\(\int \frac {x^4}{(d+e x^4) (a+b x^4+c x^8)^2} \, dx\) [116]

Optimal result

Integrand size = 27, antiderivative size = 1265 \[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx =\text {Too large to display} \] Output:

-1/4*x*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x^4)/(-4*a*c+b^2)/(a*e^2-b*d*e+ 
c*d^2)/(c*x^8+b*x^4+a)+1/16*c^(3/4)*(b^3*d*e^2+b*(8*c^2*d^3-3*a*(-4*a*c+b^ 
2)^(1/2)*e^3-c*d*e*(9*(-4*a*c+b^2)^(1/2)*d-20*a*e))-b^2*e*(13*c*d^2+e*((-4 
*a*c+b^2)^(1/2)*d-3*a*e))+2*c*(a*e^2*(11*(-4*a*c+b^2)^(1/2)*d-14*a*e)+c*d^ 
2*(3*(-4*a*c+b^2)^(1/2)*d+2*a*e)))*arctan(2^(1/4)*c^(1/4)*x/(-b-(-4*a*c+b^ 
2)^(1/2))^(1/4))*2^(3/4)/(-4*a*c+b^2)^(3/2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)/ 
(a*e^2-b*d*e+c*d^2)^2-1/16*c^(3/4)*(b^3*d*e^2-b^2*e*(13*c*d^2-e*((-4*a*c+b 
^2)^(1/2)*d+3*a*e))-2*c*(c*d^2*(3*(-4*a*c+b^2)^(1/2)*d-2*a*e)+a*e^2*(11*(- 
4*a*c+b^2)^(1/2)*d+14*a*e))+b*(8*c^2*d^3+3*a*(-4*a*c+b^2)^(1/2)*e^3+c*d*e* 
(9*(-4*a*c+b^2)^(1/2)*d+20*a*e)))*arctan(2^(1/4)*c^(1/4)*x/(-b+(-4*a*c+b^2 
)^(1/2))^(1/4))*2^(3/4)/(-4*a*c+b^2)^(3/2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)/( 
a*e^2-b*d*e+c*d^2)^2-1/4*d^(1/4)*e^(11/4)*arctan(-1+2^(1/2)*e^(1/4)*x/d^(1 
/4))*2^(1/2)/(a*e^2-b*d*e+c*d^2)^2-1/4*d^(1/4)*e^(11/4)*arctan(1+2^(1/2)*e 
^(1/4)*x/d^(1/4))*2^(1/2)/(a*e^2-b*d*e+c*d^2)^2+1/16*c^(3/4)*(b^3*d*e^2+b* 
(8*c^2*d^3-3*a*(-4*a*c+b^2)^(1/2)*e^3-c*d*e*(9*(-4*a*c+b^2)^(1/2)*d-20*a*e 
))-b^2*e*(13*c*d^2+e*((-4*a*c+b^2)^(1/2)*d-3*a*e))+2*c*(a*e^2*(11*(-4*a*c+ 
b^2)^(1/2)*d-14*a*e)+c*d^2*(3*(-4*a*c+b^2)^(1/2)*d+2*a*e)))*arctanh(2^(1/4 
)*c^(1/4)*x/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/(-4*a*c+b^2)^(3/2)/(-b- 
(-4*a*c+b^2)^(1/2))^(3/4)/(a*e^2-b*d*e+c*d^2)^2-1/16*c^(3/4)*(b^3*d*e^2-b^ 
2*e*(13*c*d^2-e*((-4*a*c+b^2)^(1/2)*d+3*a*e))-2*c*(c*d^2*(3*(-4*a*c+b^2...
 

