Integrand size = 18, antiderivative size = 610 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=-\frac {1}{a x}+\frac {\sqrt [3]{c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}} \]
-1/a/x+1/6*c^(1/3)*ln(2^(1/3)*c^(1/3)*x+(b-(-4*a*c+b^2)^(1/2))^(1/3))*(1+b /(-4*a*c+b^2)^(1/2))*2^(1/3)/a/(b-(-4*a*c+b^2)^(1/2))^(1/3)-1/12*c^(1/3)*l n(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-(-4*a*c+b^2)^(1/2))^(1/3)+(b-(- 4*a*c+b^2)^(1/2))^(2/3))*(1+b/(-4*a*c+b^2)^(1/2))*2^(1/3)/a/(b-(-4*a*c+b^2 )^(1/2))^(1/3)+1/6*c^(1/3)*arctan(1/3*(1-2*2^(1/3)*c^(1/3)*x/(b-(-4*a*c+b^ 2)^(1/2))^(1/3))*3^(1/2))*(1+b/(-4*a*c+b^2)^(1/2))*2^(1/3)/a*3^(1/2)/(b-(- 4*a*c+b^2)^(1/2))^(1/3)+1/6*c^(1/3)*ln(2^(1/3)*c^(1/3)*x+(b+(-4*a*c+b^2)^( 1/2))^(1/3))*(1-b/(-4*a*c+b^2)^(1/2))*2^(1/3)/a/(b+(-4*a*c+b^2)^(1/2))^(1/ 3)-1/12*c^(1/3)*ln(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+(-4*a*c+b^2)^( 1/2))^(1/3)+(b+(-4*a*c+b^2)^(1/2))^(2/3))*(1-b/(-4*a*c+b^2)^(1/2))*2^(1/3) /a/(b+(-4*a*c+b^2)^(1/2))^(1/3)+1/6*c^(1/3)*arctan(1/3*(1-2*2^(1/3)*c^(1/3 )*x/(b+(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*(1-b/(-4*a*c+b^2)^(1/2))*2^(1/3 )/a*3^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=-\frac {1}{a x}-\frac {\text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {b \log (x-\text {$\#$1})+c \log (x-\text {$\#$1}) \text {$\#$1}^3}{b \text {$\#$1}+2 c \text {$\#$1}^4}\&\right ]}{3 a} \]
-(1/(a*x)) - RootSum[a + b*#1^3 + c*#1^6 & , (b*Log[x - #1] + c*Log[x - #1 ]*#1^3)/(b*#1 + 2*c*#1^4) & ]/(3*a)
Time = 0.83 (sec) , antiderivative size = 532, normalized size of antiderivative = 0.87, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1704, 25, 1834, 27, 821, 16, 1142, 25, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx\) |
| \(\Big \downarrow \) 1704 |
| \(\displaystyle \frac {\int -\frac {x \left (c x^3+b\right )}{c x^6+b x^3+a}dx}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {x \left (c x^3+b\right )}{c x^6+b x^3+a}dx}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 1834 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \int \frac {2 x}{2 c x^3+b-\sqrt {b^2-4 a c}}dx+\frac {1}{2} c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {2 x}{2 c x^3+b+\sqrt {b^2-4 a c}}dx}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \int \frac {x}{2 c x^3+b-\sqrt {b^2-4 a c}}dx+c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {x}{2 c x^3+b+\sqrt {b^2-4 a c}}dx}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle -\frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b-\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\int \frac {1}{\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b-\sqrt {b^2-4 a c}}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b+\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\int \frac {1}{\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b+\sqrt {b^2-4 a c}}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b-\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b+\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {\frac {3}{2} \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {\frac {3}{2} \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {\frac {3}{2} \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}-\frac {1}{2} \int \frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}-\frac {1}{2} \int \frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {1}{a x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{2 \sqrt [3]{2} \sqrt [3]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{2 \sqrt [3]{2} \sqrt [3]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {1}{a x}\) |
-(1/(a*x)) - (c*(1 + b/Sqrt[b^2 - 4*a*c])*(-1/3*Log[(b - Sqrt[b^2 - 4*a*c] )^(1/3) + 2^(1/3)*c^(1/3)*x]/(2^(2/3)*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3 )) + (-((Sqrt[3]*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c]) ^(1/3))/Sqrt[3]])/(2^(1/3)*c^(1/3))) + Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2]/(2 *2^(1/3)*c^(1/3)))/(3*2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3))) + c* (1 - b/Sqrt[b^2 - 4*a*c])*(-1/3*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3 )*c^(1/3)*x]/(2^(2/3)*c^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)) + (-((Sqrt[3] *ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]] )/(2^(1/3)*c^(1/3))) + Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3) *(b + Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2]/(2*2^(1/3)*c^(1/3) ))/(3*2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3))))/a
3.2.49.3.1 Defintions of rubi rules used
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[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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[((d_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_ Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*d*(m + 1) )), x] - Simp[1/(a*d^n*(m + 1)) Int[(d*x)^(m + n)*(b*(m + n*(p + 1) + 1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{ a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ [{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n , 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.10
| method | result | size |
| default | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{6}+\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{4} c +\textit {\_R} b \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +b \,\textit {\_R}^{2}}}{3 a}-\frac {1}{a x}\) | \(61\) |
| risch | \(-\frac {1}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 c^{3} a^{7}-48 b^{2} c^{2} a^{6}+12 b^{4} c \,a^{5}-b^{6} a^{4}\right ) \textit {\_Z}^{6}+\left (-32 b \,c^{3} a^{3}+32 b^{3} c^{2} a^{2}-10 b^{5} c a +b^{7}\right ) \textit {\_Z}^{3}+c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (224 c^{3} a^{7}-176 b^{2} c^{2} a^{6}+46 b^{4} c \,a^{5}-4 b^{6} a^{4}\right ) \textit {\_R}^{6}+\left (-100 b \,c^{3} a^{3}+97 b^{3} c^{2} a^{2}-30 b^{5} c a +3 b^{7}\right ) \textit {\_R}^{3}+3 c^{4}\right ) x +\left (16 a^{6} c^{3}-24 a^{5} b^{2} c^{2}+9 a^{4} b^{4} c -a^{3} b^{6}\right ) \textit {\_R}^{5}-a^{2} b \,c^{3} \textit {\_R}^{2}\right )\right )}{3}\) | \(239\) |
Leaf count of result is larger than twice the leaf count of optimal. 3225 vs. \(2 (471) = 942\).
