Integrand size = 25, antiderivative size = 655 \[ \int \frac {d+e x^3}{x^3 \left (a+b x^3+c x^6\right )} \, dx=-\frac {d}{2 a x^2}+\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b+\sqrt {b^2-4 a c}\right )^{2/3}} \]
-1/2*d/a/x^2-1/6*c^(2/3)*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^(2/3)/a/(b-(-4*a*c+b^2)^(1/2))^(2/ 3)+1/12*c^(2/3)*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^(2/3)/a/(b-(-4*a*c+b^2)^(1/2))^(2/3)+1/6*c^(2/3)*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^(2/3)/a*3^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(2/3)-1/6*c^(2/3) *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^(2/3)/a/(b+(-4*a*c+b^2)^(1/2))^(2/3)+1/12*c^(2/3)*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^(2/3)/a/(b+(-4*a*c+b ^2)^(1/2))^(2/3)+1/6*c^(2/3)*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^(2/3)/a*3 ^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(2/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.14 \[ \int \frac {d+e x^3}{x^3 \left (a+b x^3+c x^6\right )} \, dx=-\frac {d}{2 a x^2}-\frac {\text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {b d \log (x-\text {$\#$1})-a e \log (x-\text {$\#$1})+c d \log (x-\text {$\#$1}) \text {$\#$1}^3}{b \text {$\#$1}^2+2 c \text {$\#$1}^5}\&\right ]}{3 a} \]
-1/2*d/(a*x^2) - RootSum[a + b*#1^3 + c*#1^6 & , (b*d*Log[x - #1] - a*e*Lo g[x - #1] + c*d*Log[x - #1]*#1^3)/(b*#1^2 + 2*c*#1^5) & ]/(3*a)
Time = 0.90 (sec) , antiderivative size = 529, normalized size of antiderivative = 0.81, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {1828, 27, 1752, 750, 16, 27, 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.
| \(\displaystyle \int \frac {d+e x^3}{x^3 \left (a+b x^3+c x^6\right )} \, dx\) |
| \(\Big \downarrow \) 1828 |
| \(\displaystyle -\frac {\int \frac {2 \left (c d x^3+b d-a e\right )}{c x^6+b x^3+a}dx}{2 a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {c d x^3+b d-a e}{c x^6+b x^3+a}dx}{a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \int \frac {1}{c x^3+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^3+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{c} x+\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{c} x+\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}-\frac {d}{2 a x^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\frac {1}{2} c \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \left (\frac {2\ 2^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} c \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{a}-\frac {d}{2 a x^2}\) |
-1/2*d/(a*x^2) - ((c*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*((2^(2/3)*Log[( b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*c^(1/3)*(b - Sqrt[b^ 2 - 4*a*c])^(2/3)) + (2*2^(2/3)*(-1/2*(Sqrt[3]*ArcTan[(1 - (2*2^(1/3)*c^(1 /3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[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]/(4*c^(1/3))))/(3*(b - Sqrt[b^2 - 4*a*c])^(2/3))))/2 + (c* (d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*((2^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c] )^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) + (2*2^(2/3)*(-1/2*(Sqrt[3]*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[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]/(4*c ^(1/3))))/(3*(b + Sqrt[b^2 - 4*a*c])^(2/3))))/2)/a
3.1.19.3.1 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[((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[((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[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.10
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (-\textit {\_R}^{3} c d +a e -b d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}}{3 a}-\frac {d}{2 a \,x^{2}}\) | \(68\) |
