Optimal. Leaf size=122 \[ -\frac{3 a^8}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac{24 a^7 \log \left (a+b \sqrt [3]{x}\right )}{b^9}+\frac{21 a^6 \sqrt [3]{x}}{b^8}-\frac{9 a^5 x^{2/3}}{b^7}+\frac{5 a^4 x}{b^6}-\frac{3 a^3 x^{4/3}}{b^5}+\frac{9 a^2 x^{5/3}}{5 b^4}-\frac{a x^2}{b^3}+\frac{3 x^{7/3}}{7 b^2} \]
[Out]
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Rubi [A] time = 0.209587, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{3 a^8}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac{24 a^7 \log \left (a+b \sqrt [3]{x}\right )}{b^9}+\frac{21 a^6 \sqrt [3]{x}}{b^8}-\frac{9 a^5 x^{2/3}}{b^7}+\frac{5 a^4 x}{b^6}-\frac{3 a^3 x^{4/3}}{b^5}+\frac{9 a^2 x^{5/3}}{5 b^4}-\frac{a x^2}{b^3}+\frac{3 x^{7/3}}{7 b^2} \]
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. \[ - \frac{3 a^{8}}{b^{9} \left (a + b \sqrt [3]{x}\right )} - \frac{24 a^{7} \log{\left (a + b \sqrt [3]{x} \right )}}{b^{9}} + \frac{21 a^{6} \sqrt [3]{x}}{b^{8}} - \frac{18 a^{5} \int ^{\sqrt [3]{x}} x\, dx}{b^{7}} + \frac{5 a^{4} x}{b^{6}} - \frac{3 a^{3} x^{\frac{4}{3}}}{b^{5}} + \frac{9 a^{2} x^{\frac{5}{3}}}{5 b^{4}} - \frac{a x^{2}}{b^{3}} + \frac{3 x^{\frac{7}{3}}}{7 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a+b*x**(1/3))**2,x)
[Out]
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Mathematica [A] time = 0.0868546, size = 122, normalized size = 1. \[ -\frac{3 a^8}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac{24 a^7 \log \left (a+b \sqrt [3]{x}\right )}{b^9}+\frac{21 a^6 \sqrt [3]{x}}{b^8}-\frac{9 a^5 x^{2/3}}{b^7}+\frac{5 a^4 x}{b^6}-\frac{3 a^3 x^{4/3}}{b^5}+\frac{9 a^2 x^{5/3}}{5 b^4}-\frac{a x^2}{b^3}+\frac{3 x^{7/3}}{7 b^2} \]
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 = 105, normalized size = 0.9 \[ -3\,{\frac{{a}^{8}}{{b}^{9} \left ( a+b\sqrt [3]{x} \right ) }}+21\,{\frac{{a}^{6}\sqrt [3]{x}}{{b}^{8}}}-9\,{\frac{{a}^{5}{x}^{2/3}}{{b}^{7}}}+5\,{\frac{{a}^{4}x}{{b}^{6}}}-3\,{\frac{{a}^{3}{x}^{4/3}}{{b}^{5}}}+{\frac{9\,{a}^{2}}{5\,{b}^{4}}{x}^{{\frac{5}{3}}}}-{\frac{a{x}^{2}}{{b}^{3}}}+{\frac{3}{7\,{b}^{2}}{x}^{{\frac{7}{3}}}}-24\,{\frac{{a}^{7}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{9}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a+b*x^(1/3))^2,x)
[Out]
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Maxima [A] time = 1.44315, size = 197, normalized size = 1.61 \[ -\frac{24 \, a^{7} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{9}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7}}{7 \, b^{9}} - \frac{4 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a}{b^{9}} + \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{2}}{5 \, b^{9}} - \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{3}}{b^{9}} + \frac{70 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{4}}{b^{9}} - \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{5}}{b^{9}} + \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{6}}{b^{9}} - \frac{3 \, a^{8}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^(1/3) + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219233, size = 170, normalized size = 1.39 \[ \frac{28 \, a^{2} b^{6} x^{2} - 140 \, a^{5} b^{3} x - 105 \, a^{8} - 840 \,{\left (a^{7} b x^{\frac{1}{3}} + a^{8}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 3 \,{\left (5 \, b^{8} x^{2} - 14 \, a^{3} b^{5} x + 140 \, a^{6} b^{2}\right )} x^{\frac{2}{3}} - 5 \,{\left (4 \, a b^{7} x^{2} - 14 \, a^{4} b^{4} x - 147 \, a^{7} b\right )} x^{\frac{1}{3}}}{35 \,{\left (b^{10} x^{\frac{1}{3}} + a b^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^(1/3) + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 63.6099, size = 343, normalized size = 2.81 \[ - \frac{840 a^{8} x^{\frac{176}{3}} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} - \frac{840 a^{7} b x^{59} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} + \frac{840 a^{7} b x^{59}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} + \frac{420 a^{6} b^{2} x^{\frac{178}{3}}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} - \frac{140 a^{5} b^{3} x^{\frac{179}{3}}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} + \frac{70 a^{4} b^{4} x^{60}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} - \frac{42 a^{3} b^{5} x^{\frac{181}{3}}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} + \frac{28 a^{2} b^{6} x^{\frac{182}{3}}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} - \frac{20 a b^{7} x^{61}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} + \frac{15 b^{8} x^{\frac{184}{3}}}{35 a b^{9} x^{\frac{176}{3}} + 35 b^{10} x^{59}} \]
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 = 0.221702, size = 150, normalized size = 1.23 \[ -\frac{24 \, a^{7}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{9}} - \frac{3 \, a^{8}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{9}} + \frac{15 \, b^{12} x^{\frac{7}{3}} - 35 \, a b^{11} x^{2} + 63 \, a^{2} b^{10} x^{\frac{5}{3}} - 105 \, a^{3} b^{9} x^{\frac{4}{3}} + 175 \, a^{4} b^{8} x - 315 \, a^{5} b^{7} x^{\frac{2}{3}} + 735 \, a^{6} b^{6} x^{\frac{1}{3}}}{35 \, b^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^(1/3) + a)^2,x, algorithm="giac")
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