Mathematica [C] (verified)

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

Time = 7.45 (sec) , antiderivative size = 531, normalized size of antiderivative = 0.42 \[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {-\frac {4 \left (c d^2+e (-b d+a e)\right ) x \left (-b^2 e+2 c \left (a e+c d x^4\right )+b c \left (d-e x^4\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+4 \sqrt {2} \sqrt [4]{d} e^{11/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )-4 \sqrt {2} \sqrt [4]{d} e^{11/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )+2 \sqrt {2} \sqrt [4]{d} e^{11/4} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {e} x^2\right )-2 \sqrt {2} \sqrt [4]{d} e^{11/4} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {e} x^2\right )+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b c^2 d^3 \log (x-\text {$\#$1})-2 b^2 c d^2 e \log (x-\text {$\#$1})+2 a c^2 d^2 e \log (x-\text {$\#$1})+b^3 d e^2 \log (x-\text {$\#$1})-a b c d e^2 \log (x-\text {$\#$1})+3 a b^2 e^3 \log (x-\text {$\#$1})-14 a^2 c e^3 \log (x-\text {$\#$1})-6 c^3 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^4+9 b c^2 d^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4+b^2 c d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4-22 a c^2 d e^2 \log (x-\text {$\#$1}) \text {$\#$1}^4+3 a b c e^3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{b^2-4 a c}}{16 \left (c d^2+e (-b d+a e)\right )^2} \] Input:

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

Output:

((-4*(c*d^2 + e*(-(b*d) + a*e))*x*(-(b^2*e) + 2*c*(a*e + c*d*x^4) + b*c*(d 
 - e*x^4)))/((b^2 - 4*a*c)*(a + b*x^4 + c*x^8)) + 4*Sqrt[2]*d^(1/4)*e^(11/ 
4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*x)/d^(1/4)] - 4*Sqrt[2]*d^(1/4)*e^(11/4)*Ar 
cTan[1 + (Sqrt[2]*e^(1/4)*x)/d^(1/4)] + 2*Sqrt[2]*d^(1/4)*e^(11/4)*Log[Sqr 
t[d] - Sqrt[2]*d^(1/4)*e^(1/4)*x + Sqrt[e]*x^2] - 2*Sqrt[2]*d^(1/4)*e^(11/ 
4)*Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*x + Sqrt[e]*x^2] + RootSum[a + b* 
#1^4 + c*#1^8 & , (b*c^2*d^3*Log[x - #1] - 2*b^2*c*d^2*e*Log[x - #1] + 2*a 
*c^2*d^2*e*Log[x - #1] + b^3*d*e^2*Log[x - #1] - a*b*c*d*e^2*Log[x - #1] + 
 3*a*b^2*e^3*Log[x - #1] - 14*a^2*c*e^3*Log[x - #1] - 6*c^3*d^3*Log[x - #1 
]*#1^4 + 9*b*c^2*d^2*e*Log[x - #1]*#1^4 + b^2*c*d*e^2*Log[x - #1]*#1^4 - 2 
2*a*c^2*d*e^2*Log[x - #1]*#1^4 + 3*a*b*c*e^3*Log[x - #1]*#1^4)/(b*#1^3 + 2 
*c*#1^7) & ]/(b^2 - 4*a*c))/(16*(c*d^2 + e*(-(b*d) + a*e))^2)
 

Rubi [A] (verified)

Time = 2.20 (sec) , antiderivative size = 1185, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1862, 1754, 1760, 25, 27, 1752, 756, 218, 221, 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.

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

\(\Big \downarrow \) 1862

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

\(\Big \downarrow \) 1754

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

\(\Big \downarrow \) 1760

\(\displaystyle \frac {-\frac {\int -\frac {a \left (-3 c (2 c d-b e) x^4+b c d+3 b^2 e-14 a c e\right )}{c x^8+b x^4+a}dx}{4 a \left (b^2-4 a c\right )}-\frac {x \left (2 a c e-b^2 e+c x^4 (2 c d-b e)+b c d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \left (\frac {e^2}{\left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}+\frac {-c e x^4+c d-b e}{\left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}\right )dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {a \left (-3 c (2 c d-b e) x^4+b c d+3 b^2 e-14 a c e\right )}{c x^8+b x^4+a}dx}{4 a \left (b^2-4 a c\right )}-\frac {x \left (2 a c e-b^2 e+c x^4 (2 c d-b e)+b c d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \left (\frac {e^2}{\left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}+\frac {-c e x^4+c d-b e}{\left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}\right )dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-3 c (2 c d-b e) x^4+b c d+3 b^2 e-14 a c e}{c x^8+b x^4+a}dx}{4 \left (b^2-4 a c\right )}-\frac {x \left (2 a c e-b^2 e+c x^4 (2 c d-b e)+b c d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \left (\frac {e^2}{\left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}+\frac {-c e x^4+c d-b e}{\left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}\right )dx}{a e^2-b d e+c d^2}\)

