Integrand size = 27, antiderivative size = 297 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=-\frac {b+c x}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+\frac {(b+c x)^3}{c}\right )^2}+\frac {5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (b \left (3 a-\frac {b^2}{c}\right )+\frac {(b+c x)^3}{c}\right )}-\frac {5 c^2 \arctan \left (\frac {\sqrt [3]{b}+\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}}{\sqrt {3} \sqrt [3]{b}}\right )}{9 \sqrt {3} b^{8/3} \left (b^2-3 a c\right )^{8/3}}+\frac {5 c^2 \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac {5 c^2 \log \left (b^{2/3} \left (b^2-3 a c\right )^{2/3}+\sqrt [3]{b} \sqrt [3]{b^2-3 a c} (b+c x)+(b+c x)^2\right )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}} \] Output:
-1/6*(c*x+b)/b/(-3*a*c+b^2)/(b*(3*a-b^2/c)+(c*x+b)^3/c)^2+5/18*c*(c*x+b)/b ^2/(-3*a*c+b^2)^2/(b*(3*a-b^2/c)+(c*x+b)^3/c)-5/27*c^2*arctan(1/3*(b^(1/3) +2*(c*x+b)/(-3*a*c+b^2)^(1/3))*3^(1/2)/b^(1/3))*3^(1/2)/b^(8/3)/(-3*a*c+b^ 2)^(8/3)+5/27*c^2*ln(b^(1/3)*(b^(2/3)-(-3*a*c+b^2)^(1/3))+c*x)/b^(8/3)/(-3 *a*c+b^2)^(8/3)-5/54*c^2*ln(b^(2/3)*(-3*a*c+b^2)^(2/3)+b^(1/3)*(-3*a*c+b^2 )^(1/3)*(c*x+b)+(c*x+b)^2)/b^(8/3)/(-3*a*c+b^2)^(8/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\frac {-\frac {3 (b+c x) \left (3 b^3-15 b^2 c x-5 c^3 x^3-3 b c \left (8 a+5 c x^2\right )\right )}{\left (3 a b+x \left (3 b^2+3 b c x+c^2 x^2\right )\right )^2}+10 c^2 \text {RootSum}\left [3 a b+3 b^2 \text {$\#$1}+3 b c \text {$\#$1}^2+c^2 \text {$\#$1}^3\&,\frac {\log (x-\text {$\#$1})}{b^2+2 b c \text {$\#$1}+c^2 \text {$\#$1}^2}\&\right ]}{54 \left (b^3-3 a b c\right )^2} \] Input:
Integrate[(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)^(-3),x]
Output:
((-3*(b + c*x)*(3*b^3 - 15*b^2*c*x - 5*c^3*x^3 - 3*b*c*(8*a + 5*c*x^2)))/( 3*a*b + x*(3*b^2 + 3*b*c*x + c^2*x^2))^2 + 10*c^2*RootSum[3*a*b + 3*b^2*#1 + 3*b*c*#1^2 + c^2*#1^3 & , Log[x - #1]/(b^2 + 2*b*c*#1 + c^2*#1^2) & ])/ (54*(b^3 - 3*a*b*c)^2)
Time = 0.54 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2458, 749, 749, 750, 16, 25, 1142, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \frac {1}{\left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^3}d\left (\frac {b}{c}+x\right )\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle -\frac {5 c \int \frac {1}{\left (c^2 \left (\frac {b}{c}+x\right )^3+b \left (3 a-\frac {b^2}{c}\right )\right )^2}d\left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle -\frac {5 c \left (-\frac {2 c \int \frac {1}{c^2 \left (\frac {b}{c}+x\right )^3+b \left (3 a-\frac {b^2}{c}\right )}d\left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\right )}{6 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {5 c \left (-\frac {2 c \left (\frac {c^{2/3} \int \frac {1}{c^{2/3} \left (\frac {b}{c}+x\right )-\frac {\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}+\frac {c^{2/3} \int -\frac {c \left (\frac {b}{c}+x\right )+2 \sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{\sqrt [3]{c} \left (c^{4/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{c^{2/3}}\right )}d\left (\frac {b}{c}+x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\right )}{6 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {5 c \left (-\frac {2 c \left (\frac {c^{2/3} \int -\frac {c \left (\frac {b}{c}+x\right )+2 \sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{\sqrt [3]{c} \left (c^{4/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{c^{2/3}}\right )}d\left (\frac {b}{c}+x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}+\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\right )}{6 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5 c \left (-\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \int \frac {c \left (\frac {b}{c}+x\right )+2 \sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\right )}{6 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {5 c \left (-\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {3}{2} \sqrt [3]{b} \sqrt [3]{b^2-3 a c} \int \frac {1}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )+\frac {\int \frac {c^{2/3} \left (2 c \left (\frac {b}{c}+x\right )+\sqrt [3]{b} \sqrt [3]{b^2-3 a c}\right )}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )}{2 c^{2/3}}\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\right )}{6 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 c \left (-\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {3}{2} \sqrt [3]{b} \sqrt [3]{b^2-3 a c} \int \frac {1}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )+\frac {1}{2} \int \frac {2 c \left (\frac {b}{c}+x\right )+\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\right )}{6 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {5 c \left (-\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {1}{2} \int \frac {2 c \left (\frac {b}{c}+x\right )+\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )-\frac {3 \int \frac {1}{-\left (\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}+1\right )^2-3}d\left (\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}+1\right )}{c^{2/3}}\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\right )}{6 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {5 c \left (-\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {1}{2} \int \frac {2 c \left (\frac {b}{c}+x\right )+\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{c^{5/3} \left (\frac {b}{c}+x\right )^2+\sqrt [3]{b} c^{2/3} \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{\sqrt [3]{c}}}d\left (\frac {b}{c}+x\right )+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}+1}{\sqrt {3}}\right )}{c^{2/3}}\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\right )}{6 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {5 c \left (-\frac {2 c \left (\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-3 a c}-c \left (\frac {b}{c}+x\right )\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {c^{2/3} \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}+1}{\sqrt {3}}\right )}{c^{2/3}}+\frac {\log \left (\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+b^{2/3} \left (b^2-3 a c\right )^{2/3}+c^2 \left (\frac {b}{c}+x\right )^2\right )}{2 c^{2/3}}\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\right )}{3 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )}\right )}{6 b \left (b^2-3 a c\right )}-\frac {c \left (\frac {b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^2}\) |
Input:
Int[(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)^(-3),x]
Output:
-1/6*(c*(b/c + x))/(b*(b^2 - 3*a*c)*(b*(3*a - b^2/c) + c^2*(b/c + x)^3)^2) - (5*c*(-1/3*(c*(b/c + x))/(b*(b^2 - 3*a*c)*(b*(3*a - b^2/c) + c^2*(b/c + x)^3)) - (2*c*(Log[b^(1/3)*(b^2 - 3*a*c)^(1/3) - c*(b/c + x)]/(3*b^(2/3)* (b^2 - 3*a*c)^(2/3)) - (c^(2/3)*((Sqrt[3]*ArcTan[(1 + (2*c*(b/c + x))/(b^( 1/3)*(b^2 - 3*a*c)^(1/3)))/Sqrt[3]])/c^(2/3) + Log[b^(2/3)*(b^2 - 3*a*c)^( 2/3) + b^(1/3)*c*(b^2 - 3*a*c)^(1/3)*(b/c + x) + c^2*(b/c + x)^2]/(2*c^(2/ 3))))/(3*b^(2/3)*(b^2 - 3*a*c)^(2/3))))/(3*b*(b^2 - 3*a*c))))/(6*b*(b^2 - 3*a*c))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.93
| method | result | size |
