Optimal. Leaf size=291 \[ \frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}}+\frac{15 a^4 b x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{2 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{30 a^3 b^2 \sqrt [3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}}-\frac{15 a b^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{\sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}-\frac{3 b^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{2 x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{30 a^2 b^3 \log \left (\sqrt [3]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}} \]
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Rubi [A] time = 0.137337, antiderivative size = 291, normalized size of antiderivative = 1., 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} \[ \frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}}+\frac{15 a^4 b x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{2 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{30 a^3 b^2 \sqrt [3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}}-\frac{15 a b^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{\sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}-\frac{3 b^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{2 x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{30 a^2 b^3 \log \left (\sqrt [3]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 1341
Rule 1355
Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}\right )^{5/2} \, dx &=3 \operatorname{Subst}\left (\int \left (a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}\right )^{5/2} 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 )^5 x^2 \, dx,x,\sqrt [3]{x}\right )}{b^4 \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 )^5}{x^3} \, dx,x,\sqrt [3]{x}\right )}{b^4 \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 (10 a^3 b^7+\frac{b^{10}}{x^3}+\frac{5 a b^9}{x^2}+\frac{10 a^2 b^8}{x}+5 a^4 b^6 x+a^5 b^5 x^2\right ) \, dx,x,\sqrt [3]{x}\right )}{b^4 \left (a b+\frac{b^2}{\sqrt [3]{x}}\right )}\\ &=-\frac{3 b^6 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}{2 \left (a b+\frac{b^2}{\sqrt [3]{x}}\right ) x^{2/3}}-\frac{15 a b^5 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}{\left (a b+\frac{b^2}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}+\frac{30 a^3 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{15 a^4 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^5 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}} x}{a+\frac{b}{\sqrt [3]{x}}}+\frac{10 a^2 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*}
Mathematica [A] time = 0.0591039, size = 99, normalized size = 0.34 \[ \frac{\left (a \sqrt [3]{x}+b\right ) \left (20 a^2 b^3 x^{2/3} \log (x)+60 a^3 b^2 x+15 a^4 b x^{4/3}+2 a^5 x^{5/3}-30 a b^4 \sqrt [3]{x}-3 b^5\right )}{2 x \sqrt{\frac{\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 91, normalized size = 0.3 \begin{align*}{\frac{x}{2} \left ({ \left ({a}^{2}{x}^{{\frac{2}{3}}}+2\,ab\sqrt [3]{x}+{b}^{2} \right ){x}^{-{\frac{2}{3}}}} \right ) ^{{\frac{5}{2}}} \left ( 15\,{a}^{4}b{x}^{4/3}+60\,{a}^{3}{b}^{2}x+20\,{a}^{2}{b}^{3}\ln \left ( x \right ){x}^{2/3}+2\,{a}^{5}{x}^{5/3}-30\,a{b}^{4}\sqrt [3]{x}-3\,{b}^{5} \right ) \left ( b+a\sqrt [3]{x} \right ) ^{-5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03974, size = 77, normalized size = 0.26 \begin{align*} 10 \, a^{2} b^{3} \log \left (x\right ) + \frac{2 \, a^{5} x^{\frac{5}{3}} + 15 \, a^{4} b x^{\frac{4}{3}} + 60 \, a^{3} b^{2} x - 30 \, a b^{4} x^{\frac{1}{3}} - 3 \, b^{5}}{2 \, x^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26051, size = 173, normalized size = 0.59 \begin{align*} a^{5} x \mathrm{sgn}\left (a x + b x^{\frac{2}{3}}\right ) \mathrm{sgn}\left (x\right ) + 10 \, a^{2} b^{3} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (a x + b x^{\frac{2}{3}}\right ) \mathrm{sgn}\left (x\right ) + \frac{15}{2} \, a^{4} b x^{\frac{2}{3}} \mathrm{sgn}\left (a x + b x^{\frac{2}{3}}\right ) \mathrm{sgn}\left (x\right ) + 30 \, a^{3} b^{2} x^{\frac{1}{3}} \mathrm{sgn}\left (a x + b x^{\frac{2}{3}}\right ) \mathrm{sgn}\left (x\right ) - \frac{3 \,{\left (10 \, a b^{4} x^{\frac{1}{3}} \mathrm{sgn}\left (a x + b x^{\frac{2}{3}}\right ) \mathrm{sgn}\left (x\right ) + b^{5} \mathrm{sgn}\left (a x + b x^{\frac{2}{3}}\right ) \mathrm{sgn}\left (x\right )\right )}}{2 \, x^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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