Optimal. Leaf size=90 \[ \frac{3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac{15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}+\frac{18 a^2 \sqrt [3]{x}}{b^5}-\frac{30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac{9 a x^{2/3}}{2 b^4}+\frac{x}{b^3} \]
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Rubi [A] time = 0.0611164, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac{15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}+\frac{18 a^2 \sqrt [3]{x}}{b^5}-\frac{30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac{9 a x^{2/3}}{2 b^4}+\frac{x}{b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b \sqrt [3]{x}\right )^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{6 a^2}{b^5}-\frac{3 a x}{b^4}+\frac{x^2}{b^3}-\frac{a^5}{b^5 (a+b x)^3}+\frac{5 a^4}{b^5 (a+b x)^2}-\frac{10 a^3}{b^5 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac{15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}+\frac{18 a^2 \sqrt [3]{x}}{b^5}-\frac{9 a x^{2/3}}{2 b^4}+\frac{x}{b^3}-\frac{30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0687078, size = 83, normalized size = 0.92 \[ \frac{\frac{3 a^5}{\left (a+b \sqrt [3]{x}\right )^2}-\frac{30 a^4}{a+b \sqrt [3]{x}}+36 a^2 b \sqrt [3]{x}-60 a^3 \log \left (a+b \sqrt [3]{x}\right )-9 a b^2 x^{2/3}+2 b^3 x}{2 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 77, normalized size = 0.9 \begin{align*}{\frac{3\,{a}^{5}}{2\,{b}^{6}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}-15\,{\frac{{a}^{4}}{{b}^{6} \left ( a+b\sqrt [3]{x} \right ) }}+18\,{\frac{{a}^{2}\sqrt [3]{x}}{{b}^{5}}}-{\frac{9\,a}{2\,{b}^{4}}{x}^{{\frac{2}{3}}}}+{\frac{x}{{b}^{3}}}-30\,{\frac{{a}^{3}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961063, size = 127, normalized size = 1.41 \begin{align*} -\frac{30 \, a^{3} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{6}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{3}}{b^{6}} - \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a}{2 \, b^{6}} + \frac{30 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{2}}{b^{6}} - \frac{15 \, a^{4}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{6}} + \frac{3 \, a^{5}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52542, size = 350, normalized size = 3.89 \begin{align*} \frac{2 \, b^{9} x^{3} + 4 \, a^{3} b^{6} x^{2} - 34 \, a^{6} b^{3} x - 27 \, a^{9} - 60 \,{\left (a^{3} b^{6} x^{2} + 2 \, a^{6} b^{3} x + a^{9}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) - 3 \,{\left (3 \, a b^{8} x^{2} + 16 \, a^{4} b^{5} x + 10 \, a^{7} b^{2}\right )} x^{\frac{2}{3}} + 3 \,{\left (12 \, a^{2} b^{7} x^{2} + 35 \, a^{5} b^{4} x + 20 \, a^{8} b\right )} x^{\frac{1}{3}}}{2 \,{\left (b^{12} x^{2} + 2 \, a^{3} b^{9} 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 = 1.13809, size = 362, normalized size = 4.02 \begin{align*} \begin{cases} - \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} - \frac{90 a^{5}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} - \frac{120 a^{4} b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} - \frac{120 a^{4} b \sqrt [3]{x}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} - \frac{60 a^{3} b^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} + \frac{20 a^{2} b^{3} x}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} - \frac{5 a b^{4} x^{\frac{4}{3}}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} + \frac{2 b^{5} x^{\frac{5}{3}}}{2 a^{2} b^{6} + 4 a b^{7} \sqrt [3]{x} + 2 b^{8} x^{\frac{2}{3}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16289, size = 107, normalized size = 1.19 \begin{align*} -\frac{30 \, a^{3} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{6}} - \frac{3 \,{\left (10 \, a^{4} b x^{\frac{1}{3}} + 9 \, a^{5}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} x - 9 \, a b^{5} x^{\frac{2}{3}} + 36 \, a^{2} b^{4} x^{\frac{1}{3}}}{2 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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