Integrand size = 18, antiderivative size = 558 \[ \int \frac {x^3}{a+b x^3+c x^6} \, dx=\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{c} \sqrt {b^2-4 a c}}-\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{c} \sqrt {b^2-4 a c}}-\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt {b^2-4 a c}}+\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt {b^2-4 a c}}+\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt {b^2-4 a c}}-\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt {b^2-4 a c}} \]
-1/6*ln(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))^(1/3)*2^(2/3)/c^(1/3)/(-4*a*c+b^2)^(1/2)+1/12*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 ))*(b-(-4*a*c+b^2)^(1/2))^(1/3)*2^(2/3)/c^(1/3)/(-4*a*c+b^2)^(1/2)+1/6*arc tan(1/3*(1-2*2^(1/3)*c^(1/3)*x/(b-(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*(b-( -4*a*c+b^2)^(1/2))^(1/3)*2^(2/3)/c^(1/3)*3^(1/2)/(-4*a*c+b^2)^(1/2)+1/6*ln (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))^(1 /3)*2^(2/3)/c^(1/3)/(-4*a*c+b^2)^(1/2)-1/12*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))*(b+( -4*a*c+b^2)^(1/2))^(1/3)*2^(2/3)/c^(1/3)/(-4*a*c+b^2)^(1/2)-1/6*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))*(b+(-4*a*c+ b^2)^(1/2))^(1/3)*2^(2/3)/c^(1/3)*3^(1/2)/(-4*a*c+b^2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.08 \[ \int \frac {x^3}{a+b x^3+c x^6} \, dx=\frac {1}{3} \text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{b+2 c \text {$\#$1}^3}\&\right ] \]
Time = 0.77 (sec) , antiderivative size = 496, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {1710, 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.
| \(\displaystyle \int \frac {x^3}{a+b x^3+c x^6} \, dx\) |
| \(\Big \downarrow \) 1710 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^3+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \int \frac {1}{c x^3+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{c} x+\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{c} x+\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \left (\frac {2\ 2^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )\) |
((1 - b/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 + ((1 + b/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
3.2.46.3.1 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_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^n/2)*(b/q + 1) Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Simp[(d^n/2)*(b/q - 1) Int[(d*x)^(m - n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] & & NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[m, n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.08
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{6}+\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +b \,\textit {\_R}^{2}}\right )}{3}\) | \(43\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{6}+\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +b \,\textit {\_R}^{2}}\right )}{3}\) | \(43\) |
Leaf count of result is larger than twice the leaf count of optimal. 1542 vs. \(2 (421) = 842\).
Time = 0.29 (sec) , antiderivative size = 1542, normalized size of antiderivative = 2.76 \[ \int \frac {x^3}{a+b x^3+c x^6} \, dx=\text {Too large to display} \]
-1/6*(1/2)^(1/3)*(sqrt(-3) + 1)*(((b^2*c - 4*a*c^2)*sqrt(b^2/(b^6*c^2 - 12 *a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)) + 1)/(b^2*c - 4*a*c^2))^(1/3)*l og(-(1/2)^(1/3)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3 + sqrt(-3)*(b^4*c - 8*a* b^2*c^2 + 16*a^2*c^3))*sqrt(b^2/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(((b^2*c - 4*a*c^2)*sqrt(b^2/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^ 2*b^2*c^4 - 64*a^3*c^5)) + 1)/(b^2*c - 4*a*c^2))^(1/3) + 2*b*x) + 1/6*(1/2 )^(1/3)*(sqrt(-3) - 1)*(((b^2*c - 4*a*c^2)*sqrt(b^2/(b^6*c^2 - 12*a*b^4*c^ 3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)) + 1)/(b^2*c - 4*a*c^2))^(1/3)*log(-(1/2) ^(1/3)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3 - sqrt(-3)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3))*sqrt(b^2/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c ^5))*(((b^2*c - 4*a*c^2)*sqrt(b^2/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)) + 1)/(b^2*c - 4*a*c^2))^(1/3) + 2*b*x) - 1/6*(1/2)^(1/3)*( sqrt(-3) + 1)*(-((b^2*c - 4*a*c^2)*sqrt(b^2/(b^6*c^2 - 12*a*b^4*c^3 + 48*a ^2*b^2*c^4 - 64*a^3*c^5)) - 1)/(b^2*c - 4*a*c^2))^(1/3)*log((1/2)^(1/3)*(b ^4*c - 8*a*b^2*c^2 + 16*a^2*c^3 + sqrt(-3)*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c ^3))*sqrt(b^2/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(-(( b^2*c - 4*a*c^2)*sqrt(b^2/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^ 3*c^5)) - 1)/(b^2*c - 4*a*c^2))^(1/3) + 2*b*x) + 1/6*(1/2)^(1/3)*(sqrt(-3) - 1)*(-((b^2*c - 4*a*c^2)*sqrt(b^2/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c ^4 - 64*a^3*c^5)) - 1)/(b^2*c - 4*a*c^2))^(1/3)*log((1/2)^(1/3)*(b^4*c ...
