\(\int \frac {x^2}{(d+e x^4) (a+b x^4+c x^8)^2} \, dx\) [117]

Optimal result

Integrand size = 27, antiderivative size = 1702 \[ \int \frac {x^2}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx =\text {Too large to display} \] Output:

-1/5*e^2*(-5*b*e+c*d)*x^3/(a*e^2-b*d*e+c*d^2)^2/(c*x^8+b*x^4+a)+c*e^3*x^7/ 
(a*e^2-b*d*e+c*d^2)^2/(c*x^8+b*x^4+a)+1/20*x^3*(a*b*c*e*(5*c*d^2-9*e*(-5*a 
*e+b*d))+(-2*a*c+b^2)*(5*c^2*d^3-c*d*e*(-13*a*e+10*b*d)+5*b*e^2*(-5*a*e+b* 
d))+5*c*(b^3*d*e^2+b*c*d*(-a*e^2+c*d^2)+2*a*c*e*(9*a*e^2+c*d^2)-b^2*(5*a*e 
^3+2*c*d^2*e))*x^4)/a/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(c*x^8+b*x^4+a)-1 
/16*c^(1/4)*(b^4*d*e^2+b^2*(c^2*d^3+5*a*(-4*a*c+b^2)^(1/2)*e^3+c*d*e*(2*(- 
4*a*c+b^2)^(1/2)*d-3*a*e))-b^3*e*(2*c*d^2+e*((-4*a*c+b^2)^(1/2)*d+5*a*e))- 
2*a*c*(10*c^2*d^3+9*a*(-4*a*c+b^2)^(1/2)*e^3+c*d*e*((-4*a*c+b^2)^(1/2)*d+2 
6*a*e))-b*c*(c*d^2*((-4*a*c+b^2)^(1/2)*d-32*a*e)-a*e^2*((-4*a*c+b^2)^(1/2) 
*d+28*a*e)))*arctan(2^(1/4)*c^(1/4)*x/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(1/ 
4)/a/(-4*a*c+b^2)^(3/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)/(c*d^2-e*(-a*e+b*d)) 
^2+1/16*c^(1/4)*(b^4*d*e^2-2*a*c*(10*c^2*d^3-9*a*(-4*a*c+b^2)^(1/2)*e^3-c* 
d*e*((-4*a*c+b^2)^(1/2)*d-26*a*e))-b^3*e*(2*c*d^2-e*((-4*a*c+b^2)^(1/2)*d- 
5*a*e))+b^2*(c^2*d^3-5*a*(-4*a*c+b^2)^(1/2)*e^3-c*d*e*(2*(-4*a*c+b^2)^(1/2 
)*d+3*a*e))-b*c*(a*e^2*((-4*a*c+b^2)^(1/2)*d-28*a*e)-c*d^2*((-4*a*c+b^2)^( 
1/2)*d+32*a*e)))*arctan(2^(1/4)*c^(1/4)*x/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*2 
^(1/4)/a/(-4*a*c+b^2)^(3/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)/(a*e^2-b*d*e+c*d 
^2)^2+1/4*e^(13/4)*arctan(-1+2^(1/2)*e^(1/4)*x/d^(1/4))*2^(1/2)/d^(1/4)/(a 
*e^2-b*d*e+c*d^2)^2+1/4*e^(13/4)*arctan(1+2^(1/2)*e^(1/4)*x/d^(1/4))*2^(1/ 
2)/d^(1/4)/(a*e^2-b*d*e+c*d^2)^2+1/16*c^(1/4)*(b^4*d*e^2+b^2*(c^2*d^3+5...
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 8.58 (sec) , antiderivative size = 645, normalized size of antiderivative = 0.38 \[ \int \frac {x^2}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {-\frac {4 \left (c d^2+e (-b d+a e)\right ) x^3 \left (-b^3 e+b c \left (3 a e+c d x^4\right )+b^2 c \left (d-e x^4\right )-2 a c^2 \left (d-e x^4\right )\right )}{a \left (-b^2+4 a c\right ) \left (a+b x^4+c x^8\right )}-\frac {4 \sqrt {2} e^{13/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{\sqrt [4]{d}}+\frac {4 \sqrt {2} e^{13/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{\sqrt [4]{d}}+\frac {2 \sqrt {2} e^{13/4} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {e} x^2\right )}{\sqrt [4]{d}}-\frac {2 \sqrt {2} e^{13/4} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {e} x^2\right )}{\sqrt [4]{d}}+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^2 c^2 d^3 \log (x-\text {$\#$1})-10 a c^3 d^3 \log (x-\text {$\#$1})-2 b^3 c d^2 e \log (x-\text {$\#$1})+17 a b c^2 d^2 e \log (x-\text {$\#$1})+b^4 d e^2 \log (x-\text {$\#$1})-2 a b^2 c d e^2 \log (x-\text {$\#$1})-26 a^2 c^2 d e^2 \log (x-\text {$\#$1})-5 a b^3 e^3 \log (x-\text {$\#$1})+23 a^2 b c e^3 \log (x-\text {$\#$1})+b c^3 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^4-2 b^2 c^2 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+2 a c^3 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+b^3 c d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-a b c^2 d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-5 a b^2 c e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4+18 a^2 c^2 e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{a \left (b^2-4 a c\right )}}{16 \left (c d^2+e (-b d+a e)\right )^2} \] Input:

