Integrand size = 25, antiderivative size = 653 \[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=-\frac {d}{a x}+\frac {\sqrt [3]{c} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}} \]
-d/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))*(d+( -2*a*e+b*d)/(-4*a*c+b^2)^(1/2))*2^(1/3)/a/(b-(-4*a*c+b^2)^(1/2))^(1/3)-1/1 2*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))*(d+(-2*a*e+b*d)/(-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))*(d+(-2*a*e+b*d)/(-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))*(d+(2*a*e-b*d)/(-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))*(d+(2*a*e-b*d)/(-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))*(d+(2*a*e-b*d)/(-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 = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.13 \[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=-\frac {d}{a x}-\frac {\text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {b d \log (x-\text {$\#$1})-a e \log (x-\text {$\#$1})+c d \log (x-\text {$\#$1}) \text {$\#$1}^3}{b \text {$\#$1}+2 c \text {$\#$1}^4}\&\right ]}{3 a} \]
-(d/(a*x)) - RootSum[a + b*#1^3 + c*#1^6 & , (b*d*Log[x - #1] - a*e*Log[x - #1] + c*d*Log[x - #1]*#1^3)/(b*#1 + 2*c*#1^4) & ]/(3*a)
Time = 0.92 (sec) , antiderivative size = 547, normalized size of antiderivative = 0.84, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1828, 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.
| \(\displaystyle \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx\) |
| \(\Big \downarrow \) 1828 |
| \(\displaystyle -\frac {\int \frac {x \left (c d x^3+b d-a e\right )}{c x^6+b x^3+a}dx}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 1834 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \int \frac {2 x}{2 c x^3+b-\sqrt {b^2-4 a c}}dx+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \int \frac {2 x}{2 c x^3+b+\sqrt {b^2-4 a c}}dx}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \int \frac {x}{2 c x^3+b-\sqrt {b^2-4 a c}}dx+c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \int \frac {x}{2 c x^3+b+\sqrt {b^2-4 a c}}dx}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle -\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b-\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\int \frac {1}{\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b-\sqrt {b^2-4 a c}}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b+\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\int \frac {1}{\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b+\sqrt {b^2-4 a c}}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b-\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b+\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}-\frac {1}{2} \int \frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}-\frac {1}{2} \int \frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {d}{a x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{2 \sqrt [3]{2} \sqrt [3]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )+c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{2 \sqrt [3]{2} \sqrt [3]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{a}-\frac {d}{a x}\) |
-(d/(a*x)) - (c*(d + (b*d - 2*a*e)/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*(d - (b*d - 2*a*e)/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
3.1.18.3.1 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[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^n + c*x^ (2*n))^(p + 1)/(a*f*(m + 1))), x] + Simp[1/(a*f^n*(m + 1)) Int[(f*x)^(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x ] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && Int egerQ[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 = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.11
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (-c d \,\textit {\_R}^{4}+\left (a e -b d \right ) \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}}{3 a}-\frac {d}{a x}\) | \(71\) |