| risch | \(\frac {\frac {5 c^{4} x^{4}}{18 \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right ) b^{2}}+\frac {10 c^{3} x^{3}}{9 b \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {5 c^{2} x^{2}}{3 \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {2 \left (2 a c +b^{2}\right ) c x}{3 b \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {9 \left (8 a c -b^{2}\right )}{486 a^{2} c^{2}-324 a \,b^{2} c +54 b^{4}}}{\left (c^{2} x^{3}+3 b c \,x^{2}+3 b^{2} x +3 a b \right )^{2}}+\frac {5 c^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{3}+3 b c \,\textit {\_Z}^{2}+3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right ) \left (\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}\right )}\right )}{27 b^{2}}\) | \(277\) |
| default | \(\frac {\frac {5 c^{4} x^{4}}{18 \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right ) b^{2}}+\frac {10 c^{3} x^{3}}{9 b \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {5 c^{2} x^{2}}{3 \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {2 \left (2 a c +b^{2}\right ) c x}{3 b \left (9 a^{2} c^{2}-6 a \,b^{2} c +b^{4}\right )}+\frac {9 \left (8 a c -b^{2}\right )}{486 a^{2} c^{2}-324 a \,b^{2} c +54 b^{4}}}{\left (c^{2} x^{3}+3 b c \,x^{2}+3 b^{2} x +3 a b \right )^{2}}+\frac {5 c^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{3}+3 b c \,\textit {\_Z}^{2}+3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}}\right )}{3 \left (81 a^{2} c^{2}-54 a \,b^{2} c +9 b^{4}\right ) b^{2}}\) | \(279\) |
Input:
int(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^3,x,method=_RETURNVERBOSE)
Output:
9*(5/162*c^4/(9*a^2*c^2-6*a*b^2*c+b^4)/b^2*x^4+10/81/b*c^3/(9*a^2*c^2-6*a* b^2*c+b^4)*x^3+5/27*c^2/(9*a^2*c^2-6*a*b^2*c+b^4)*x^2+2/27*(2*a*c+b^2)*c/b /(9*a^2*c^2-6*a*b^2*c+b^4)*x+1/54*(8*a*c-b^2)/(9*a^2*c^2-6*a*b^2*c+b^4))/( c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^2+5/27*c^2/b^2*sum(1/(9*a^2*c^2-6*a*b^2*c +b^4)/(_R^2*c^2+2*_R*b*c+b^2)*ln(x-_R),_R=RootOf(_Z^3*c^2+3*_Z^2*b*c+3*_Z* b^2+3*a*b))
Leaf count of result is larger than twice the leaf count of optimal. 1264 vs. \(2 (252) = 504\).
Time = 0.09 (sec) , antiderivative size = 1264, normalized size of antiderivative = 4.26 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^3,x, algorithm="fricas")
Output:
-1/54*(9*b^10 - 126*a*b^8*c + 513*a^2*b^6*c^2 - 648*a^3*b^4*c^3 - 15*(b^6* c^4 - 6*a*b^4*c^5 + 9*a^2*b^2*c^6)*x^4 - 60*(b^7*c^3 - 6*a*b^5*c^4 + 9*a^2 *b^3*c^5)*x^3 - 90*(b^8*c^2 - 6*a*b^6*c^3 + 9*a^2*b^4*c^4)*x^2 + 30*sqrt(1 /3)*(9*a^2*b^5*c^2 - 27*a^3*b^3*c^3 + (b^3*c^6 - 3*a*b*c^7)*x^6 + 6*(b^4*c ^5 - 3*a*b^2*c^6)*x^5 + 15*(b^5*c^4 - 3*a*b^3*c^5)*x^4 + 6*(3*b^6*c^3 - 8* a*b^4*c^4 - 3*a^2*b^2*c^5)*x^3 + 9*(b^7*c^2 - a*b^5*c^3 - 6*a^2*b^3*c^4)*x ^2 + 18*(a*b^6*c^2 - 3*a^2*b^4*c^3)*x)*(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^( 1/6)*arctan(sqrt(1/3)*(2*(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(2/3)*(c*x + b) + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(1/3)*(b^3 - 3*a*b*c))/(b^6 - 6*a*b^4 *c + 9*a^2*b^2*c^2)^(5/6)) + 5*(c^6*x^6 + 6*b*c^5*x^5 + 15*b^2*c^4*x^4 + 1 8*a*b^3*c^2*x + 9*a^2*b^2*c^2 + 6*(3*b^3*c^3 + a*b*c^4)*x^3 + 9*(b^4*c^2 + 2*a*b^2*c^3)*x^2)*(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(2/3)*log(-b^5 + 3*a* b^3*c - (b^3*c^2 - 3*a*b*c^3)*x^2 - 2*(b^4*c - 3*a*b^2*c^2)*x - (b^6 - 6*a *b^4*c + 9*a^2*b^2*c^2)^(2/3)*(c*x + b) - (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 )^(1/3)*(b^3 - 3*a*b*c)) - 10*(c^6*x^6 + 6*b*c^5*x^5 + 15*b^2*c^4*x^4 + 18 *a*b^3*c^2*x + 9*a^2*b^2*c^2 + 6*(3*b^3*c^3 + a*b*c^4)*x^3 + 9*(b^4*c^2 + 2*a*b^2*c^3)*x^2)*(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(2/3)*log(-b^4 + 3*a*b ^2*c - (b^3*c - 3*a*b*c^2)*x + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)^(2/3)) - 36*(b^9*c - 4*a*b^7*c^2 - 3*a^2*b^5*c^3 + 18*a^3*b^3*c^4)*x)/(9*a^2*b^14 - 108*a^3*b^12*c + 486*a^4*b^10*c^2 - 972*a^5*b^8*c^3 + 729*a^6*b^6*c^4 ...