\(\Big \downarrow \) 1752

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {-\frac {1}{2} c \left (3 (2 c d-b e)+\frac {3 e b^2+8 c d b-28 a c e}{\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 )}{\sqrt [4]{2} \sqrt [4]{c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}\right )-\frac {1}{2} c \left (3 (2 c d-b e)-\frac {3 e b^2+8 c d b-28 a c e}{\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 )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{4 \left (b^2-4 a c\right )}-\frac {x \left (c (2 c d-b e) x^4+b c d-b^2 e+2 a c e\right )}{4 \left (b^2-4 a c\right ) \left (c x^8+b x^4+a\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^{7/4}}{2 \sqrt {2} d^{3/4} \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^{7/4}}{2 \sqrt {2} d^{3/4} \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^{7/4}}{4 \sqrt {2} d^{3/4} \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^{7/4}}{4 \sqrt {2} d^{3/4} \left (c d^2-b e d+a e^2\right )}+\frac {c^{3/4} \left (e+\frac {2 c d-b 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 \sqrt [4]{2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right )}+\frac {c^{3/4} \left (e-\frac {2 c d-b 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 \sqrt [4]{2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right )}+\frac {c^{3/4} \left (e+\frac {2 c d-b 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 \sqrt [4]{2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right )}+\frac {c^{3/4} \left (e-\frac {2 c d-b 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 \sqrt [4]{2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right )}\right )}{c d^2-b e d+a e^2}\)

Input:

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

Output:

(-1/4*(x*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x^4))/((b^2 - 4*a*c)*( 
a + b*x^4 + c*x^8)) + (-1/2*(c*(3*(2*c*d - b*e) + (8*b*c*d + 3*b^2*e - 28* 
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^(1/4)*c^(1/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTanh[ 
(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b - 
 Sqrt[b^2 - 4*a*c])^(3/4)))) - (c*(3*(2*c*d - b*e) - (8*b*c*d + 3*b^2*e - 
28*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^(1/4)*c^(1/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTa 
nh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(- 
b + Sqrt[b^2 - 4*a*c])^(3/4))))/2)/(4*(b^2 - 4*a*c)))/(c*d^2 - b*d*e + a*e 
^2) - (d*e*((c^(3/4)*(e + (2*c*d - b*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^(1/4)*(-b - Sqrt[b^2 - 4 
*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)) + (c^(3/4)*(e - (2*c*d - b*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^(1/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)) - (e^(7/ 
4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(2*Sqrt[2]*d^(3/4)*(c*d^2 - b* 
d*e + a*e^2)) + (e^(7/4)*ArcTan[1 + (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(2*Sqrt[ 
2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)) + (c^(3/4)*(e + (2*c*d - b*e)/Sqrt[b^2 
 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2 
*2^(1/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)) + (c^(...
 

