Integrand size = 25, antiderivative size = 723 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\frac {e x^2}{2 c}-\frac {\left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} c^{5/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} c^{5/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}}-\frac {\left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{5/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{5/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}}+\frac {\left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{5/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{5/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}} \] Output:
1/2*e*x^2/c-1/6*(c*d-b*e-(2*a*c*e-b^2*e+b*c*d)/(-4*a*c+b^2)^(1/2))*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))*2^(1/3)* 3^(1/2)/c^(5/3)/(b-(-4*a*c+b^2)^(1/2))^(1/3)-1/6*(c*d-b*e+(2*a*c*e-b^2*e+b *c*d)/(-4*a*c+b^2)^(1/2))*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))*2^(1/3)*3^(1/2)/c^(5/3)/(b+(-4*a*c+b^2)^(1/2))^(1 /3)-1/6*(c*d-b*e-(2*a*c*e-b^2*e+b*c*d)/(-4*a*c+b^2)^(1/2))*ln((b-(-4*a*c+b ^2)^(1/2))^(1/3)+2^(1/3)*c^(1/3)*x)*2^(1/3)/c^(5/3)/(b-(-4*a*c+b^2)^(1/2)) ^(1/3)-1/6*(c*d-b*e+(2*a*c*e-b^2*e+b*c*d)/(-4*a*c+b^2)^(1/2))*ln((b+(-4*a* c+b^2)^(1/2))^(1/3)+2^(1/3)*c^(1/3)*x)*2^(1/3)/c^(5/3)/(b+(-4*a*c+b^2)^(1/ 2))^(1/3)+1/12*(c*d-b*e-(2*a*c*e-b^2*e+b*c*d)/(-4*a*c+b^2)^(1/2))*ln((b-(- 4*a*c+b^2)^(1/2))^(2/3)-2^(1/3)*c^(1/3)*(b-(-4*a*c+b^2)^(1/2))^(1/3)*x+2^( 2/3)*c^(2/3)*x^2)*2^(1/3)/c^(5/3)/(b-(-4*a*c+b^2)^(1/2))^(1/3)+1/12*(c*d-b *e+(2*a*c*e-b^2*e+b*c*d)/(-4*a*c+b^2)^(1/2))*ln((b+(-4*a*c+b^2)^(1/2))^(2/ 3)-2^(1/3)*c^(1/3)*(b+(-4*a*c+b^2)^(1/2))^(1/3)*x+2^(2/3)*c^(2/3)*x^2)*2^( 1/3)/c^(5/3)/(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 = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.12 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\frac {3 e x^2-2 \text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {a e \log (x-\text {$\#$1})-c d \log (x-\text {$\#$1}) \text {$\#$1}^3+b e \log (x-\text {$\#$1}) \text {$\#$1}^3}{b \text {$\#$1}+2 c \text {$\#$1}^4}\&\right ]}{6 c} \] Input:
Integrate[(x^4*(d + e*x^3))/(a + b*x^3 + c*x^6),x]
Output:
(3*e*x^2 - 2*RootSum[a + b*#1^3 + c*#1^6 & , (a*e*Log[x - #1] - c*d*Log[x - #1]*#1^3 + b*e*Log[x - #1]*#1^3)/(b*#1 + 2*c*#1^4) & ])/(6*c)
Time = 1.16 (sec) , antiderivative size = 577, normalized size of antiderivative = 0.80, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {1826, 27, 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.
\(\displaystyle \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx\) |
\(\Big \downarrow \) 1826 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {\int \frac {2 x \left (a e-(c d-b e) x^3\right )}{c x^6+b x^3+a}dx}{2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {\int \frac {x \left (a e-(c d-b e) x^3\right )}{c x^6+b x^3+a}dx}{c}\) |
\(\Big \downarrow \) 1834 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {-\frac {1}{2} \left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \int \frac {2 x}{2 c x^3+b-\sqrt {b^2-4 a c}}dx-\frac {1}{2} \left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \int \frac {2 x}{2 c x^3+b+\sqrt {b^2-4 a c}}dx}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {-\left (\left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \int \frac {x}{2 c x^3+b-\sqrt {b^2-4 a c}}dx\right )-\left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \int \frac {x}{2 c x^3+b+\sqrt {b^2-4 a c}}dx}{c}\) |
\(\Big \downarrow \) 821 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {-\left (\left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b-\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\int \frac {1}{\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b-\sqrt {b^2-4 a c}}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )\right )-\left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b+\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\int \frac {1}{\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b+\sqrt {b^2-4 a c}}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{c}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {-\left (\left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b-\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )\right )-\left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b+\sqrt {b^2-4 a c}}}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{c}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {-\left (\left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )\right )-\left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {-\left (\left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )\right )-\left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{c} \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x\right )}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {-\left (\left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )\right )-\left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{c}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {-\left (\left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}-\frac {1}{2} \int \frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )\right )-\left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}-\frac {1}{2} \int \frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{c}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {-\left (\left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )\right )-\left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}-2 \sqrt [3]{2} \sqrt [3]{c} x}{2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{c}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {e x^2}{2 c}-\frac {-\left (\left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{2 \sqrt [3]{2} \sqrt [3]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )\right )-\left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \left (\frac {\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{2 \sqrt [3]{2} \sqrt [3]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{c}}}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}}\right )}{c}\) |
Input:
Int[(x^4*(d + e*x^3))/(a + b*x^3 + c*x^6),x]
Output:
(e*x^2)/(2*c) - (-((c*d - b*e - (b*c*d - b^2*e + 2*a*c*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*e + (b*c*d - b^2*e + 2*a*c*e)/S qrt[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 + Sq rt[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
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[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*( m + n*(2*p + 1) + 1)) Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a* e*(m - n + 1) + (b*e*(m + n*p + 1) - c*d*(m + n*(2*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] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && Intege rQ[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 = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.10
method | result | size |
default | \(\frac {e \,x^{2}}{2 c}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (\left (e b -c d \right ) \textit {\_R}^{4}+a e \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}}{3 c}\) | \(70\) |
risch | \(\frac {e \,x^{2}}{2 c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (\left (-e b +c d \right ) \textit {\_R}^{4}-a e \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}}{3 c}\) | \(71\) |
Input:
int(x^4*(e*x^3+d)/(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)
Output:
1/2*e*x^2/c-1/3/c*sum(((b*e-c*d)*_R^4+a*e*_R)/(2*_R^5*c+_R^2*b)*ln(x-_R),_ R=RootOf(_Z^6*c+_Z^3*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 13535 vs. \(2 (583) = 1166\).
