Integrand size = 27, antiderivative size = 245 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {2 c \arctan \left (\frac {\sqrt [3]{b}+\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {2 c \log \left (b-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c \log \left (b^{2/3} \left (b^2-3 a c\right )^{2/3}+\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+c^2 \left (\frac {b}{c}+x\right )^2\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}} \]
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Time = 0.19 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2092, 205, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=\frac {2 c \arctan \left (\frac {\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}+\sqrt [3]{b}}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {c \log \left (\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+b^{2/3} \left (b^2-3 a c\right )^{2/3}+c^2 \left (\frac {b}{c}+x\right )^2\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {2 c \log \left (-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+b+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}} \]
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Rule 31
Rule 205
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 2092
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (b \left (3 a-\frac {b^2}{c}\right )+c^2 x^3\right )^2} \, dx,x,\frac {b}{c}+x\right ) \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {(2 c) \text {Subst}\left (\int \frac {1}{b \left (3 a-\frac {b^2}{c}\right )+c^2 x^3} \, dx,x,\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right )} \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {\left (2 c^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-\frac {\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{\sqrt [3]{c}}+c^{2/3} x} \, dx,x,\frac {b}{c}+x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {\left (2 c^{5/3}\right ) \text {Subst}\left (\int \frac {-\frac {2 \sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{\sqrt [3]{c}}-c^{2/3} x}{\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{c^{2/3}}+\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c} x+c^{4/3} x^2} \, dx,x,\frac {b}{c}+x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}} \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {2 c \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c \text {Subst}\left (\int \frac {\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c}+2 c^{4/3} x}{\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{c^{2/3}}+\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c} x+c^{4/3} x^2} \, dx,x,\frac {b}{c}+x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c^{4/3} \text {Subst}\left (\int \frac {1}{\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{c^{2/3}}+\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c} x+c^{4/3} x^2} \, dx,x,\frac {b}{c}+x\right )}{3 b^{4/3} \left (b^2-3 a c\right )^{4/3}} \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {2 c \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c \log \left (b^{2/3} \left (b^2-3 a c\right )^{2/3}+\sqrt [3]{b} \sqrt [3]{b^2-3 a c} (b+c x)+(b+c x)^2\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {(2 c) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}\right )}{3 b^{5/3} \left (b^2-3 a c\right )^{5/3}} \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {2 c \tan ^{-1}\left (\frac {1+\frac {2 (b+c x)}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {2 c \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c \log \left (b^{2/3} \left (b^2-3 a c\right )^{2/3}+\sqrt [3]{b} \sqrt [3]{b^2-3 a c} (b+c x)+(b+c x)^2\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {\frac {3 (b+c x)}{3 a b+x \left (3 b^2+3 b c x+c^2 x^2\right )}+2 c \text {RootSum}\left [3 a b+3 b^2 \text {$\#$1}+3 b c \text {$\#$1}^2+c^2 \text {$\#$1}^3\&,\frac {\log (x-\text {$\#$1})}{b^2+2 b c \text {$\#$1}+c^2 \text {$\#$1}^2}\&\right ]}{9 \left (b^3-3 a b c\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.55
method | result | size |
default | \(\frac {\frac {c x}{9 b \left (3 a c -b^{2}\right )}+\frac {1}{27 a c -9 b^{2}}}{\frac {1}{3} c^{2} x^{3}+b c \,x^{2}+b^{2} x +a b}+\frac {2 c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{3}+3 b \,\textit {\_Z}^{2} c +3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}}\right )}{9 b \left (3 a c -b^{2}\right )}\) | \(134\) |
risch | \(\frac {\frac {c x}{9 b \left (3 a c -b^{2}\right )}+\frac {1}{27 a c -9 b^{2}}}{\frac {1}{3} c^{2} x^{3}+b c \,x^{2}+b^{2} x +a b}+\frac {2 c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{3}+3 b \,\textit {\_Z}^{2} c +3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\left (3 a c -b^{2}\right ) \left (\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}\right )}\right )}{9 b}\) | \(134\) |
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Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (204) = 408\).
