Optimal. Leaf size=100 \[ \frac{12 b^3}{a^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{3 b^3}{2 a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )^2}+\frac{18 b^2 \sqrt [3]{x}}{a^5}-\frac{30 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}-\frac{9 b x^{2/3}}{2 a^4}+\frac{x}{a^3} \]
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Rubi [A] time = 0.0683635, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 44} \[ \frac{12 b^3}{a^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{3 b^3}{2 a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )^2}+\frac{18 b^2 \sqrt [3]{x}}{a^5}-\frac{30 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}-\frac{9 b x^{2/3}}{2 a^4}+\frac{x}{a^3} \]
Antiderivative was successfully verified.
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Rule 190
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3} \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^3} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^4}-\frac{3 b}{a^4 x^3}+\frac{6 b^2}{a^5 x^2}-\frac{10 b^3}{a^6 x}+\frac{b^4}{a^4 (a+b x)^3}+\frac{4 b^4}{a^5 (a+b x)^2}+\frac{10 b^4}{a^6 (a+b x)}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=\frac{3 b^3}{2 a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )^2}+\frac{12 b^3}{a^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{18 b^2 \sqrt [3]{x}}{a^5}-\frac{9 b x^{2/3}}{2 a^4}+\frac{x}{a^3}-\frac{30 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}\\ \end{align*}
Mathematica [A] time = 0.0740524, size = 83, normalized size = 0.83 \[ \frac{-9 a^2 b x^{2/3}+2 a^3 x+\frac{3 b^5}{\left (a \sqrt [3]{x}+b\right )^2}-\frac{30 b^4}{a \sqrt [3]{x}+b}+36 a b^2 \sqrt [3]{x}-60 b^3 \log \left (a \sqrt [3]{x}+b\right )}{2 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 77, normalized size = 0.8 \begin{align*}{\frac{x}{{a}^{3}}}-{\frac{9\,b}{2\,{a}^{4}}{x}^{{\frac{2}{3}}}}+18\,{\frac{{b}^{2}\sqrt [3]{x}}{{a}^{5}}}-15\,{\frac{{b}^{4}}{{a}^{6} \left ( b+a\sqrt [3]{x} \right ) }}+{\frac{3\,{b}^{5}}{2\,{a}^{6}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}-30\,{\frac{{b}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04606, size = 136, normalized size = 1.36 \begin{align*} \frac{2 \, a^{4} - \frac{5 \, a^{3} b}{x^{\frac{1}{3}}} + \frac{20 \, a^{2} b^{2}}{x^{\frac{2}{3}}} + \frac{90 \, a b^{3}}{x} + \frac{60 \, b^{4}}{x^{\frac{4}{3}}}}{2 \,{\left (\frac{a^{7}}{x} + \frac{2 \, a^{6} b}{x^{\frac{4}{3}}} + \frac{a^{5} b^{2}}{x^{\frac{5}{3}}}\right )}} - \frac{30 \, b^{3} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{6}} - \frac{10 \, b^{3} \log \left (x\right )}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53351, size = 350, normalized size = 3.5 \begin{align*} \frac{2 \, a^{9} x^{3} + 4 \, a^{6} b^{3} x^{2} - 34 \, a^{3} b^{6} x - 27 \, b^{9} - 60 \,{\left (a^{6} b^{3} x^{2} + 2 \, a^{3} b^{6} x + b^{9}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 3 \,{\left (3 \, a^{8} b x^{2} + 16 \, a^{5} b^{4} x + 10 \, a^{2} b^{7}\right )} x^{\frac{2}{3}} + 3 \,{\left (12 \, a^{7} b^{2} x^{2} + 35 \, a^{4} b^{5} x + 20 \, a b^{8}\right )} x^{\frac{1}{3}}}{2 \,{\left (a^{12} x^{2} + 2 \, a^{9} b^{3} x + a^{6} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.09564, size = 362, normalized size = 3.62 \begin{align*} \begin{cases} \frac{2 a^{5} x^{\frac{5}{3}}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{5 a^{4} b x^{\frac{4}{3}}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} + \frac{20 a^{3} b^{2} x}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{60 a^{2} b^{3} x^{\frac{2}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{120 a b^{4} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{120 a b^{4} \sqrt [3]{x}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{60 b^{5} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{90 b^{5}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} & \text{for}\: a \neq 0 \\\frac{x^{2}}{2 b^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17975, size = 107, normalized size = 1.07 \begin{align*} -\frac{30 \, b^{3} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{6}} - \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}} + \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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