Optimal. Leaf size=300 \[ \frac{3 b^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac{15 b^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}+\frac{18 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{9 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{30 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]
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Rubi [A] time = 0.187466, antiderivative size = 300, 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} \[ \frac{3 b^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac{15 b^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}+\frac{18 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{9 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{30 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]
Antiderivative was successfully verified.
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Rule 1341
Rule 1355
Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\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 \frac{x^2}{\left (a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{\left (3 b^2 \left (a b+\frac{b^2}{\sqrt [3]{x}}\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a b+\frac{b^2}{x}\right )^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}\\ &=\frac{\left (3 b^2 \left (a b+\frac{b^2}{\sqrt [3]{x}}\right )\right ) \operatorname{Subst}\left (\int \frac{x^5}{\left (b^2+a b x\right )^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}\\ &=\frac{\left (3 b^2 \left (a b+\frac{b^2}{\sqrt [3]{x}}\right )\right ) \operatorname{Subst}\left (\int \left (\frac{6}{a^5 b}-\frac{3 x}{a^4 b^2}+\frac{x^2}{a^3 b^3}-\frac{b^2}{a^5 (b+a x)^3}+\frac{5 b}{a^5 (b+a x)^2}-\frac{10}{a^5 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}\\ &=\frac{3 \left (a b^5+\frac{b^6}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )^2}-\frac{15 \left (a b^4+\frac{b^5}{\sqrt [3]{x}}\right )}{a^6 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}} \left (b+a \sqrt [3]{x}\right )}+\frac{18 \left (a b^2+\frac{b^3}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}{a^5 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}-\frac{9 \left (a b+\frac{b^2}{\sqrt [3]{x}}\right ) x^{2/3}}{2 a^4 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}+\frac{\left (a+\frac{b}{\sqrt [3]{x}}\right ) x}{a^3 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}-\frac{30 \left (a b^3+\frac{b^4}{\sqrt [3]{x}}\right ) \log \left (b+a \sqrt [3]{x}\right )}{a^6 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}\\ \end{align*}
Mathematica [A] time = 0.0897873, size = 126, normalized size = 0.42 \[ \frac{\left (a \sqrt [3]{x}+b\right ) \left (63 a^2 b^3 x^{2/3}+20 a^3 b^2 x-5 a^4 b x^{4/3}+2 a^5 x^{5/3}+6 a b^4 \sqrt [3]{x}-60 b^3 \left (a \sqrt [3]{x}+b\right )^2 \log \left (a \sqrt [3]{x}+b\right )-27 b^5\right )}{2 a^6 x \left (\frac{\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 141, normalized size = 0.5 \begin{align*}{\frac{1}{2\,x{a}^{6}} \left ( 2\,{a}^{5}{x}^{5/3}-5\,{a}^{4}b{x}^{4/3}-60\,{x}^{2/3}\ln \left ( b+a\sqrt [3]{x} \right ){a}^{2}{b}^{3}+63\,{x}^{2/3}{a}^{2}{b}^{3}-120\,\sqrt [3]{x}\ln \left ( b+a\sqrt [3]{x} \right ) a{b}^{4}+6\,a{b}^{4}\sqrt [3]{x}-60\,\ln \left ( b+a\sqrt [3]{x} \right ){b}^{5}+20\,{a}^{3}{b}^{2}x-27\,{b}^{5} \right ) \left ( b+a\sqrt [3]{x} \right ) \left ({ \left ({a}^{2}{x}^{{\frac{2}{3}}}+2\,ab\sqrt [3]{x}+{b}^{2} \right ){x}^{-{\frac{2}{3}}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02366, size = 131, normalized size = 0.44 \begin{align*} \frac{2 \, a^{5} x^{\frac{5}{3}} - 5 \, a^{4} b x^{\frac{4}{3}} + 20 \, a^{3} b^{2} x + 63 \, a^{2} b^{3} x^{\frac{2}{3}} + 6 \, a b^{4} x^{\frac{1}{3}} - 27 \, b^{5}}{2 \,{\left (a^{8} x^{\frac{2}{3}} + 2 \, a^{7} b x^{\frac{1}{3}} + a^{6} b^{2}\right )}} - \frac{30 \, b^{3} \log \left (a x^{\frac{1}{3}} + b\right )}{a^{6}} \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 \frac{1}{\left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22877, size = 163, normalized size = 0.54 \begin{align*} -\frac{30 \, b^{3} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{6} \mathrm{sgn}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} - \frac{3 \,{\left (10 \, a b^{4} x^{\frac{1}{3}} + 9 \, b^{5}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} a^{6} \mathrm{sgn}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} + \frac{2 \, a^{6} x - 9 \, a^{5} b x^{\frac{2}{3}} + 36 \, a^{4} b^{2} x^{\frac{1}{3}}}{2 \, a^{9} \mathrm{sgn}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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