right )}{b \left (a+b \sqrt [3]{x}\right )}\\ &=\frac{\left (3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}\right ) \operatorname{Subst}\left (\int \left (a b x^2+b^2 x^3\right ) \, dx,x,\sqrt [3]{x}\right )}{b \left (a+b \sqrt [3]{x}\right )}\\ &=\frac{a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} x}{a+b \sqrt [3]{x}}+\frac{3 b \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} x^{4/3}}{4 \left (a+b \sqrt [3]{x}\right )}\\ \end{align*}
Mathematica [A] time = 0.0091801, size = 43, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b \sqrt [3]{x}\right )^2} \left (4 a x+3 b x^{4/3}\right )}{4 \left (a+b \sqrt [3]{x}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 43, normalized size = 0.5 \begin{align*}{\frac{1}{4}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 3\,b{x}^{4/3}+4\,ax \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01913, size = 28, normalized size = 0.32 \begin{align*} \frac{3}{4} \, b x^{\frac{4}{3}} + a x \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 \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11565, size = 35, normalized size = 0.4 \begin{align*} \frac{3}{4} \, b x^{\frac{4}{3}} \mathrm{sgn}\left (b x^{\frac{1}{3}} + a\right ) + a x \mathrm{sgn}\left (b x^{\frac{1}{3}} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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