Optimal. Leaf size=150 \[ -\frac{3 b^{10}}{a^{11} \left (a \sqrt [3]{x}+b\right )}-\frac{30 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{11}}+\frac{27 b^8 \sqrt [3]{x}}{a^{10}}-\frac{12 b^7 x^{2/3}}{a^9}+\frac{7 b^6 x}{a^8}-\frac{9 b^5 x^{4/3}}{2 a^7}+\frac{3 b^4 x^{5/3}}{a^6}-\frac{2 b^3 x^2}{a^5}+\frac{9 b^2 x^{7/3}}{7 a^4}-\frac{3 b x^{8/3}}{4 a^3}+\frac{x^3}{3 a^2} \]
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Rubi [A] time = 0.281659, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 b^{10}}{a^{11} \left (a \sqrt [3]{x}+b\right )}-\frac{30 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{11}}+\frac{27 b^8 \sqrt [3]{x}}{a^{10}}-\frac{12 b^7 x^{2/3}}{a^9}+\frac{7 b^6 x}{a^8}-\frac{9 b^5 x^{4/3}}{2 a^7}+\frac{3 b^4 x^{5/3}}{a^6}-\frac{2 b^3 x^2}{a^5}+\frac{9 b^2 x^{7/3}}{7 a^4}-\frac{3 b x^{8/3}}{4 a^3}+\frac{x^3}{3 a^2} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a + b/x^(1/3))^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{3 a^{2}} - \frac{3 b x^{\frac{8}{3}}}{4 a^{3}} + \frac{9 b^{2} x^{\frac{7}{3}}}{7 a^{4}} - \frac{2 b^{3} x^{2}}{a^{5}} + \frac{3 b^{4} x^{\frac{5}{3}}}{a^{6}} - \frac{9 b^{5} x^{\frac{4}{3}}}{2 a^{7}} + \frac{7 b^{6} x}{a^{8}} - \frac{24 b^{7} \int ^{\sqrt [3]{x}} x\, dx}{a^{9}} + \frac{27 b^{8} \sqrt [3]{x}}{a^{10}} - \frac{3 b^{10}}{a^{11} \left (a \sqrt [3]{x} + b\right )} - \frac{30 b^{9} \log{\left (a \sqrt [3]{x} + b \right )}}{a^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a+b/x**(1/3))**2,x)
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Mathematica [A] time = 0.104455, size = 150, normalized size = 1. \[ -\frac{3 b^{10}}{a^{11} \left (a \sqrt [3]{x}+b\right )}-\frac{30 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{11}}+\frac{27 b^8 \sqrt [3]{x}}{a^{10}}-\frac{12 b^7 x^{2/3}}{a^9}+\frac{7 b^6 x}{a^8}-\frac{9 b^5 x^{4/3}}{2 a^7}+\frac{3 b^4 x^{5/3}}{a^6}-\frac{2 b^3 x^2}{a^5}+\frac{9 b^2 x^{7/3}}{7 a^4}-\frac{3 b x^{8/3}}{4 a^3}+\frac{x^3}{3 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a + b/x^(1/3))^2,x]
[Out]
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Maple [A] time = 0.012, size = 127, normalized size = 0.9 \[ -3\,{\frac{{b}^{10}}{{a}^{11} \left ( b+a\sqrt [3]{x} \right ) }}+27\,{\frac{{b}^{8}\sqrt [3]{x}}{{a}^{10}}}-12\,{\frac{{b}^{7}{x}^{2/3}}{{a}^{9}}}+7\,{\frac{{b}^{6}x}{{a}^{8}}}-{\frac{9\,{b}^{5}}{2\,{a}^{7}}{x}^{{\frac{4}{3}}}}+3\,{\frac{{b}^{4}{x}^{5/3}}{{a}^{6}}}-2\,{\frac{{b}^{3}{x}^{2}}{{a}^{5}}}+{\frac{9\,{b}^{2}}{7\,{a}^{4}}{x}^{{\frac{7}{3}}}}-{\frac{3\,b}{4\,{a}^{3}}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}}{3\,{a}^{2}}}-30\,{\frac{{b}^{9}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{11}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a+b/x^(1/3))^2,x)
