\(\int \frac {x^4 (d+e x^3)}{a+b x^3+c x^6} \, dx\) [41]

Optimal result

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)
 

Mathematica [C] (verified)

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)
 

Rubi [A] (verified)

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
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 821

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]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 1826

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]
 

rule 1834

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]
 

Maple [C] (verified)

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))
 

Fricas [B] (verification not implemented)

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
 

Sympy [F(-1)]

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
 

Maxima [F]

\[ \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\int { \frac {{\left (e x^{3} + d\right )} x^{4}}{c x^{6} + b x^{3} + a} \,d x } \] 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
 

Giac [F(-1)]

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
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

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*...
 

Reduce [F]

\[ \int \frac {x^4 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx=\frac {-2 \left (\int \frac {x^{4}}{c \,x^{6}+b \,x^{3}+a}d x \right ) b e +2 \left (\int \frac {x^{4}}{c \,x^{6}+b \,x^{3}+a}d x \right ) c d -2 \left (\int \frac {x}{c \,x^{6}+b \,x^{3}+a}d x \right ) a e +e \,x^{2}}{2 c} \] 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)