Integrand size = 27, antiderivative size = 305 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2}+\frac {5 c^2 \left (\frac {b}{c}+x\right )}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {5 c^2 \arctan \left (\frac {\sqrt [3]{b}+\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}}{\sqrt {3} \sqrt [3]{b}}\right )}{9 \sqrt {3} b^{8/3} \left (b^2-3 a c\right )^{8/3}}+\frac {5 c^2 \log \left (b-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac {5 c^2 \log \left (b^{2/3} \left (b^2-3 a c\right )^{2/3}+\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+c^2 \left (\frac {b}{c}+x\right )^2\right )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}} \]
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Time = 0.26 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=-\frac {5 c^2 \arctan \left (\frac {\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}+\sqrt [3]{b}}{\sqrt {3} \sqrt [3]{b}}\right )}{9 \sqrt {3} b^{8/3} \left (b^2-3 a c\right )^{8/3}}+\frac {5 c^2 \left (\frac {b}{c}+x\right )}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2}+\frac {5 c^2 \log \left (-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+b+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac {5 c^2 \log \left (\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+b^{2/3} \left (b^2-3 a c\right )^{2/3}+c^2 \left (\frac {b}{c}+x\right )^2\right )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}} \]
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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 )^3} \, dx,x,\frac {b}{c}+x\right ) \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{6 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 )^2}-\frac {(5 c) \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 )}{6 b \left (b^2-3 a c\right )} \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{6 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 )^2}+\frac {5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {\left (5 c^2\right ) \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 )}{9 b^2 \left (b^2-3 a c\right )^2} \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{6 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 )^2}+\frac {5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {\left (5 c^{8/3}\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 )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}+\frac {\left (5 c^{8/3}\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 )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}} \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{6 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 )^2}+\frac {5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {5 c^2 \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac {\left (5 c^2\right ) \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 )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac {\left (5 c^{7/3}\right ) \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 )}{18 b^{7/3} \left (b^2-3 a c\right )^{7/3}} \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{6 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 )^2}+\frac {5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {5 c^2 \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac {5 c^2 \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 )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}}+\frac {\left (5 c^2\right ) \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 )}{9 b^{8/3} \left (b^2-3 a c\right )^{8/3}} \\ & = -\frac {c \left (\frac {b}{c}+x\right )}{6 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 )^2}+\frac {5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {5 c^2 \tan ^{-1}\left (\frac {1+\frac {2 (b+c x)}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{8/3} \left (b^2-3 a c\right )^{8/3}}+\frac {5 c^2 \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac {5 c^2 \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 )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\frac {-\frac {3 (b+c x) \left (3 b^3-15 b^2 c x-5 c^3 x^3-3 b c \left (8 a+5 c x^2\right )\right )}{\left (3 a b+x \left (3 b^2+3 b c x+c^2 x^2\right )\right )^2}+10 c^2 \text {RootSum}\left [3 a b+3 b^2 \text {$\#$1}+3 b c \text {$\#$1}^2+c^2 \text {$\#$1}^3\&,\frac {\log (x-\text {$\#$1})}{b^2+2 b c \text {$\#$1}+c^2 \text {$\#$1}^2}\&\right ]}{54 \left (b^3-3 a b c\right )^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {\frac {5 c^{4} x^{4}}{18 \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right ) b^{2}}+\frac {10 c^{3} x^{3}}{9 b \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {5 c^{2} x^{2}}{3 \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {2 \left (2 a c +b^{2}\right ) c x}{3 b \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {9 \left (8 a c -b^{2}\right )}{486 a^{2} c^{2}-324 a \,b^{2} c +54 b^{4}}}{\left (c^{2} x^{3}+3 b c \,x^{2}+3 b^{2} x +3 a b \right )^{2}}+\frac {5 c^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{3}+3 b \,\textit {\_Z}^{2} c +3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right ) \left (\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}\right )}\right )}{27 b^{2}}\) | \(277\) |
default | \(\frac {\frac {5 c^{4} x^{4}}{18 \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right ) b^{2}}+\frac {10 c^{3} x^{3}}{9 b \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {5 c^{2} x^{2}}{3 \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {2 \left (2 a c +b^{2}\right ) c x}{3 b \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {9 \left (8 a c -b^{2}\right )}{486 a^{2} c^{2}-324 a \,b^{2} c +54 b^{4}}}{\left (c^{2} x^{3}+3 b c \,x^{2}+3 b^{2} x +3 a b \right )^{2}}+\frac {5 c^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{3}+3 b \,\textit {\_Z}^{2} c +3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}}\right )}{3 \left (81 a^{2} c^{2}-54 a \,b^{2} c +9 b^{4}\right ) b^{2}}\) | \(279\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1268 vs. \(2 (262) = 524\).