Time = 0.93 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.22 \[ \int \frac {x^3}{a+b x^3+c x^6} \, dx=\operatorname {RootSum} {\left (t^{6} \cdot \left (46656 a^{3} c^{4} - 34992 a^{2} b^{2} c^{3} + 8748 a b^{4} c^{2} - 729 b^{6} c\right ) + t^{3} \cdot \left (432 a^{2} c^{2} - 216 a b^{2} c + 27 b^{4}\right ) + a, \left ( t \mapsto t \log {\left (x + \frac {2592 t^{4} a^{2} c^{3} - 1296 t^{4} a b^{2} c^{2} + 162 t^{4} b^{4} c + 12 t a c - 3 t b^{2}}{b} \right )} \right )\right )} \]
RootSum(_t**6*(46656*a**3*c**4 - 34992*a**2*b**2*c**3 + 8748*a*b**4*c**2 - 729*b**6*c) + _t**3*(432*a**2*c**2 - 216*a*b**2*c + 27*b**4) + a, Lambda( _t, _t*log(x + (2592*_t**4*a**2*c**3 - 1296*_t**4*a*b**2*c**2 + 162*_t**4* b**4*c + 12*_t*a*c - 3*_t*b**2)/b)))
\[ \int \frac {x^3}{a+b x^3+c x^6} \, dx=\int { \frac {x^{3}}{c x^{6} + b x^{3} + a} \,d x } \]
\[ \int \frac {x^3}{a+b x^3+c x^6} \, dx=\int { \frac {x^{3}}{c x^{6} + b x^{3} + a} \,d x } \]
Time = 12.04 (sec) , antiderivative size = 2129, normalized size of antiderivative = 3.82 \[ \int \frac {x^3}{a+b x^3+c x^6} \, dx=\text {Too large to display} \]
log((2^(2/3)*(-(b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c) /(c*(4*a*c - b^2)^3))^(1/3)*(9*a*b^3*c^2 - 36*a^2*b*c^3 + (9*2^(1/3)*a*c^3 *(4*a*c - b^2)^2*(x - (2^(2/3)*b*(-(b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16* a^2*c^2 - 8*a*b^2*c)/(c*(4*a*c - b^2)^3))^(1/3))/2)*(-(b*(-(4*a*c - b^2)^3 )^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c)/(c*(4*a*c - b^2)^3))^(2/3))/2))/6 + 3*a*c^2*x*(2*a*c - b^2))*((b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c)/(54*(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3)))^( 1/3) + log((2^(2/3)*(9*a*b^3*c^2 - 36*a^2*b*c^3 + (9*2^(1/3)*a*c^3*(x - (2 ^(2/3)*b*((b*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c)/(c*( 4*a*c - b^2)^3))^(1/3))/2)*(4*a*c - b^2)^2*((b*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c)/(c*(4*a*c - b^2)^3))^(2/3))/2)*((b*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c)/(c*(4*a*c - b^2)^3))^(1/3) )/6 + 3*a*c^2*x*(2*a*c - b^2))*(-(b*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^ 2*c^2 + 8*a*b^2*c)/(54*(b^6*c - 64*a^3*c^4 - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 )))^(1/3) + log((2^(2/3)*(3^(1/2)*1i - 1)*(36*a^2*b*c^3 - 9*a*b^3*c^2 + (2 ^(1/3)*(3^(1/2)*1i + 1)*(81*a*c^3*x*(4*a*c - b^2)^2 - (81*2^(2/3)*a*b*c^3* (3^(1/2)*1i - 1)*(4*a*c - b^2)^2*(-(b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16* a^2*c^2 - 8*a*b^2*c)/(c*(4*a*c - b^2)^3))^(1/3))/4)*(-(b*(-(4*a*c - b^2)^3 )^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c)/(c*(4*a*c - b^2)^3))^(2/3))/36)*(- (b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c)/(c*(4*a*c -...