Integrand size = 46, antiderivative size = 334 \[ \int \frac {x^2}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\frac {2 (-1)^{2/3} \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {2 \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{11/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {2 (-1)^{2/3} \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}} \]
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Time = 0.50 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2122, 632, 210} \[ \int \frac {x^2}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\frac {2 (-1)^{2/3} \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac {2 \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{11/6} c^{2/3} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {2 (-1)^{2/3} \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]
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Rule 210
Rule 632
Rule 2122
Rubi steps \begin{align*} \text {integral}& = \left (19683 a^6\right ) \int \left (\frac {(-1)^{2/3}}{177147 \left (1+\sqrt [3]{-1}\right )^2 a^{22/3} c^{2/3} \left (-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2\right )}+\frac {1}{531441 a^{22/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}-\frac {(-1)^{2/3}}{177147 \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{22/3} c^{2/3} \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{27 a^{4/3} c^{2/3}}+\frac {(-1)^{2/3} \int \frac {1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{27 a^{4/3} c^{2/3}}+\frac {(-1)^{2/3} \int \frac {1}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{9 \left (1+\sqrt [3]{-1}\right )^2 a^{4/3} c^{2/3}} \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{-3 a \left (4 b-3 \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 a^{2/3} \sqrt [3]{c}+2 b x\right )}{27 a^{4/3} c^{2/3}}-\frac {\left (2 (-1)^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3 a \left (4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x\right )}{27 a^{4/3} c^{2/3}}-\frac {\left (2 (-1)^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3 a \left (4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x\right )}{9 \left (1+\sqrt [3]{-1}\right )^2 a^{4/3} c^{2/3}} \\ & = \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} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} 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} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} 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 {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{11/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.29 \[ \int \frac {x^2}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\frac {1}{3} \text {RootSum}\left [27 a^3+27 a^2 b \text {$\#$1}^2+27 a^2 c \text {$\#$1}^3+9 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{18 a^2 b+27 a^2 c \text {$\#$1}+12 a b^2 \text {$\#$1}^2+2 b^3 \text {$\#$1}^4}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.28
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\) | \(93\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\) | \(93\) |
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Result contains complex when optimal does not.
Time = 2.43 (sec) , antiderivative size = 27094, normalized size of antiderivative = 81.12 \[ \int \frac {x^2}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {x^2}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int { \frac {x^{2}}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}} \,d x } \]
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\[ \int \frac {x^2}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int { \frac {x^{2}}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}} \,d x } \]
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Time = 9.51 (sec) , antiderivative size = 825, normalized size of antiderivative = 2.47 \[ \int \frac {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} \, dx=\sum _{k=1}^6\ln \left (-a^3\,b^9\,\left (-{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^2\,a^4\,c^2\,13122-{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^3\,a^6\,c^3\,1062882+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^4\,a^8\,c^4\,43046721+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^5\,a^{10}\,c^5\,3486784401+\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )\,a^2\,c\,81+\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )\,a\,b^2\,x\,18-{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^4\,a^7\,b^3\,c^2\,25509168-{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^5\,a^9\,b^3\,c^3\,6198727824+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^2\,a^3\,b^2\,c\,x\,5832+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^3\,a^5\,b^2\,c^2\,x\,708588+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^4\,a^7\,b^2\,c^3\,x\,38263752+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^5\,a^9\,b^2\,c^4\,x\,774840978+1\right )\,27\right )\,\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right ) \]
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