| risch | \(-\frac {d}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{7} c^{3}-48 b^{2} c^{2} a^{6}+12 a^{5} b^{4} c -b^{6} a^{4}\right ) \textit {\_Z}^{6}+\left (-16 a^{5} c^{2} e^{3}+8 a^{4} b^{2} c \,e^{3}+48 a^{4} b \,c^{2} d \,e^{2}+48 a^{4} c^{3} d^{2} e -a^{3} b^{4} e^{3}-24 a^{3} b^{3} c d \,e^{2}-72 a^{3} b^{2} c^{2} d^{2} e -32 a^{3} b \,c^{3} d^{3}+3 a^{2} b^{5} d \,e^{2}+27 a^{2} b^{4} c \,d^{2} e +32 a^{2} b^{3} c^{2} d^{3}-3 a \,b^{6} d^{2} e -10 a \,b^{5} c \,d^{3}+b^{7} d^{3}\right ) \textit {\_Z}^{3}+a^{3} c \,e^{6}-3 a^{2} b c d \,e^{5}+3 a^{2} c^{2} d^{2} e^{4}+3 a \,b^{2} c \,d^{2} e^{4}-6 a b \,c^{2} d^{3} e^{3}+3 a \,c^{3} d^{4} e^{2}-b^{3} c \,d^{3} e^{3}+3 b^{2} c^{2} d^{4} e^{2}-3 b \,c^{3} d^{5} e +c^{4} d^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (224 a^{7} c^{3}-176 b^{2} c^{2} a^{6}+46 a^{5} b^{4} c -4 b^{6} a^{4}\right ) \textit {\_R}^{6}+\left (-52 a^{5} c^{2} e^{3}+25 a^{4} b^{2} c \,e^{3}+144 a^{4} b \,c^{2} d \,e^{2}+156 a^{4} c^{3} d^{2} e -3 a^{3} b^{4} e^{3}-72 a^{3} b^{3} c d \,e^{2}-219 a^{3} b^{2} c^{2} d^{2} e -100 a^{3} b \,c^{3} d^{3}+9 a^{2} b^{5} d \,e^{2}+81 a^{2} b^{4} c \,d^{2} e +97 a^{2} b^{3} c^{2} d^{3}-9 a \,b^{6} d^{2} e -30 a \,b^{5} c \,d^{3}+3 b^{7} d^{3}\right ) \textit {\_R}^{3}+3 a^{3} c \,e^{6}-9 a^{2} b c d \,e^{5}+9 a^{2} c^{2} d^{2} e^{4}+9 a \,b^{2} c \,d^{2} e^{4}-18 a b \,c^{2} d^{3} e^{3}+9 a \,c^{3} d^{4} e^{2}-3 b^{3} c \,d^{3} e^{3}+9 b^{2} c^{2} d^{4} e^{2}-9 b \,c^{3} d^{5} e +3 c^{4} d^{6}\right ) x +\left (16 a^{6} b \,c^{2} e +16 a^{6} c^{3} d -8 a^{5} b^{3} c e -24 a^{5} b^{2} c^{2} d +a^{4} b^{5} e +9 a^{4} b^{4} c d -a^{3} b^{6} d \right ) \textit {\_R}^{5}+\left (-a^{4} b c \,e^{4}+4 a^{4} c^{2} d \,e^{3}+a^{3} b^{2} c d \,e^{3}-6 a^{3} b \,c^{2} d^{2} e^{2}+4 a^{3} c^{3} d^{3} e +a^{2} b^{2} c^{2} d^{3} e -a^{2} b \,c^{3} d^{4}\right ) \textit {\_R}^{2}\right )\right )}{3}\) | \(843\) |
1/3/a*sum((-c*d*_R^4+(a*e-b*d)*_R)/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=RootOf(_Z ^6*c+_Z^3*b+a))-d/a/x
Leaf count of result is larger than twice the leaf count of optimal. 11285 vs. \(2 (517) = 1034\).
Time = 37.04 (sec) , antiderivative size = 11285, normalized size of antiderivative = 17.28 \[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {e x^{3} + d}{{\left (c x^{6} + b x^{3} + a\right )} x^{2}} \,d x } \]
\[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {e x^{3} + d}{{\left (c x^{6} + b x^{3} + a\right )} x^{2}} \,d x } \]
Time = 36.13 (sec) , antiderivative size = 11174, normalized size of antiderivative = 17.11 \[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\text {Too large to display} \]
log((2^(1/3)*(-(b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 - a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 + 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3 )^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 - 6 *a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3) ^(1/2) - 72*a^3*b^2*c^2*d^2*e + 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/ (a^4*(4*a*c - b^2)^3))^(2/3)*((2^(2/3)*(27*a^7*c^3*x*(4*a*c - b^2)*(b^4*d^ 2 - 2*a^3*c*e^2 + a^2*b^2*e^2 + 2*a^2*c^2*d^2 - 2*a*b^3*d*e - 4*a*b^2*c*d^ 2 + 6*a^2*b*c*d*e) - (27*2^(1/3)*a^10*b*c^3*(4*a*c - b^2)^2*(-(b^7*d^3 - a ^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b* c^3*d^3 - a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5 *d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 + 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2* e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 - 6*a^3*c*d*e^2*(-(4*a*c - b^2 )^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2 *e + 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3 ))/2)*(-(b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*...