Time = 0.28 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.87 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {3 \, b^{7} - 18 \, a b^{5} c + 27 \, a^{2} b^{3} c^{2} - 2 \, \sqrt {3} {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{6}} {\left (3 \, a b^{4} c - 9 \, a^{2} b^{2} c^{2} + {\left (b^{3} c^{3} - 3 \, a b c^{4}\right )} x^{3} + 3 \, {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} x^{2} + 3 \, {\left (b^{5} c - 3 \, a b^{3} c^{2}\right )} x\right )} \arctan \left (\frac {2 \, \sqrt {3} {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c x + b\right )} + \sqrt {3} {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}} {\left (b^{3} - 3 \, a b c\right )}}{3 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {5}{6}}}\right ) - {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + 3 \, a b c\right )} \log \left (-b^{5} + 3 \, a b^{3} c - {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} x^{2} - 2 \, {\left (b^{4} c - 3 \, a b^{2} c^{2}\right )} x - {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c x + b\right )} - {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}} {\left (b^{3} - 3 \, a b c\right )}\right ) + 2 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + 3 \, a b c\right )} \log \left (-b^{4} + 3 \, a b^{2} c - {\left (b^{3} c - 3 \, a b c^{2}\right )} x + {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}}\right ) + 3 \, {\left (b^{6} c - 6 \, a b^{4} c^{2} + 9 \, a^{2} b^{2} c^{3}\right )} x}{9 \, {\left (3 \, a b^{10} - 27 \, a^{2} b^{8} c + 81 \, a^{3} b^{6} c^{2} - 81 \, a^{4} b^{4} c^{3} + {\left (b^{9} c^{2} - 9 \, a b^{7} c^{3} + 27 \, a^{2} b^{5} c^{4} - 27 \, a^{3} b^{3} c^{5}\right )} x^{3} + 3 \, {\left (b^{10} c - 9 \, a b^{8} c^{2} + 27 \, a^{2} b^{6} c^{3} - 27 \, a^{3} b^{4} c^{4}\right )} x^{2} + 3 \, {\left (b^{11} - 9 \, a b^{9} c + 27 \, a^{2} b^{7} c^{2} - 27 \, a^{3} b^{5} c^{3}\right )} x\right )}} \]
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Time = 0.66 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=\frac {b + c x}{27 a^{2} b^{2} c - 9 a b^{4} + x^{3} \cdot \left (9 a b c^{3} - 3 b^{3} c^{2}\right ) + x^{2} \cdot \left (27 a b^{2} c^{2} - 9 b^{4} c\right ) + x \left (27 a b^{3} c - 9 b^{5}\right )} + \operatorname {RootSum} {\left (t^{3} \cdot \left (177147 a^{5} b^{5} c^{5} - 295245 a^{4} b^{7} c^{4} + 196830 a^{3} b^{9} c^{3} - 65610 a^{2} b^{11} c^{2} + 10935 a b^{13} c - 729 b^{15}\right ) - 8 c^{3}, \left ( t \mapsto t \log {\left (x + \frac {81 t a^{2} b^{2} c^{2} - 54 t a b^{4} c + 9 t b^{6} + 2 b c}{2 c^{2}} \right )} \right )\right )} \]
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\[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )}^{2}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=-\frac {2 \, \sqrt {3} \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}{c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}\right ) - \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left ({\left | c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}} \right |}\right )}{9 \, {\left (b^{3} - 3 \, a b c\right )}} - \frac {c x + b}{3 \, {\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )} {\left (b^{3} - 3 \, a b c\right )}} \]
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Time = 10.17 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx=\frac {\frac {1}{3\,\left (3\,a\,c-b^2\right )}+\frac {c\,x}{3\,b\,\left (3\,a\,c-b^2\right )}}{3\,b^2\,x+3\,b\,c\,x^2+3\,a\,b+c^2\,x^3}+\frac {2\,c\,\ln \left (b+b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+c\,x\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}}-\frac {\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x-\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (c+\sqrt {3}\,c\,1{}\mathrm {i}\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}}-\frac {\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x+\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (c-\sqrt {3}\,c\,1{}\mathrm {i}\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}} \]
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