Time = 1.55 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\frac {24 a b^{2} c - 3 b^{4} + 30 b^{2} c^{2} x^{2} + 20 b c^{3} x^{3} + 5 c^{4} x^{4} + x \left (24 a b c^{2} + 12 b^{3} c\right )}{1458 a^{4} b^{4} c^{2} - 972 a^{3} b^{6} c + 162 a^{2} b^{8} + x^{6} \cdot \left (162 a^{2} b^{2} c^{6} - 108 a b^{4} c^{5} + 18 b^{6} c^{4}\right ) + x^{5} \cdot \left (972 a^{2} b^{3} c^{5} - 648 a b^{5} c^{4} + 108 b^{7} c^{3}\right ) + x^{4} \cdot \left (2430 a^{2} b^{4} c^{4} - 1620 a b^{6} c^{3} + 270 b^{8} c^{2}\right ) + x^{3} \cdot \left (972 a^{3} b^{3} c^{4} + 2268 a^{2} b^{5} c^{3} - 1836 a b^{7} c^{2} + 324 b^{9} c\right ) + x^{2} \cdot \left (2916 a^{3} b^{4} c^{3} - 486 a^{2} b^{6} c^{2} - 648 a b^{8} c + 162 b^{10}\right ) + x \left (2916 a^{3} b^{5} c^{2} - 1944 a^{2} b^{7} c + 324 a b^{9}\right )} + \operatorname {RootSum} {\left (t^{3} \cdot \left (129140163 a^{8} b^{8} c^{8} - 344373768 a^{7} b^{10} c^{7} + 401769396 a^{6} b^{12} c^{6} - 267846264 a^{5} b^{14} c^{5} + 111602610 a^{4} b^{16} c^{4} - 29760696 a^{3} b^{18} c^{3} + 4960116 a^{2} b^{20} c^{2} - 472392 a b^{22} c + 19683 b^{24}\right ) - 125 c^{6}, \left ( t \mapsto t \log {\left (x + \frac {729 t a^{3} b^{3} c^{3} - 729 t a^{2} b^{5} c^{2} + 243 t a b^{7} c - 27 t b^{9} + 5 b c^{2}}{5 c^{3}} \right )} \right )\right )} \] Input:
integrate(1/(c**2*x**3+3*b*c*x**2+3*b**2*x+3*a*b)**3,x)
Output:
(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*(24*a*b*c**2 + 12*b**3*c))/(1458*a**4*b**4*c**2 - 972*a**3*b**6*c + 162 *a**2*b**8 + x**6*(162*a**2*b**2*c**6 - 108*a*b**4*c**5 + 18*b**6*c**4) + x**5*(972*a**2*b**3*c**5 - 648*a*b**5*c**4 + 108*b**7*c**3) + x**4*(2430*a **2*b**4*c**4 - 1620*a*b**6*c**3 + 270*b**8*c**2) + x**3*(972*a**3*b**3*c* *4 + 2268*a**2*b**5*c**3 - 1836*a*b**7*c**2 + 324*b**9*c) + x**2*(2916*a** 3*b**4*c**3 - 486*a**2*b**6*c**2 - 648*a*b**8*c + 162*b**10) + x*(2916*a** 3*b**5*c**2 - 1944*a**2*b**7*c + 324*a*b**9)) + RootSum(_t**3*(129140163*a **8*b**8*c**8 - 344373768*a**7*b**10*c**7 + 401769396*a**6*b**12*c**6 - 26 7846264*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) - 125*c** 6, Lambda(_t, _t*log(x + (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))))
\[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\int { \frac {1}{{\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )}^{3}} \,d x } \] Input:
integrate(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^3,x, algorithm="maxima")
Output:
5/9*c^2*integrate(1/(c^2*x^3 + 3*b*c*x^2 + 3*b^2*x + 3*a*b), x)/(b^6 - 6*a *b^4*c + 9*a^2*b^2*c^2) + 1/18*(5*c^4*x^4 + 20*b*c^3*x^3 + 30*b^2*c^2*x^2 - 3*b^4 + 24*a*b^2*c + 12*(b^3*c + 2*a*b*c^2)*x)/(9*a^2*b^8 - 54*a^3*b^6*c + 81*a^4*b^4*c^2 + (b^6*c^4 - 6*a*b^4*c^5 + 9*a^2*b^2*c^6)*x^6 + 6*(b^7*c ^3 - 6*a*b^5*c^4 + 9*a^2*b^3*c^5)*x^5 + 15*(b^8*c^2 - 6*a*b^6*c^3 + 9*a^2* b^4*c^4)*x^4 + 6*(3*b^9*c - 17*a*b^7*c^2 + 21*a^2*b^5*c^3 + 9*a^3*b^3*c^4) *x^3 + 9*(b^10 - 4*a*b^8*c - 3*a^2*b^6*c^2 + 18*a^3*b^4*c^3)*x^2 + 18*(a*b ^9 - 6*a^2*b^7*c + 9*a^3*b^5*c^2)*x)