Time = 0.28 (sec) , antiderivative size = 1268, normalized size of antiderivative = 4.16 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\text {Too large to display} \]
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Time = 1.39 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.55 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\frac {24 a b^{2} c - 3 b^{4} + 30 b^{2} c^{2} x^{2} + 20 b c^{3} x^{3} + 5 c^{4} x^{4} + x \left (24 a b c^{2} + 12 b^{3} c\right )}{1458 a^{4} b^{4} c^{2} - 972 a^{3} b^{6} c + 162 a^{2} b^{8} + x^{6} \cdot \left (162 a^{2} b^{2} c^{6} - 108 a b^{4} c^{5} + 18 b^{6} c^{4}\right ) + x^{5} \cdot \left (972 a^{2} b^{3} c^{5} - 648 a b^{5} c^{4} + 108 b^{7} c^{3}\right ) + x^{4} \cdot \left (2430 a^{2} b^{4} c^{4} - 1620 a b^{6} c^{3} + 270 b^{8} c^{2}\right ) + x^{3} \cdot \left (972 a^{3} b^{3} c^{4} + 2268 a^{2} b^{5} c^{3} - 1836 a b^{7} c^{2} + 324 b^{9} c\right ) + x^{2} \cdot \left (2916 a^{3} b^{4} c^{3} - 486 a^{2} b^{6} c^{2} - 648 a b^{8} c + 162 b^{10}\right ) + x \left (2916 a^{3} b^{5} c^{2} - 1944 a^{2} b^{7} c + 324 a b^{9}\right )} + \operatorname {RootSum} {\left (t^{3} \cdot \left (129140163 a^{8} b^{8} c^{8} - 344373768 a^{7} b^{10} c^{7} + 401769396 a^{6} b^{12} c^{6} - 267846264 a^{5} b^{14} c^{5} + 111602610 a^{4} b^{16} c^{4} - 29760696 a^{3} b^{18} c^{3} + 4960116 a^{2} b^{20} c^{2} - 472392 a b^{22} c + 19683 b^{24}\right ) - 125 c^{6}, \left ( t \mapsto t \log {\left (x + \frac {729 t a^{3} b^{3} c^{3} - 729 t a^{2} b^{5} c^{2} + 243 t a b^{7} c - 27 t b^{9} + 5 b c^{2}}{5 c^{3}} \right )} \right )\right )} \]
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\[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\int { \frac {1}{{\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )}^{3}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\frac {5 \, {\left (2 \, \sqrt {3} \left (\frac {c^{6}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}{c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}\right ) - \left (\frac {c^{6}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac {c^{6}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left ({\left | c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}} \right |}\right )\right )}}{54 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}} + \frac {5 \, c^{4} x^{4} + 20 \, b c^{3} x^{3} + 30 \, b^{2} c^{2} x^{2} + 12 \, b^{3} c x + 24 \, a b c^{2} x - 3 \, b^{4} + 24 \, a b^{2} c}{18 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )} {\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )}^{2}} \]
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Time = 10.19 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\frac {\frac {8\,a\,c-b^2}{6\,\left (9\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}+\frac {5\,c^2\,x^2}{3\,\left (9\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}+\frac {10\,c^3\,x^3}{9\,b\,\left (9\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}+\frac {5\,c^4\,x^4}{18\,b^2\,\left (9\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}+\frac {2\,c\,x\,\left (b^2+2\,a\,c\right )}{3\,b\,\left (9\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}}{x^2\,\left (9\,b^4+18\,a\,c\,b^2\right )+9\,a^2\,b^2+c^4\,x^6+x^3\,\left (18\,b^3\,c+6\,a\,b\,c^2\right )+6\,b\,c^3\,x^5+15\,b^2\,c^2\,x^4+18\,a\,b^3\,x}+\frac {5\,c^2\,\ln \left (b\,{\left (3\,a\,c-b^2\right )}^{8/3}-b^{19/3}+c\,x\,{\left (3\,a\,c-b^2\right )}^{8/3}+27\,a^3\,b^{1/3}\,c^3-27\,a^2\,b^{7/3}\,c^2+9\,a\,b^{13/3}\,c\right )}{27\,b^{8/3}\,{\left (3\,a\,c-b^2\right )}^{8/3}}-\frac {5\,c^2\,\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x-\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,b^{8/3}\,{\left (3\,a\,c-b^2\right )}^{8/3}}+\frac {5\,c^2\,\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x+\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,b^{8/3}\,{\left (3\,a\,c-b^2\right )}^{8/3}} \]
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