Integrate[x^2/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

((-4*(c*d^2 + e*(-(b*d) + a*e))*x^3*(-(b^3*e) + b*c*(3*a*e + c*d*x^4) + b^ 
2*c*(d - e*x^4) - 2*a*c^2*(d - e*x^4)))/(a*(-b^2 + 4*a*c)*(a + b*x^4 + c*x 
^8)) - (4*Sqrt[2]*e^(13/4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/d^(1/4 
) + (4*Sqrt[2]*e^(13/4)*ArcTan[1 + (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/d^(1/4) + 
 (2*Sqrt[2]*e^(13/4)*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*x + Sqrt[e]*x^2 
])/d^(1/4) - (2*Sqrt[2]*e^(13/4)*Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*x + 
 Sqrt[e]*x^2])/d^(1/4) + RootSum[a + b*#1^4 + c*#1^8 & , (b^2*c^2*d^3*Log[ 
x - #1] - 10*a*c^3*d^3*Log[x - #1] - 2*b^3*c*d^2*e*Log[x - #1] + 17*a*b*c^ 
2*d^2*e*Log[x - #1] + b^4*d*e^2*Log[x - #1] - 2*a*b^2*c*d*e^2*Log[x - #1] 
- 26*a^2*c^2*d*e^2*Log[x - #1] - 5*a*b^3*e^3*Log[x - #1] + 23*a^2*b*c*e^3* 
Log[x - #1] + b*c^3*d^3*Log[x - #1]*#1^4 - 2*b^2*c^2*d^2*e*Log[x - #1]*#1^ 
4 + 2*a*c^3*d^2*e*Log[x - #1]*#1^4 + b^3*c*d*e^2*Log[x - #1]*#1^4 - a*b*c^ 
2*d*e^2*Log[x - #1]*#1^4 - 5*a*b^2*c*e^3*Log[x - #1]*#1^4 + 18*a^2*c^2*e^3 
*Log[x - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(a*(b^2 - 4*a*c)))/(16*(c*d^2 + e 
*(-(b*d) + a*e))^2)
 

Rubi [A] (verified)

Time = 2.53 (sec) , antiderivative size = 1320, normalized size of antiderivative = 0.78, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1862, 1824, 27, 1828, 1834, 27, 827, 218, 221, 1836, 2009}

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.

\(\displaystyle \int \frac {x^2}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx\)

\(\Big \downarrow \) 1862

\(\displaystyle \frac {\int \frac {c d x^4+a e}{x^2 \left (c x^8+b x^4+a\right )^2}dx}{a e^2-b d e+c d^2}-\frac {d e \int \frac {1}{x^2 \left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 1824

\(\displaystyle \frac {-\frac {\int \frac {a \left (5 c (2 c d-b e) x^4+b c d-5 b^2 e+18 a c e\right )}{x^2 \left (c x^8+b x^4+a\right )}dx}{4 a \left (b^2-4 a c\right )}-\frac {-e \left (b^2-2 a c\right )+c x^4 (2 c d-b e)+b c d}{4 x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \frac {1}{x^2 \left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {5 c (2 c d-b e) x^4+b c d-5 b^2 e+18 a c e}{x^2 \left (c x^8+b x^4+a\right )}dx}{4 \left (b^2-4 a c\right )}-\frac {-e \left (b^2-2 a c\right )+c x^4 (2 c d-b e)+b c d}{4 x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \frac {1}{x^2 \left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 1828