| risch | \(-\frac {d}{2 a \,x^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 c^{3} a^{8}-48 b^{2} c^{2} a^{7}+12 b^{4} c \,a^{6}-b^{6} a^{5}\right ) \textit {\_Z}^{6}+\left (16 a^{5} b \,c^{2} e^{3}+48 a^{5} c^{3} d \,e^{2}-8 a^{4} b^{3} c \,e^{3}-72 a^{4} b^{2} c^{2} d \,e^{2}-96 a^{4} b \,c^{3} d^{2} e -16 a^{4} c^{4} d^{3}+a^{3} b^{5} e^{3}+27 a^{3} b^{4} c d \,e^{2}+96 a^{3} b^{3} c^{2} d^{2} e +56 a^{3} b^{2} c^{3} d^{3}-3 a^{2} b^{6} d \,e^{2}-30 a^{2} b^{5} c \,d^{2} e -41 a^{2} b^{4} c^{2} d^{3}+3 a \,b^{7} d^{2} e +11 a \,b^{6} c \,d^{3}-b^{8} d^{3}\right ) \textit {\_Z}^{3}+a^{3} c^{2} e^{6}-3 a^{2} b \,c^{2} d \,e^{5}+3 a^{2} c^{3} d^{2} e^{4}+3 a \,b^{2} c^{2} d^{2} e^{4}-6 a b \,c^{3} d^{3} e^{3}+3 a \,c^{4} d^{4} e^{2}-b^{3} c^{2} d^{3} e^{3}+3 b^{2} c^{3} d^{4} e^{2}-3 b \,c^{4} d^{5} e +c^{5} d^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (224 c^{3} a^{8}-176 b^{2} c^{2} a^{7}+46 b^{4} c \,a^{6}-4 b^{6} a^{5}\right ) \textit {\_R}^{6}+\left (48 a^{5} b \,c^{2} e^{3}+156 a^{5} c^{3} d \,e^{2}-24 a^{4} b^{3} c \,e^{3}-219 a^{4} b^{2} c^{2} d \,e^{2}-300 a^{4} b \,c^{3} d^{2} e -52 a^{4} c^{4} d^{3}+3 a^{3} b^{5} e^{3}+81 a^{3} b^{4} c d \,e^{2}+291 a^{3} b^{3} c^{2} d^{2} e +173 a^{3} b^{2} c^{3} d^{3}-9 a^{2} b^{6} d \,e^{2}-90 a^{2} b^{5} c \,d^{2} e -124 a^{2} b^{4} c^{2} d^{3}+9 a \,b^{7} d^{2} e +33 a \,b^{6} c \,d^{3}-3 b^{8} d^{3}\right ) \textit {\_R}^{3}+3 a^{3} c^{2} e^{6}-9 a^{2} b \,c^{2} d \,e^{5}+9 a^{2} c^{3} d^{2} e^{4}+9 a \,b^{2} c^{2} d^{2} e^{4}-18 a b \,c^{3} d^{3} e^{3}+9 a \,c^{4} d^{4} e^{2}-3 b^{3} c^{2} d^{3} e^{3}+9 b^{2} c^{3} d^{4} e^{2}-9 b \,c^{4} d^{5} e +3 c^{5} d^{6}\right ) x +\left (-12 a^{6} b \,c^{2} e^{2}-16 a^{6} c^{3} d e +7 a^{5} b^{3} c \,e^{2}+36 a^{5} b^{2} c^{2} d e +20 a^{5} b \,c^{3} d^{2}-a^{4} b^{5} e^{2}-16 a^{4} b^{4} c d e -25 a^{4} b^{3} c^{2} d^{2}+2 a^{3} b^{6} d e +9 a^{3} b^{5} c \,d^{2}-a^{2} b^{7} d^{2}\right ) \textit {\_R}^{4}+\left (-a^{4} c^{2} e^{5}+2 a^{3} b \,c^{2} d \,e^{4}-2 a^{3} c^{3} d^{2} e^{3}-a^{2} b^{2} c^{2} d^{2} e^{3}+2 a^{2} b \,c^{3} d^{3} e^{2}-a^{2} c^{4} d^{4} e \right ) \textit {\_R} \right )\right )}{3}\) | \(970\) |
1/3/a*sum((-_R^3*c*d+a*e-b*d)/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=RootOf(_Z^6*c+ _Z^3*b+a))-1/2*d/a/x^2
Leaf count of result is larger than twice the leaf count of optimal. 11459 vs. \(2 (517) = 1034\).
Time = 18.76 (sec) , antiderivative size = 11459, normalized size of antiderivative = 17.49 \[ \int \frac {d+e x^3}{x^3 \left (a+b x^3+c x^6\right )} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {d+e x^3}{x^3 \left (a+b x^3+c x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {d+e x^3}{x^3 \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {e x^{3} + d}{{\left (c x^{6} + b x^{3} + a\right )} x^{3}} \,d x } \]
\[ \int \frac {d+e x^3}{x^3 \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {e x^{3} + d}{{\left (c x^{6} + b x^{3} + a\right )} x^{3}} \,d x } \]
Time = 35.79 (sec) , antiderivative size = 13466, normalized size of antiderivative = 20.56 \[ \int \frac {d+e x^3}{x^3 \left (a+b x^3+c x^6\right )} \, dx=\text {Too large to display} \]
log(- (2^(2/3)*((b^8*d^3 - a^3*b^5*e^3 + 16*a^4*c^4*d^3 + b^5*d^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^3*c*e^3 - 16*a^5*b*c^2*e^3 + 2*a^4*c*e^3*(-(4*a *c - b^2)^3)^(1/2) + 3*a^2*b^6*d*e^2 - 48*a^5*c^3*d*e^2 + 41*a^2*b^4*c^2*d ^3 - 56*a^3*b^2*c^3*d^3 - a^3*b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^6* c*d^3 - 3*a*b^7*d^2*e - 5*a*b^3*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^4*d ^2*e*(-(4*a*c - b^2)^3)^(1/2) + 30*a^2*b^5*c*d^2*e - 27*a^3*b^4*c*d*e^2 + 96*a^4*b*c^3*d^2*e + 5*a^2*b*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^3* d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 96*a^3*b^3*c^2*d^2*e + 72*a^4*b^2*c^2*d*e ^2 - 6*a^3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b^2*c*d^2*e*(-(4*a* c - b^2)^3)^(1/2) - 9*a^3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2))/(a^5*(4*a*c - b^2)^3))^(1/3)*((2^(1/3)*(81*a^8*c^3*x*(4*a*c - b^2)^2*(a*b*e - b^2*d + a*c*d) + (81*2^(2/3)*a^10*b*c^3*(4*a*c - b^2)^2*((b^8*d^3 - a^3*b^5*e^3 + 16*a^4*c^4*d^3 + b^5*d^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^3*c*e^3 - 16*a ^5*b*c^2*e^3 + 2*a^4*c*e^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^6*d*e^2 - 48 *a^5*c^3*d*e^2 + 41*a^2*b^4*c^2*d^3 - 56*a^3*b^2*c^3*d^3 - a^3*b^2*e^3*(-( 4*a*c - b^2)^3)^(1/2) - 11*a*b^6*c*d^3 - 3*a*b^7*d^2*e - 5*a*b^3*c*d^3*(-( 4*a*c - b^2)^3)^(1/2) - 3*a*b^4*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 30*a^2*b^ 5*c*d^2*e - 27*a^3*b^4*c*d*e^2 + 96*a^4*b*c^3*d^2*e + 5*a^2*b*c^2*d^3*(-(4 *a*c - b^2)^3)^(1/2) + 3*a^2*b^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 96*a^3*b ^3*c^2*d^2*e + 72*a^4*b^2*c^2*d*e^2 - 6*a^3*c^2*d^2*e*(-(4*a*c - b^2)^3...