Optimal. Leaf size=113 \[ \frac{15 b^4 x^{2/3}}{2 a^6}+\frac{9 b^2 x^{4/3}}{4 a^4}+\frac{3 b^7}{a^8 \left (a \sqrt [3]{x}+b\right )}-\frac{18 b^5 \sqrt [3]{x}}{a^7}-\frac{4 b^3 x}{a^5}+\frac{21 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^8}-\frac{6 b x^{5/3}}{5 a^3}+\frac{x^2}{2 a^2} \]
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Rubi [A] time = 0.085341, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac{15 b^4 x^{2/3}}{2 a^6}+\frac{9 b^2 x^{4/3}}{4 a^4}+\frac{3 b^7}{a^8 \left (a \sqrt [3]{x}+b\right )}-\frac{18 b^5 \sqrt [3]{x}}{a^7}-\frac{4 b^3 x}{a^5}+\frac{21 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^8}-\frac{6 b x^{5/3}}{5 a^3}+\frac{x^2}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2} \, dx &=\int \frac{x^{5/3}}{\left (b+a \sqrt [3]{x}\right )^2} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x^7}{(b+a x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{6 b^5}{a^7}+\frac{5 b^4 x}{a^6}-\frac{4 b^3 x^2}{a^5}+\frac{3 b^2 x^3}{a^4}-\frac{2 b x^4}{a^3}+\frac{x^5}{a^2}-\frac{b^7}{a^7 (b+a x)^2}+\frac{7 b^6}{a^7 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 b^7}{a^8 \left (b+a \sqrt [3]{x}\right )}-\frac{18 b^5 \sqrt [3]{x}}{a^7}+\frac{15 b^4 x^{2/3}}{2 a^6}-\frac{4 b^3 x}{a^5}+\frac{9 b^2 x^{4/3}}{4 a^4}-\frac{6 b x^{5/3}}{5 a^3}+\frac{x^2}{2 a^2}+\frac{21 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^8}\\ \end{align*}
Mathematica [A] time = 0.134864, size = 111, normalized size = 0.98 \[ \frac{a \left (45 a^3 b^2 x^{4/3}-80 a^2 b^3 x-24 a^4 b x^{5/3}+10 a^5 x^2+150 a b^4 x^{2/3}-\frac{60 b^6}{a+\frac{b}{\sqrt [3]{x}}}-360 b^5 \sqrt [3]{x}\right )+420 b^6 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )+140 b^6 \log (x)}{20 a^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 94, normalized size = 0.8 \begin{align*} 3\,{\frac{{b}^{7}}{{a}^{8} \left ( b+a\sqrt [3]{x} \right ) }}-18\,{\frac{{b}^{5}\sqrt [3]{x}}{{a}^{7}}}+{\frac{15\,{b}^{4}}{2\,{a}^{6}}{x}^{{\frac{2}{3}}}}-4\,{\frac{{b}^{3}x}{{a}^{5}}}+{\frac{9\,{b}^{2}}{4\,{a}^{4}}{x}^{{\frac{4}{3}}}}-{\frac{6\,b}{5\,{a}^{3}}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}}{2\,{a}^{2}}}+21\,{\frac{{b}^{6}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96935, size = 151, normalized size = 1.34 \begin{align*} \frac{10 \, a^{6} - \frac{14 \, a^{5} b}{x^{\frac{1}{3}}} + \frac{21 \, a^{4} b^{2}}{x^{\frac{2}{3}}} - \frac{35 \, a^{3} b^{3}}{x} + \frac{70 \, a^{2} b^{4}}{x^{\frac{4}{3}}} - \frac{210 \, a b^{5}}{x^{\frac{5}{3}}} - \frac{420 \, b^{6}}{x^{2}}}{20 \,{\left (\frac{a^{8}}{x^{2}} + \frac{a^{7} b}{x^{\frac{7}{3}}}\right )}} + \frac{21 \, b^{6} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{8}} + \frac{7 \, b^{6} \log \left (x\right )}{a^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58512, size = 312, normalized size = 2.76 \begin{align*} \frac{10 \, a^{9} x^{3} - 70 \, a^{6} b^{3} x^{2} - 80 \, a^{3} b^{6} x + 60 \, b^{9} + 420 \,{\left (a^{3} b^{6} x + b^{9}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 6 \,{\left (4 \, a^{8} b x^{2} - 21 \, a^{5} b^{4} x - 35 \, a^{2} b^{7}\right )} x^{\frac{2}{3}} + 15 \,{\left (3 \, a^{7} b^{2} x^{2} - 21 \, a^{4} b^{5} x - 28 \, a b^{8}\right )} x^{\frac{1}{3}}}{20 \,{\left (a^{11} x + a^{8} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.03926, size = 277, normalized size = 2.45 \begin{align*} \begin{cases} \frac{10 a^{7} x^{\frac{7}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac{14 a^{6} b x^{2}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac{21 a^{5} b^{2} x^{\frac{5}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac{35 a^{4} b^{3} x^{\frac{4}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac{70 a^{3} b^{4} x}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac{210 a^{2} b^{5} x^{\frac{2}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac{420 a b^{6} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac{420 b^{7} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac{420 b^{7}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{8}{3}}}{8 b^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14153, size = 135, normalized size = 1.19 \begin{align*} \frac{21 \, b^{6} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{8}} + \frac{3 \, b^{7}}{{\left (a x^{\frac{1}{3}} + b\right )} a^{8}} + \frac{10 \, a^{10} x^{2} - 24 \, a^{9} b x^{\frac{5}{3}} + 45 \, a^{8} b^{2} x^{\frac{4}{3}} - 80 \, a^{7} b^{3} x + 150 \, a^{6} b^{4} x^{\frac{2}{3}} - 360 \, a^{5} b^{5} x^{\frac{1}{3}}}{20 \, a^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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