\(\displaystyle \frac {-\frac {-\frac {\int \frac {x^2 \left (c \left (-5 e b^2+c d b+18 a c e\right ) x^4-10 a c^2 d+b^2 c d-5 b^3 e+23 a b c e\right )}{c x^8+b x^4+a}dx}{a}-\frac {18 a c e-5 b^2 e+b c d}{a x}}{4 \left (b^2-4 a c\right )}-\frac {-e \left (b^2-2 a c\right )+c x^4 (2 c d-b e)+b c d}{4 x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \frac {1}{x^2 \left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 1834

\(\displaystyle \frac {-\frac {-\frac {\frac {1}{2} c \left (\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \int \frac {2 x^2}{2 c x^4+b-\sqrt {b^2-4 a c}}dx+\frac {1}{2} c \left (-\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \int \frac {2 x^2}{2 c x^4+b+\sqrt {b^2-4 a c}}dx}{a}-\frac {18 a c e-5 b^2 e+b c d}{a x}}{4 \left (b^2-4 a c\right )}-\frac {-e \left (b^2-2 a c\right )+c x^4 (2 c d-b e)+b c d}{4 x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \frac {1}{x^2 \left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {c \left (\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \int \frac {x^2}{2 c x^4+b-\sqrt {b^2-4 a c}}dx+c \left (-\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \int \frac {x^2}{2 c x^4+b+\sqrt {b^2-4 a c}}dx}{a}-\frac {18 a c e-5 b^2 e+b c d}{a x}}{4 \left (b^2-4 a c\right )}-\frac {-e \left (b^2-2 a c\right )+c x^4 (2 c d-b e)+b c d}{4 x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \frac {1}{x^2 \left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {-\frac {-\frac {c \left (-\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {-b-\sqrt {b^2-4 a c}}}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )+c \left (\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {\sqrt {b^2-4 a c}-b}}dx}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )}{a}-\frac {18 a c e-5 b^2 e+b c d}{a x}}{4 \left (b^2-4 a c\right )}-\frac {-e \left (b^2-2 a c\right )+c x^4 (2 c d-b e)+b c d}{4 x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \frac {1}{x^2 \left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {-\frac {c \left (-\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )+c \left (\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{2 \sqrt {2} \sqrt {c}}\right )}{a}-\frac {18 a c e-5 b^2 e+b c d}{a x}}{4 \left (b^2-4 a c\right )}-\frac {-e \left (b^2-2 a c\right )+c x^4 (2 c d-b e)+b c d}{4 x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \frac {1}{x^2 \left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {-\frac {c \left (-\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )+c \left (\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{a}-\frac {18 a c e-5 b^2 e+b c d}{a x}}{4 \left (b^2-4 a c\right )}-\frac {-e \left (b^2-2 a c\right )+c x^4 (2 c d-b e)+b c d}{4 x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \frac {1}{x^2 \left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 1836