Time = 0.15 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\frac {5 \, {\left (2 \, \sqrt {3} \left (\frac {c^{6}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}{c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}\right ) - \left (\frac {c^{6}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac {c^{6}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left ({\left | c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}} \right |}\right )\right )}}{54 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}} + \frac {5 \, c^{4} x^{4} + 20 \, b c^{3} x^{3} + 30 \, b^{2} c^{2} x^{2} + 12 \, b^{3} c x + 24 \, a b c^{2} x - 3 \, b^{4} + 24 \, a b^{2} c}{18 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )} {\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )}^{2}} \] Input:
integrate(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^3,x, algorithm="giac")
Output:
5/54*(2*sqrt(3)*(c^6/(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2))^(1/3)*arctan((sqrt (3)*c*x + sqrt(3)*b - sqrt(3)*(-b^3 + 3*a*b*c)^(1/3))/(c*x + b + (-b^3 + 3 *a*b*c)^(1/3))) - (c^6/(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2))^(1/3)*log(4*(sqr t(3)*c*x + sqrt(3)*b - sqrt(3)*(-b^3 + 3*a*b*c)^(1/3))^2 + 4*(c*x + b + (- b^3 + 3*a*b*c)^(1/3))^2) + 2*(c^6/(b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2))^(1/3) *log(abs(c*x + b + (-b^3 + 3*a*b*c)^(1/3))))/(b^6 - 6*a*b^4*c + 9*a^2*b^2* c^2) + 1/18*(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)/((b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2)*(c^2*x^3 + 3*b*c*x^2 + 3*b^2*x + 3*a*b)^2)
Time = 13.59 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.63 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx=\frac {\frac {8\,a\,c-b^2}{6\,\left (9\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}+\frac {5\,c^2\,x^2}{3\,\left (9\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}+\frac {10\,c^3\,x^3}{9\,b\,\left (9\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}+\frac {5\,c^4\,x^4}{18\,b^2\,\left (9\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}+\frac {2\,c\,x\,\left (b^2+2\,a\,c\right )}{3\,b\,\left (9\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}}{x^2\,\left (9\,b^4+18\,a\,c\,b^2\right )+9\,a^2\,b^2+c^4\,x^6+x^3\,\left (18\,b^3\,c+6\,a\,b\,c^2\right )+6\,b\,c^3\,x^5+15\,b^2\,c^2\,x^4+18\,a\,b^3\,x}+\frac {5\,c^2\,\ln \left (b\,{\left (3\,a\,c-b^2\right )}^{8/3}-b^{19/3}+c\,x\,{\left (3\,a\,c-b^2\right )}^{8/3}+27\,a^3\,b^{1/3}\,c^3-27\,a^2\,b^{7/3}\,c^2+9\,a\,b^{13/3}\,c\right )}{27\,b^{8/3}\,{\left (3\,a\,c-b^2\right )}^{8/3}}-\frac {5\,c^2\,\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x-\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,b^{8/3}\,{\left (3\,a\,c-b^2\right )}^{8/3}}+\frac {5\,c^2\,\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x+\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,b^{8/3}\,{\left (3\,a\,c-b^2\right )}^{8/3}} \] Input:
int(1/(3*a*b + 3*b^2*x + c^2*x^3 + 3*b*c*x^2)^3,x)
Output:
((8*a*c - b^2)/(6*(b^4 + 9*a^2*c^2 - 6*a*b^2*c)) + (5*c^2*x^2)/(3*(b^4 + 9 *a^2*c^2 - 6*a*b^2*c)) + (10*c^3*x^3)/(9*b*(b^4 + 9*a^2*c^2 - 6*a*b^2*c)) + (5*c^4*x^4)/(18*b^2*(b^4 + 9*a^2*c^2 - 6*a*b^2*c)) + (2*c*x*(2*a*c + b^2 ))/(3*b*(b^4 + 9*a^2*c^2 - 6*a*b^2*c)))/(x^2*(9*b^4 + 18*a*b^2*c) + 9*a^2* b^2 + c^4*x^6 + x^3*(18*b^3*c + 6*a*b*c^2) + 6*b*c^3*x^5 + 15*b^2*c^2*x^4 + 18*a*b^3*x) + (5*c^2*log(b*(3*a*c - b^2)^(8/3) - b^(19/3) + c*x*(3*a*c - b^2)^(8/3) + 27*a^3*b^(1/3)*c^3 - 27*a^2*b^(7/3)*c^2 + 9*a*b^(13/3)*c))/( 27*b^(8/3)*(3*a*c - b^2)^(8/3)) - (5*c^2*log(2*b - b^(1/3)*(3*a*c - b^2)^( 1/3) + 2*c*x - 3^(1/2)*b^(1/3)*(3*a*c - b^2)^(1/3)*1i)*((3^(1/2)*1i)/2 + 1 /2))/(27*b^(8/3)*(3*a*c - b^2)^(8/3)) + (5*c^2*log(2*b - b^(1/3)*(3*a*c - b^2)^(1/3) + 2*c*x + 3^(1/2)*b^(1/3)*(3*a*c - b^2)^(1/3)*1i)*((3^(1/2)*1i) /2 - 1/2))/(27*b^(8/3)*(3*a*c - b^2)^(8/3))
Time = 0.25 (sec) , antiderivative size = 2343, normalized size of antiderivative = 7.89 \[ \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^3,x)
Output:
( - 90*b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)*atan((b**(1/3)*(3*a*c - b**2 )**(1/3) - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)))*a**2*b** 2*c**2 - 180*b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)*atan((b**(1/3)*(3*a*c - b**2)**(1/3) - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)))*a* b**3*c**2*x - 180*b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)*atan((b**(1/3)*(3 *a*c - b**2)**(1/3) - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3) ))*a*b**2*c**3*x**2 - 60*b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)*atan((b**( 1/3)*(3*a*c - b**2)**(1/3) - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b**2)**(1/3)* sqrt(3)))*a*b*c**4*x**3 - 90*b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)*atan(( b**(1/3)*(3*a*c - b**2)**(1/3) - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b**2)**(1 /3)*sqrt(3)))*b**4*c**2*x**2 - 180*b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)* atan((b**(1/3)*(3*a*c - b**2)**(1/3) - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b** 2)**(1/3)*sqrt(3)))*b**3*c**3*x**3 - 150*b**(1/3)*(3*a*c - b**2)**(1/3)*sq rt(3)*atan((b**(1/3)*(3*a*c - b**2)**(1/3) - 2*b - 2*c*x)/(b**(1/3)*(3*a*c - b**2)**(1/3)*sqrt(3)))*b**2*c**4*x**4 - 60*b**(1/3)*(3*a*c - b**2)**(1/ 3)*sqrt(3)*atan((b**(1/3)*(3*a*c - b**2)**(1/3) - 2*b - 2*c*x)/(b**(1/3)*( 3*a*c - b**2)**(1/3)*sqrt(3)))*b*c**5*x**5 - 10*b**(1/3)*(3*a*c - b**2)**( 1/3)*sqrt(3)*atan((b**(1/3)*(3*a*c - b**2)**(1/3) - 2*b - 2*c*x)/(b**(1/3) *(3*a*c - b**2)**(1/3)*sqrt(3)))*c**6*x**6 - 45*b**(1/3)*(3*a*c - b**2)**( 1/3)*log(b**(2/3)*(3*a*c - b**2)**(2/3) - b**(1/3)*(3*a*c - b**2)**(1/3...