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Maxima [A] time = 1.44173, size = 196, normalized size = 1.31 \[ \frac{28 \, a^{9} - \frac{35 \, a^{8} b}{x^{\frac{1}{3}}} + \frac{45 \, a^{7} b^{2}}{x^{\frac{2}{3}}} - \frac{60 \, a^{6} b^{3}}{x} + \frac{84 \, a^{5} b^{4}}{x^{\frac{4}{3}}} - \frac{126 \, a^{4} b^{5}}{x^{\frac{5}{3}}} + \frac{210 \, a^{3} b^{6}}{x^{2}} - \frac{420 \, a^{2} b^{7}}{x^{\frac{7}{3}}} + \frac{1260 \, a b^{8}}{x^{\frac{8}{3}}} + \frac{2520 \, b^{9}}{x^{3}}}{84 \,{\left (\frac{a^{11}}{x^{3}} + \frac{a^{10} b}{x^{\frac{10}{3}}}\right )}} - \frac{30 \, b^{9} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{11}} - \frac{10 \, b^{9} \log \left (x\right )}{a^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a + b/x^(1/3))^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22898, size = 200, normalized size = 1.33 \[ -\frac{35 \, a^{9} b x^{3} - 84 \, a^{6} b^{4} x^{2} + 420 \, a^{3} b^{7} x + 252 \, b^{10} + 2520 \,{\left (a b^{9} x^{\frac{1}{3}} + b^{10}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 9 \,{\left (5 \, a^{8} b^{2} x^{2} - 14 \, a^{5} b^{5} x + 140 \, a^{2} b^{8}\right )} x^{\frac{2}{3}} - 2 \,{\left (14 \, a^{10} x^{3} - 30 \, a^{7} b^{3} x^{2} + 105 \, a^{4} b^{6} x + 1134 \, a b^{9}\right )} x^{\frac{1}{3}}}{84 \,{\left (a^{12} x^{\frac{1}{3}} + a^{11} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a + b/x^(1/3))^2,x, algorithm="fricas")
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Sympy [A] time = 23.5188, size = 367, normalized size = 2.45 \[ \begin{cases} \frac{28 a^{10} x^{\frac{10}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{35 a^{9} b x^{3}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{45 a^{8} b^{2} x^{\frac{8}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{60 a^{7} b^{3} x^{\frac{7}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{84 a^{6} b^{4} x^{2}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{126 a^{5} b^{5} x^{\frac{5}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{210 a^{4} b^{6} x^{\frac{4}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{420 a^{3} b^{7} x}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} + \frac{1260 a^{2} b^{8} x^{\frac{2}{3}}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{2520 a b^{9} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{2520 b^{10} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} - \frac{2520 b^{10}}{84 a^{12} \sqrt [3]{x} + 84 a^{11} b} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{11}{3}}}{11 b^{2}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a+b/x**(1/3))**2,x)
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GIAC/XCAS [A] time = 0.214604, size = 180, normalized size = 1.2 \[ -\frac{30 \, b^{9}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{11}} - \frac{3 \, b^{10}}{{\left (a x^{\frac{1}{3}} + b\right )} a^{11}} + \frac{28 \, a^{16} x^{3} - 63 \, a^{15} b x^{\frac{8}{3}} + 108 \, a^{14} b^{2} x^{\frac{7}{3}} - 168 \, a^{13} b^{3} x^{2} + 252 \, a^{12} b^{4} x^{\frac{5}{3}} - 378 \, a^{11} b^{5} x^{\frac{4}{3}} + 588 \, a^{10} b^{6} x - 1008 \, a^{9} b^{7} x^{\frac{2}{3}} + 2268 \, a^{8} b^{8} x^{\frac{1}{3}}}{84 \, a^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a + b/x^(1/3))^2,x, algorithm="giac")
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