Optimal. Leaf size=147 \[ -\frac{3 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{b^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{2 b \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rubi [A] time = 0.0703464, antiderivative size = 147, 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} \[ -\frac{3 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{b^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{2 b \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
Antiderivative was successfully verified.
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Rule 1341
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{\left (3 b \left (a+b \sqrt [3]{x}\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+b^2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=\frac{\left (3 b \left (a+b \sqrt [3]{x}\right )\right ) \operatorname{Subst}\left (\int \left (-\frac{a}{b^3}+\frac{x}{b^2}+\frac{a^2}{b^3 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=-\frac{3 a \left (a+b \sqrt [3]{x}\right ) \sqrt [3]{x}}{b^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 \left (a+b \sqrt [3]{x}\right ) x^{2/3}}{2 b \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ \end{align*}
Mathematica [A] time = 0.0380569, size = 65, normalized size = 0.44 \[ \frac{3 \left (a+b \sqrt [3]{x}\right ) \left (2 a^2 \log \left (a+b \sqrt [3]{x}\right )+b \sqrt [3]{x} \left (b \sqrt [3]{x}-2 a\right )\right )}{2 b^3 \sqrt{\left (a+b \sqrt [3]{x}\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 103, normalized size = 0.7 \begin{align*}{\frac{1}{2\,{b}^{3}}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 3\,{b}^{2}{x}^{2/3}+2\,{a}^{2}\ln \left ({b}^{3}x+{a}^{3} \right ) -2\,{a}^{2}\ln \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) +4\,{a}^{2}\ln \left ( a+b\sqrt [3]{x} \right ) -6\,ab\sqrt [3]{x} \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 [A] time = 1.06445, size = 62, normalized size = 0.42 \begin{align*} \frac{3 \, a^{2} b^{2} \log \left (x^{\frac{1}{3}} + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{3 \, a b x^{\frac{1}{3}}}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \, x^{\frac{2}{3}}}{2 \, \sqrt{b^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06321, size = 89, normalized size = 0.61 \begin{align*} \frac{3 \,{\left (2 \, a^{2} \log \left (b x^{\frac{1}{3}} + a\right ) + b^{2} x^{\frac{2}{3}} - 2 \, a b x^{\frac{1}{3}}\right )}}{2 \, b^{3}} \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 \frac{1}{\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 = 1.13357, size = 82, normalized size = 0.56 \begin{align*} \frac{3 \,{\left (b x^{\frac{2}{3}} \mathrm{sgn}\left (b x^{\frac{1}{3}} + a\right ) - 2 \, a x^{\frac{1}{3}} \mathrm{sgn}\left (b x^{\frac{1}{3}} + a\right )\right )}}{2 \, b^{2}} + \frac{3 \, a^{2} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{3} \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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