Time = 45.78 (sec) , antiderivative size = 13535, normalized size of antiderivative = 18.72 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\text {Too large to display} \] Input:
integrate(x^4*(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\text {Timed out} \] Input:
integrate(x**4*(e*x**3+d)/(c*x**6+b*x**3+a),x)
Output:
Timed out
\[ \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\int { \frac {{\left (e x^{3} + d\right )} x^{4}}{c x^{6} + b x^{3} + a} \,d x } \] Input:
integrate(x^4*(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="maxima")
Output:
1/2*e*x^2/c - integrate(-((c*d - b*e)*x^4 - a*e*x)/(c*x^6 + b*x^3 + a), x) /c
Timed out. \[ \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\text {Timed out} \] Input:
integrate(x^4*(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="giac")
Output:
Timed out
Time = 69.11 (sec) , antiderivative size = 13112, normalized size of antiderivative = 18.14 \[ \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\text {Too large to display} \] Input:
int((x^4*(d + e*x^3))/(a + b*x^3 + c*x^6),x)
Output:
log((2^(1/3)*((2^(2/3)*(27*a^2*c*x*(4*a*c - b^2)*(b^2*e^2 + 2*c^2*d^2 - 2* a*c*e^2 - 2*b*c*d*e) - (27*2^(1/3)*a*b*c^3*(4*a*c - b^2)^2*(-(b^8*e^3 + 16 *a^4*c^4*e^3 - b^5*c^3*d^3 + b^5*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^3*c^ 4*d^3 - 16*a^2*b*c^5*d^3 + 2*a*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 48*a^3*c ^5*d^2*e + 3*b^6*c^2*d^2*e + 41*a^2*b^4*c^2*e^3 - 56*a^3*b^2*c^3*e^3 - b^2 *c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^6*c*e^3 - 3*b^7*c*d*e^2 - 5*a*b ^3*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^4*c^3*d^2*e + 30*a*b^5*c^2*d*e^ 2 + 96*a^3*b*c^4*d*e^2 - 3*b^4*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 5*a^2*b* c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 72*a^2*b^2*c^4*d^2*e - 96*a^2*b^3*c^3*d *e^2 - 6*a^2*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*b^3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 12*a*b^2*c^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b*c^3 *d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(c^5*(4*a*c - b^2)^3))^(2/3))/2)*(-(b^8*e ^3 + 16*a^4*c^4*e^3 - b^5*c^3*d^3 + b^5*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a *b^3*c^4*d^3 - 16*a^2*b*c^5*d^3 + 2*a*c^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 4 8*a^3*c^5*d^2*e + 3*b^6*c^2*d^2*e + 41*a^2*b^4*c^2*e^3 - 56*a^3*b^2*c^3*e^ 3 - b^2*c^3*d^3*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^6*c*e^3 - 3*b^7*c*d*e^2 - 5*a*b^3*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 27*a*b^4*c^3*d^2*e + 30*a*b^5*c ^2*d*e^2 + 96*a^3*b*c^4*d*e^2 - 3*b^4*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 5 *a^2*b*c^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 72*a^2*b^2*c^4*d^2*e - 96*a^2*b^ 3*c^3*d*e^2 - 6*a^2*c^3*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*b^3*c^2*d^2*...
\[ \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\frac {-2 \left (\int \frac {x^{4}}{c \,x^{6}+b \,x^{3}+a}d x \right ) b e +2 \left (\int \frac {x^{4}}{c \,x^{6}+b \,x^{3}+a}d x \right ) c d -2 \left (\int \frac {x}{c \,x^{6}+b \,x^{3}+a}d x \right ) a e +e \,x^{2}}{2 c} \] Input:
int(x^4*(e*x^3+d)/(c*x^6+b*x^3+a),x)
Output:
( - 2*int(x**4/(a + b*x**3 + c*x**6),x)*b*e + 2*int(x**4/(a + b*x**3 + c*x **6),x)*c*d - 2*int(x/(a + b*x**3 + c*x**6),x)*a*e + e*x**2)/(2*c)