Optimal. Leaf size=134 \[ -\frac{3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac{24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac{84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{63 a^5 \sqrt [3]{x}}{b^8}+\frac{45 a^4 x^{2/3}}{2 b^7}-\frac{10 a^3 x}{b^6}+\frac{9 a^2 x^{4/3}}{2 b^5}-\frac{9 a x^{5/3}}{5 b^4}+\frac{x^2}{2 b^3} \]
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Rubi [A] time = 0.225814, antiderivative size = 134, 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 \[ -\frac{3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac{24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac{84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{63 a^5 \sqrt [3]{x}}{b^8}+\frac{45 a^4 x^{2/3}}{2 b^7}-\frac{10 a^3 x}{b^6}+\frac{9 a^2 x^{4/3}}{2 b^5}-\frac{9 a x^{5/3}}{5 b^4}+\frac{x^2}{2 b^3} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a + b*x^(1/3))^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{3 a^{8}}{2 b^{9} \left (a + b \sqrt [3]{x}\right )^{2}} + \frac{24 a^{7}}{b^{9} \left (a + b \sqrt [3]{x}\right )} + \frac{84 a^{6} \log{\left (a + b \sqrt [3]{x} \right )}}{b^{9}} - \frac{63 a^{5} \sqrt [3]{x}}{b^{8}} + \frac{45 a^{4} \int ^{\sqrt [3]{x}} x\, dx}{b^{7}} - \frac{10 a^{3} x}{b^{6}} + \frac{9 a^{2} x^{\frac{4}{3}}}{2 b^{5}} - \frac{9 a x^{\frac{5}{3}}}{5 b^{4}} + \frac{x^{2}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a+b*x**(1/3))**3,x)
[Out]
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Mathematica [A] time = 0.08003, size = 120, normalized size = 0.9 \[ \frac{-\frac{15 a^8}{\left (a+b \sqrt [3]{x}\right )^2}+\frac{240 a^7}{a+b \sqrt [3]{x}}+840 a^6 \log \left (a+b \sqrt [3]{x}\right )-630 a^5 b \sqrt [3]{x}+225 a^4 b^2 x^{2/3}-100 a^3 b^3 x+45 a^2 b^4 x^{4/3}-18 a b^5 x^{5/3}+5 b^6 x^2}{10 b^9} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a + b*x^(1/3))^3,x]
[Out]
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Maple [A] time = 0.013, size = 111, normalized size = 0.8 \[ -{\frac{3\,{a}^{8}}{2\,{b}^{9}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}+24\,{\frac{{a}^{7}}{{b}^{9} \left ( a+b\sqrt [3]{x} \right ) }}-63\,{\frac{{a}^{5}\sqrt [3]{x}}{{b}^{8}}}+{\frac{45\,{a}^{4}}{2\,{b}^{7}}{x}^{{\frac{2}{3}}}}-10\,{\frac{{a}^{3}x}{{b}^{6}}}+{\frac{9\,{a}^{2}}{2\,{b}^{5}}{x}^{{\frac{4}{3}}}}-{\frac{9\,a}{5\,{b}^{4}}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}}{2\,{b}^{3}}}+84\,{\frac{{a}^{6}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{9}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a+b*x^(1/3))^3,x)
[Out]
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Maxima [A] time = 1.45562, size = 197, normalized size = 1.47 \[ \frac{84 \, a^{6} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{9}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{6}}{2 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a}{5 \, b^{9}} + \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{2}}{b^{9}} - \frac{56 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{3}}{b^{9}} + \frac{105 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{4}}{b^{9}} - \frac{168 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{5}}{b^{9}} + \frac{24 \, a^{7}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{9}} - \frac{3 \, a^{8}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^(1/3) + a)^3,x, algorithm="maxima")
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Fricas [A] time = 0.226016, size = 198, normalized size = 1.48 \[ \frac{14 \, a^{2} b^{6} x^{2} - 280 \, a^{5} b^{3} x + 225 \, a^{8} + 840 \,{\left (a^{6} b^{2} x^{\frac{2}{3}} + 2 \, a^{7} b x^{\frac{1}{3}} + a^{8}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) +{\left (5 \, b^{8} x^{2} - 28 \, a^{3} b^{5} x - 1035 \, a^{6} b^{2}\right )} x^{\frac{2}{3}} - 2 \,{\left (4 \, a b^{7} x^{2} - 35 \, a^{4} b^{4} x + 195 \, a^{7} b\right )} x^{\frac{1}{3}}}{10 \,{\left (b^{11} x^{\frac{2}{3}} + 2 \, a b^{10} x^{\frac{1}{3}} + a^{2} b^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^(1/3) + a)^3,x, algorithm="fricas")
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Sympy [A] time = 7.33795, size = 493, normalized size = 3.68 \[ \begin{cases} \frac{840 a^{8} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{1260 a^{8}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{1680 a^{7} b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{1680 a^{7} b \sqrt [3]{x}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{840 a^{6} b^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} - \frac{280 a^{5} b^{3} x}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{70 a^{4} b^{4} x^{\frac{4}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} - \frac{28 a^{3} b^{5} x^{\frac{5}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{14 a^{2} b^{6} x^{2}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} - \frac{8 a b^{7} x^{\frac{7}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} + \frac{5 b^{8} x^{\frac{8}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac{2}{3}}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{3}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a+b*x**(1/3))**3,x)
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GIAC/XCAS [A] time = 0.223357, size = 151, normalized size = 1.13 \[ \frac{84 \, a^{6}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{9}} + \frac{3 \,{\left (16 \, a^{7} b x^{\frac{1}{3}} + 15 \, a^{8}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{9}} + \frac{5 \, b^{15} x^{2} - 18 \, a b^{14} x^{\frac{5}{3}} + 45 \, a^{2} b^{13} x^{\frac{4}{3}} - 100 \, a^{3} b^{12} x + 225 \, a^{4} b^{11} x^{\frac{2}{3}} - 630 \, a^{5} b^{10} x^{\frac{1}{3}}}{10 \, b^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*x^(1/3) + a)^3,x, algorithm="giac")
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