Optimal. Leaf size=334 \[ \frac{2 (-1)^{2/3} \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{2 \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{11/6} c^{2/3} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{2 (-1)^{2/3} \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]
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Rubi [A] time = 2.26823, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{2 (-1)^{2/3} \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{2 \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{11/6} c^{2/3} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{2 (-1)^{2/3} \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(27*a^3 + 27*a^2*b*x^2 + 27*a^2*c*x^3 + 9*a*b^2*x^4 + b^3*x^6),x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b**3*x**6+9*a*b**2*x**4+27*a**2*c*x**3+27*a**2*b*x**2+27*a**3),x)
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Mathematica [C] time = 0.0670611, size = 97, normalized size = 0.29 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6 b^3+9 \text{$\#$1}^4 a b^2+27 \text{$\#$1}^3 a^2 c+27 \text{$\#$1}^2 a^2 b+27 a^3\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 b^3+12 \text{$\#$1}^2 a b^2+27 \text{$\#$1} a^2 c+18 a^2 b}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(27*a^3 + 27*a^2*b*x^2 + 27*a^2*c*x^3 + 9*a*b^2*x^4 + b^3*x^6),x]
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Maple [C] time = 0.006, size = 93, normalized size = 0.3 \[{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({b}^{3}{{\it \_Z}}^{6}+9\,a{b}^{2}{{\it \_Z}}^{4}+27\,{a}^{2}c{{\it \_Z}}^{3}+27\,{a}^{2}b{{\it \_Z}}^{2}+27\,{a}^{3} \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}{b}^{3}+12\,{{\it \_R}}^{3}a{b}^{2}+27\,{{\it \_R}}^{2}{a}^{2}c+18\,{\it \_R}\,{a}^{2}b}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*c*x^3 + 27*a^2*b*x^2 + 27*a^3),x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*c*x^3 + 27*a^2*b*x^2 + 27*a^3),x, algorithm="fricas")
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Sympy [A] time = 81.7248, size = 167, normalized size = 0.5 \[ \operatorname{RootSum}{\left (t^{6} \left (282429536481 a^{12} c^{6} - 669462604992 a^{11} b^{3} c^{4}\right ) - 129140163 t^{4} a^{8} c^{4} + 19683 t^{2} a^{4} c^{2} - 1, \left ( t \mapsto t \log{\left (x + \frac{62762119218 t^{5} a^{11} c^{6} - 148769467776 t^{5} a^{10} b^{3} c^{4} - 387420489 t^{4} a^{9} c^{5} + 918330048 t^{4} a^{8} b^{3} c^{3} - 23914845 t^{3} a^{7} c^{4} - 11337408 t^{3} a^{6} b^{3} c^{2} + 177147 t^{2} a^{5} c^{3} + 2187 t a^{3} c^{2} - 18 a c}{8 b^{2}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b**3*x**6+9*a*b**2*x**4+27*a**2*c*x**3+27*a**2*b*x**2+27*a**3),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*c*x^3 + 27*a^2*b*x^2 + 27*a^3),x, algorithm="giac")
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