Defintions of rubi rules used

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 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 756

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

rule 1752

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x 
_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) 
   Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   I 
nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 
, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 
 - 4*a*c] ||  !IGtQ[n/2, 0])
 

rule 1754

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
)), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), 
 x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 
 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
 

rule 1760

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p 
_), x_Symbol] :> Simp[(-x)*(d*b^2 - a*b*e - 2*a*c*d + (b*d - 2*a*e)*c*x^n)* 
((a + b*x^n + c*x^(2*n))^(p + 1)/(a*n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/ 
(a*n*(p + 1)*(b^2 - 4*a*c))   Int[Simp[(n*p + n + 1)*d*b^2 - a*b*e - 2*a*c* 
d*(2*n*p + 2*n + 1) + (2*n*p + 3*n + 1)*(d*b - 2*a*e)*c*x^n, x]*(a + b*x^n 
+ c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n 
] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]
 

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 = 3.52 (sec) , antiderivative size = 456, normalized size of antiderivative = 0.36

method	result	size
default	\(\frac {\frac {-\frac {c \left (a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}\right ) x^{5}}{4 \left (4 a c -b^{2}\right )}+\frac {\left (2 a^{2} c \,e^{3}-a \,b^{2} e^{3}-a b c d \,e^{2}+2 a \,c^{2} d^{2} e +b^{3} d \,e^{2}-2 b^{2} c \,d^{2} e +b \,c^{2} d^{3}\right ) x}{16 a c -4 b^{2}}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (-3 a b \,e^{3}+22 a c d \,e^{2}-b^{2} d \,e^{2}-9 b c \,d^{2} e +6 c^{2} d^{3}\right ) \textit {\_R}^{4}+14 a^{2} c \,e^{3}-3 a \,b^{2} e^{3}+a b c d \,e^{2}-2 a \,c^{2} d^{2} e -b^{3} d \,e^{2}+2 b^{2} c \,d^{2} e -b \,c^{2} d^{3}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a c -16 b^{2}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {e^{3} \left (\frac {d}{e}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {d}{e}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{e}}}{x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{e}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{e}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{e}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}\)	\(456\)
risch	\(\text {Expression too large to display}\)	\(65913\)

Input:

int(x^4/(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*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2* 
e-2*c^2*d^3)/(4*a*c-b^2)*x^5+1/4*(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)/(4*a*c-b^2)*x)/(c*x^8+b*x^4+a)+ 
1/16/(4*a*c-b^2)*sum((c*(-3*a*b*e^3+22*a*c*d*e^2-b^2*d*e^2-9*b*c*d^2*e+6*c 
^2*d^3)*_R^4+14*a^2*c*e^3-3*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)/(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*arct 
an(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. \[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/8*(sqrt(2)*e^(11/4)*log(sqrt(e)*x^2 + sqrt(2)*d^(1/4)*e^(1/4)*x + sqrt( 
d))/d^(3/4) - sqrt(2)*e^(11/4)*log(sqrt(e)*x^2 - sqrt(2)*d^(1/4)*e^(1/4)*x 
 + sqrt(d))/d^(3/4) + sqrt(2)*e^3*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(d)*sqrt(-sqrt(d)*sqrt(e))) + sq 
rt(2)*e^3*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(d)*sqrt(-sqrt(d)*sqrt(e))))*d/(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*((2*c^2*d - b*c*e)*x^ 
5 + (b*c*d - (b^2 - 2*a*c)*e)*x)/(((b^2*c^2 - 4*a*c^3)*d^2 - (b^3*c - 4*a* 
b*c^2)*d*e + (a*b^2*c - 4*a^2*c^2)*e^2)*x^8 + ((b^3*c - 4*a*b*c^2)*d^2 - ( 
b^4 - 4*a*b^2*c)*d*e + (a*b^3 - 4*a^2*b*c)*e^2)*x^4 + (a*b^2*c - 4*a^2*c^2 
)*d^2 - (a*b^3 - 4*a^2*b*c)*d*e + (a^2*b^2 - 4*a^3*c)*e^2) + 1/4*integrate 
((b*c^2*d^3 - (6*c^3*d^3 - 9*b*c^2*d^2*e - 3*a*b*c*e^3 - (b^2*c - 22*a*c^2 
)*d*e^2)*x^4 - 2*(b^2*c - a*c^2)*d^2*e + (b^3 - a*b*c)*d*e^2 + (3*a*b^2 - 
14*a^2*c)*e^3)/(c*x^8 + b*x^4 + a), x)/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c 
 - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4 
*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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