\(\displaystyle \frac {-\frac {-\frac {c \left (-\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )+c \left (\frac {28 a b c e-20 a c^2 d-5 b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}+18 a c e-5 b^2 e+b c d\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{a}-\frac {18 a c e-5 b^2 e+b c d}{a x}}{4 \left (b^2-4 a c\right )}-\frac {-e \left (b^2-2 a c\right )+c x^4 (2 c d-b e)+b c d}{4 x \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \left (-\frac {x^2 e^3}{d \left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}+\frac {x^2 \left (-c (c d-b e) x^4-b c d+b^2 e-a c e\right )}{a \left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}+\frac {1}{a d x^2}\right )dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {c (2 c d-b e) x^4+b c d-\left (b^2-2 a c\right ) e}{4 \left (b^2-4 a c\right ) x \left (c x^8+b x^4+a\right )}-\frac {-\frac {-5 e b^2+c d b+18 a c e}{a x}-\frac {c \left (-5 e b^2+c d b+18 a c e-\frac {-5 e b^3+c d b^2+28 a c e b-20 a c^2 d}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )+c \left (-5 e b^2+c d b+18 a c e+\frac {-5 e b^3+c d b^2+28 a c e b-20 a c^2 d}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{a}}{4 \left (b^2-4 a c\right )}}{c d^2-b e d+a e^2}-\frac {d e \left (\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right ) e^{9/4}}{2 \sqrt {2} d^{5/4} \left (c d^2-b e d+a e^2\right )}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}+1\right ) e^{9/4}}{2 \sqrt {2} d^{5/4} \left (c d^2-b e d+a e^2\right )}-\frac {\log \left (\sqrt {e} x^2-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {d}\right ) e^{9/4}}{4 \sqrt {2} d^{5/4} \left (c d^2-b e d+a e^2\right )}+\frac {\log \left (\sqrt {e} x^2+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {d}\right ) e^{9/4}}{4 \sqrt {2} d^{5/4} \left (c d^2-b e d+a e^2\right )}-\frac {\sqrt [4]{c} \left (c d-b e-\frac {-e b^2+c d b+2 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} a \sqrt [4]{-b-\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )}-\frac {\sqrt [4]{c} \left (c d-b e+\frac {-e b^2+c d b+2 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt {b^2-4 a c}-b} \left (c d^2-b e d+a e^2\right )}+\frac {\sqrt [4]{c} \left (c d-b e-\frac {-e b^2+c d b+2 a c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2\ 2^{3/4} a \sqrt [4]{-b-\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )}+\frac {\sqrt [4]{c} \left (c d-b e+\frac {-e b^2+c d b+2 a c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt {b^2-4 a c}-b} \left (c d^2-b e d+a e^2\right )}-\frac {1}{a d x}\right )}{c d^2-b e d+a e^2}\)

Input:

Int[x^2/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

(-1/4*(b*c*d - (b^2 - 2*a*c)*e + c*(2*c*d - b*e)*x^4)/((b^2 - 4*a*c)*x*(a 
+ b*x^4 + c*x^8)) - (-((b*c*d - 5*b^2*e + 18*a*c*e)/(a*x)) - (c*(b*c*d - 5 
*b^2*e + 18*a*c*e - (b^2*c*d - 20*a*c^2*d - 5*b^3*e + 28*a*b*c*e)/Sqrt[b^2 
 - 4*a*c])*(ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2* 
2^(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - ArcTanh[(2^(1/4)*c^(1/4) 
*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4* 
a*c])^(1/4))) + c*(b*c*d - 5*b^2*e + 18*a*c*e + (b^2*c*d - 20*a*c^2*d - 5* 
b^3*e + 28*a*b*c*e)/Sqrt[b^2 - 4*a*c])*(ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + S 
qrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b + Sqrt[b^2 - 4*a*c])^(1/4) 
) - ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4) 
*c^(3/4)*(-b + Sqrt[b^2 - 4*a*c])^(1/4))))/a)/(4*(b^2 - 4*a*c)))/(c*d^2 - 
b*d*e + a*e^2) - (d*e*(-(1/(a*d*x)) - (c^(1/4)*(c*d - b*e - (b*c*d - b^2*e 
 + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 
 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(1/4)*(c*d^2 - b*d* 
e + a*e^2)) - (c^(1/4)*(c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4 
*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3 
/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(1/4)*(c*d^2 - b*d*e + a*e^2)) + (e^(9/4)*A 
rcTan[1 - (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(2*Sqrt[2]*d^(5/4)*(c*d^2 - b*d*e 
+ a*e^2)) - (e^(9/4)*ArcTan[1 + (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(2*Sqrt[2]*d 
^(5/4)*(c*d^2 - b*d*e + a*e^2)) + (c^(1/4)*(c*d - b*e - (b*c*d - b^2*e ...
 

