Integrand size = 42, antiderivative size = 522 \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=-\frac {\sqrt [3]{-1} \left (2 \sqrt [3]{-1} b+3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt {3} a^{17/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac {\log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}} \]
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Time = 0.92 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2095, 648, 632, 210, 642} \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=-\frac {\sqrt [3]{-1} \left (3 \sqrt [3]{a} c^{2/3}+2 \sqrt [3]{-1} b\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} c^{2/3} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt {3} a^{17/6} c^{2/3} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}-\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{27 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} c^{2/3} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}+\frac {\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac {\log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 2095
Rubi steps \begin{align*} \text {integral}& = \left (19683 a^6\right ) \int \left (\frac {-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{531441 \left (1+\sqrt [3]{-1}\right )^2 a^{26/3} c^{2/3} \left (-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2\right )}+\frac {-\sqrt [3]{a} b+3 a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{1594323 a^{26/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac {(-1)^{2/3} \sqrt [3]{a} b-3 a^{2/3} c^{2/3}+\sqrt [3]{-1} b \sqrt [3]{c} x}{531441 \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{26/3} 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 {-\sqrt [3]{a} b+3 a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{8/3} c^{2/3}}-\frac {\int \frac {(-1)^{2/3} \sqrt [3]{a} b-3 a^{2/3} c^{2/3}+\sqrt [3]{-1} b \sqrt [3]{c} x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{8/3} c^{2/3}}+\frac {\int \frac {-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{27 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} c^{2/3}} \\ & = -\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \int \frac {1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 a^{7/3} c^{2/3}}-\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \int \frac {1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 a^{7/3} c^{2/3}}-\frac {\left (-2 b \left (-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}\right )-3 \sqrt [3]{-1} a^{2/3} b c^{2/3}\right ) \int \frac {1}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} b c^{2/3}}+\frac {\int \frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \int \frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 a^{8/3} \sqrt [3]{c}}-\frac {\int \frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}} \\ & = \frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac {\log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}+\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \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 )}{81 a^{7/3} c^{2/3}}+\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{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 )}{81 a^{7/3} c^{2/3}}+\frac {\left (-2 b \left (-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}\right )-3 \sqrt [3]{-1} a^{2/3} b c^{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 )}{27 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} b c^{2/3}} \\ & = -\frac {\left (2 (-1)^{2/3} b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) \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 )}{27 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \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 )}{81 \sqrt {3} a^{17/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \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 )}{81 \sqrt {3} a^{17/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac {\log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.19 \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\frac {1}{3} \text {RootSum}\left [27 a^3+27 a^2 b \text {$\#$1}^2+27 a^2 c \text {$\#$1}^3+9 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{18 a^2 b \text {$\#$1}+27 a^2 c \text {$\#$1}^2+12 a b^2 \text {$\#$1}^3+2 b^3 \text {$\#$1}^5}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.17
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\) | \(90\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\) | \(90\) |
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Exception generated. \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int { \frac {1}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}} \,d x } \]
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\[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\int { \frac {1}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}} \,d x } \]
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Time = 9.50 (sec) , antiderivative size = 1394, normalized size of antiderivative = 2.67 \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Too large to display} \]
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