Time = 0.39 (sec) , antiderivative size = 3225, normalized size of antiderivative = 5.29 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\text {Too large to display} \]
1/6*(2*(1/2)^(1/3)*a*x*((b^3 - 2*a*b*c + (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8 *a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9* b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c))^(1/3)*log((1 /2)^(2/3)*(b^9 - 11*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c ^4 - (a^4*b^8 - 13*a^5*b^6*c + 60*a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c ^4)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/( a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))*((b^3 - 2*a*b*c + (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2* c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)) )/(a^4*b^2 - 4*a^5*c))^(2/3) + 2*(b^4*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*x) + 2*(1/2)^(1/3)*a*x*((b^3 - 2*a*b*c - (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^ 6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c))^(1/3)*log((1/2)^( 2/3)*(b^9 - 11*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c^4 + (a^4*b^8 - 13*a^5*b^6*c + 60*a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c^4)*s qrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b ^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))*((b^3 - 2*a*b*c - (a^ 4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(a^ 4*b^2 - 4*a^5*c))^(2/3) + 2*(b^4*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*x) + (1...
Time = 2.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.41 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\operatorname {RootSum} {\left (t^{6} \cdot \left (46656 a^{7} c^{3} - 34992 a^{6} b^{2} c^{2} + 8748 a^{5} b^{4} c - 729 a^{4} b^{6}\right ) + t^{3} \left (- 864 a^{3} b c^{3} + 864 a^{2} b^{3} c^{2} - 270 a b^{5} c + 27 b^{7}\right ) + c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 15552 t^{5} a^{8} c^{4} + 27216 t^{5} a^{7} b^{2} c^{3} - 14580 t^{5} a^{6} b^{4} c^{2} + 3159 t^{5} a^{5} b^{6} c - 243 t^{5} a^{4} b^{8} + 252 t^{2} a^{4} b c^{4} - 567 t^{2} a^{3} b^{3} c^{3} + 378 t^{2} a^{2} b^{5} c^{2} - 99 t^{2} a b^{7} c + 9 t^{2} b^{9}}{2 a^{2} c^{5} - 4 a b^{2} c^{4} + b^{4} c^{3}} \right )} \right )\right )} - \frac {1}{a x} \]
RootSum(_t**6*(46656*a**7*c**3 - 34992*a**6*b**2*c**2 + 8748*a**5*b**4*c - 729*a**4*b**6) + _t**3*(-864*a**3*b*c**3 + 864*a**2*b**3*c**2 - 270*a*b** 5*c + 27*b**7) + c**4, Lambda(_t, _t*log(x + (-15552*_t**5*a**8*c**4 + 272 16*_t**5*a**7*b**2*c**3 - 14580*_t**5*a**6*b**4*c**2 + 3159*_t**5*a**5*b** 6*c - 243*_t**5*a**4*b**8 + 252*_t**2*a**4*b*c**4 - 567*_t**2*a**3*b**3*c* *3 + 378*_t**2*a**2*b**5*c**2 - 99*_t**2*a*b**7*c + 9*_t**2*b**9)/(2*a**2* c**5 - 4*a*b**2*c**4 + b**4*c**3)))) - 1/(a*x)
\[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + b x^{3} + a\right )} x^{2}} \,d x } \]
\[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + b x^{3} + a\right )} x^{2}} \,d x } \]
Time = 11.56 (sec) , antiderivative size = 2978, normalized size of antiderivative = 4.88 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\text {Too large to display} \]
log(36*a^9*c^6 + 9*a^7*b^4*c^4 - 45*a^8*b^2*c^5 - (2^(2/3)*(27*a^7*c^3*x*( b^6 - 8*a^3*c^3 + 18*a^2*b^2*c^2 - 8*a*b^4*c) + (27*2^(1/3)*a^10*b*c^3*(4* a*c - b^2)^2*(-(b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2 *b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-( 4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3))/2)*(-(b^7 + b^4*(-(4* a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4* a*c - b^2)^3))^(1/3))/6)*((b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c ^3 + 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4* a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2)))^(1/3) + log(36*a^9*c^6 + 9*a^7*b^4*c^4 - 45*a^8*b^2*c ^5 - (2^(2/3)*(27*a^7*c^3*x*(b^6 - 8*a^3*c^3 + 18*a^2*b^2*c^2 - 8*a*b^4*c) + (27*2^(1/3)*a^10*b*c^3*(4*a*c - b^2)^2*((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^ (2/3))/2)*((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2*b^3 *c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a* c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(1/3))/6)*(-(b^4*(-(4*a*c - b^2) ^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2 )^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(a^4*...