Defintions of rubi rules used

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 1824

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( 
c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(a + b*x^n + c*x^ 
(2*n))^(p + 1)*((d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^n)/(a*f*n*(p + 
 1)*(b^2 - 4*a*c))), x] + Simp[1/(a*n*(p + 1)*(b^2 - 4*a*c))   Int[(f*x)^m* 
(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m 
+ 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + n*(2*p + 3) + 1)*(b*d - 2*a*e) 
*x^n, x], 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] && LtQ[p, -1] && IntegerQ[p]
 

rule 1828

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]
 

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]
 

rule 1836

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.))/((a_) + (c_.)*(x_)^ 
(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e 
*x^n)^q/(a + b*x^n + c*x^(2*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] && IntegerQ[q] && Int 
egerQ[m]
 

rule 1862

Int[(((f_.)*(x_))^(m_.)*((a_.) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)) 
/((d_.) + (e_.)*(x_)^(n_)), x_Symbol] :> Simp[f^n/(c*d^2 - b*d*e + a*e^2) 
 Int[(f*x)^(m - n)*(a*e + c*d*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] - Simp 
[d*e*(f^n/(c*d^2 - b*d*e + a*e^2))   Int[(f*x)^(m - n)*((a + b*x^n + c*x^(2 
*n))^(p + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2 
, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.77 (sec) , antiderivative size = 573, normalized size of antiderivative = 0.34

method	result	size
default	\(-\frac {\frac {\frac {c \left (2 a^{2} c \,e^{3}-a \,b^{2} e^{3}-a b c d \,e^{2}+2 a \,c^{2} d^{2} e +b^{3} d \,e^{2}-2 b^{2} c \,d^{2} e +b \,c^{2} d^{3}\right ) x^{7}}{4 a \left (4 a c -b^{2}\right )}+\frac {\left (3 a^{2} b c \,e^{3}-2 a^{2} c^{2} e^{2} d -a \,b^{3} e^{3}-2 a \,b^{2} c d \,e^{2}+5 a b \,c^{2} d^{2} e -2 a \,c^{3} d^{3}+b^{4} d \,e^{2}-2 b^{3} c \,d^{2} e +b^{2} c^{2} d^{3}\right ) x^{3}}{4 a \left (4 a c -b^{2}\right )}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (18 a^{2} c \,e^{3}-5 a \,b^{2} e^{3}-a b c d \,e^{2}+2 a \,c^{2} d^{2} e +b^{3} d \,e^{2}-2 b^{2} c \,d^{2} e +b \,c^{2} d^{3}\right ) \textit {\_R}^{6}+\left (23 a^{2} b c \,e^{3}-26 a^{2} c^{2} e^{2} d -5 a \,b^{3} e^{3}-2 a \,b^{2} c d \,e^{2}+17 a b \,c^{2} d^{2} e -10 a \,c^{3} d^{3}+b^{4} d \,e^{2}-2 b^{3} c \,d^{2} e +b^{2} c^{2} d^{3}\right ) \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{16 a \left (4 a c -b^{2}\right )}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {e^{3} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{e}}}{x^{2}+\left (\frac {d}{e}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{e}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{e}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{e}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (\frac {d}{e}\right )^{\frac {1}{4}}}\)	\(573\)
risch	\(\text {Expression too large to display}\)	\(63005\)

Input:

int(x^2/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/(a*e^2-b*d*e+c*d^2)^2*((1/4*c*(2*a^2*c*e^3-a*b^2*e^3-a*b*c*d*e^2+2*a*c^ 
2*d^2*e+b^3*d*e^2-2*b^2*c*d^2*e+b*c^2*d^3)/a/(4*a*c-b^2)*x^7+1/4*(3*a^2*b* 
c*e^3-2*a^2*c^2*d*e^2-a*b^3*e^3-2*a*b^2*c*d*e^2+5*a*b*c^2*d^2*e-2*a*c^3*d^ 
3+b^4*d*e^2-2*b^3*c*d^2*e+b^2*c^2*d^3)/a/(4*a*c-b^2)*x^3)/(c*x^8+b*x^4+a)+ 
1/16/a/(4*a*c-b^2)*sum((c*(18*a^2*c*e^3-5*a*b^2*e^3-a*b*c*d*e^2+2*a*c^2*d^ 
2*e+b^3*d*e^2-2*b^2*c*d^2*e+b*c^2*d^3)*_R^6+(23*a^2*b*c*e^3-26*a^2*c^2*d*e 
^2-5*a*b^3*e^3-2*a*b^2*c*d*e^2+17*a*b*c^2*d^2*e-10*a*c^3*d^3+b^4*d*e^2-2*b 
^3*c*d^2*e+b^2*c^2*d^3)*_R^2)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*c+ 
_Z^4*b+a)))+1/8*e^3/(a*e^2-b*d*e+c*d^2)^2/(d/e)^(1/4)*2^(1/2)*(ln((x^2-(d/ 
e)^(1/4)*x*2^(1/2)+(d/e)^(1/2))/(x^2+(d/e)^(1/4)*x*2^(1/2)+(d/e)^(1/2)))+2 
*arctan(2^(1/2)/(d/e)^(1/4)*x+1)+2*arctan(2^(1/2)/(d/e)^(1/4)*x-1))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x^2/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**2/(e*x**4+d)/(c*x**8+b*x**4+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^2}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\int { \frac {x^{2}}{{\left (c x^{8} + b x^{4} + a\right )}^{2} {\left (e x^{4} + d\right )}} \,d x } \] Input:

integrate(x^2/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="maxima")
 

Output:

-1/8*e^4*(sqrt(2)*log(sqrt(e)*x^2 + sqrt(2)*d^(1/4)*e^(1/4)*x + sqrt(d))/( 
d^(1/4)*e^(3/4)) - sqrt(2)*log(sqrt(e)*x^2 - sqrt(2)*d^(1/4)*e^(1/4)*x + s 
qrt(d))/(d^(1/4)*e^(3/4)) - sqrt(2)*log((2*sqrt(e)*x - sqrt(2)*sqrt(-sqrt( 
d)*sqrt(e)) + sqrt(2)*d^(1/4)*e^(1/4))/(2*sqrt(e)*x + sqrt(2)*sqrt(-sqrt(d 
)*sqrt(e)) + sqrt(2)*d^(1/4)*e^(1/4)))/(sqrt(-sqrt(d)*sqrt(e))*sqrt(e)) - 
sqrt(2)*log((2*sqrt(e)*x - sqrt(2)*sqrt(-sqrt(d)*sqrt(e)) - sqrt(2)*d^(1/4 
)*e^(1/4))/(2*sqrt(e)*x + sqrt(2)*sqrt(-sqrt(d)*sqrt(e)) - sqrt(2)*d^(1/4) 
*e^(1/4)))/(sqrt(-sqrt(d)*sqrt(e))*sqrt(e)))/(c^2*d^4 - 2*b*c*d^3*e - 2*a* 
b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2) + 1/4*((b*c^2*d - (b^2*c - 2*a* 
c^2)*e)*x^7 + ((b^2*c - 2*a*c^2)*d - (b^3 - 3*a*b*c)*e)*x^3)/(((a*b^2*c^2 
- 4*a^2*c^3)*d^2 - (a*b^3*c - 4*a^2*b*c^2)*d*e + (a^2*b^2*c - 4*a^3*c^2)*e 
^2)*x^8 + ((a*b^3*c - 4*a^2*b*c^2)*d^2 - (a*b^4 - 4*a^2*b^2*c)*d*e + (a^2* 
b^3 - 4*a^3*b*c)*e^2)*x^4 + (a^2*b^2*c - 4*a^3*c^2)*d^2 - (a^2*b^3 - 4*a^3 
*b*c)*d*e + (a^3*b^2 - 4*a^4*c)*e^2) + 1/4*integrate(((b*c^3*d^3 - 2*(b^2* 
c^2 - a*c^3)*d^2*e + (b^3*c - a*b*c^2)*d*e^2 - (5*a*b^2*c - 18*a^2*c^2)*e^ 
3)*x^6 + ((b^2*c^2 - 10*a*c^3)*d^3 - (2*b^3*c - 17*a*b*c^2)*d^2*e + (b^4 - 
 2*a*b^2*c - 26*a^2*c^2)*d*e^2 - (5*a*b^3 - 23*a^2*b*c)*e^3)*x^2)/(c*x^8 + 
 b*x^4 + a), x)/((a*b^2*c^2 - 4*a^2*c^3)*d^4 - 2*(a*b^3*c - 4*a^2*b*c^2)*d 
^3*e + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*e^2 - 2*(a^2*b^3 - 4*a^3*b*c) 
*d*e^3 + (a^3*b^2 - 4*a^4*c)*e^4)
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x^2/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Hanged} \] Input:

int(x^2/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {x^2}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {x^{2}}{\left (e \,x^{4}+d \right ) \left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:

int(x^2/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
 

Output:

int(x^2/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)