\(\int \frac {x^{13}}{(d+e x^4) (a+b x^4+c x^8)^2} \, dx\) [110]

Optimal result

Integrand size = 27, antiderivative size = 633 \[ \int \frac {x^{13}}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=-\frac {x^2 \left (a (b d-2 a e)+\left (b^2 d-2 a c d-a b e\right ) x^4\right )}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^4+c x^8\right )}+\frac {\left (b^3 d^2 e-a b e \left (c d^2-a e^2\right )-2 a c d \left (3 c d^2-a e^2\right )+b^2 \left (c d^3-2 a d e^2\right )-\frac {b^4 d^2 e-a b^2 e \left (3 c d^2-a e^2\right )-4 a b c d \left (2 c d^2+a e^2\right )+4 a^2 c e \left (5 c d^2+a e^2\right )+b^3 \left (c d^3-2 a d e^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (b^3 d^2 e-a b e \left (c d^2-a e^2\right )-2 a c d \left (3 c d^2-a e^2\right )+b^2 \left (c d^3-2 a d e^2\right )+\frac {b^4 d^2 e-a b^2 e \left (3 c d^2-a e^2\right )-4 a b c d \left (2 c d^2+a e^2\right )+4 a^2 c e \left (5 c d^2+a e^2\right )+b^3 \left (c d^3-2 a d e^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}-\frac {d^{5/2} \sqrt {e} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{2 \left (c d^2-b d e+a e^2\right )^2} \] Output:

-1/4*x^2*(a*(-2*a*e+b*d)+(-a*b*e-2*a*c*d+b^2*d)*x^4)/(-4*a*c+b^2)/(a*e^2-b 
*d*e+c*d^2)/(c*x^8+b*x^4+a)+1/8*(b^3*d^2*e-a*b*e*(-a*e^2+c*d^2)-2*a*c*d*(- 
a*e^2+3*c*d^2)+b^2*(-2*a*d*e^2+c*d^3)-(b^4*d^2*e-a*b^2*e*(-a*e^2+3*c*d^2)- 
4*a*b*c*d*(a*e^2+2*c*d^2)+4*a^2*c*e*(a*e^2+5*c*d^2)+b^3*(-2*a*d*e^2+c*d^3) 
)/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x^2/(b-(-4*a*c+b^2)^(1/2))^(1 
/2))*2^(1/2)/c^(1/2)/(-4*a*c+b^2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(a*e^2-b*d* 
e+c*d^2)^2+1/8*(b^3*d^2*e-a*b*e*(-a*e^2+c*d^2)-2*a*c*d*(-a*e^2+3*c*d^2)+b^ 
2*(-2*a*d*e^2+c*d^3)+(b^4*d^2*e-a*b^2*e*(-a*e^2+3*c*d^2)-4*a*b*c*d*(a*e^2+ 
2*c*d^2)+4*a^2*c*e*(a*e^2+5*c*d^2)+b^3*(-2*a*d*e^2+c*d^3))/(-4*a*c+b^2)^(1 
/2))*arctan(2^(1/2)*c^(1/2)*x^2/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/c^(1 
/2)/(-4*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(a*e^2-b*d*e+c*d^2)^2-1/2*d^ 
(5/2)*e^(1/2)*arctan(e^(1/2)*x^2/d^(1/2))/(a*e^2-b*d*e+c*d^2)^2
 

Mathematica [A] (verified)

Time = 2.98 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.15 \[ \int \frac {x^{13}}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {1}{8} \left (\frac {2 x^2 \left (-2 a^2 e+b^2 d x^4-2 a c d x^4+a b \left (d-e x^4\right )\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \left (a+b x^4+c x^8\right )}+\frac {\sqrt {2} \left (-b^4 d^2 e+b^3 \left (-c d^3+d e \left (\sqrt {b^2-4 a c} d+2 a e\right )\right )+b^2 \left (-a e^2 \left (2 \sqrt {b^2-4 a c} d+a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+3 a e\right )\right )+a b \left (8 c^2 d^3+a \sqrt {b^2-4 a c} e^3+c d e \left (-\sqrt {b^2-4 a c} d+4 a e\right )\right )-2 a c \left (a e^2 \left (-\sqrt {b^2-4 a c} d+2 a e\right )+c d^2 \left (3 \sqrt {b^2-4 a c} d+10 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2+e (-b d+a e)\right )^2}+\frac {\sqrt {2} \left (b^4 d^2 e+b^3 \left (c d^3+d e \left (\sqrt {b^2-4 a c} d-2 a e\right )\right )+b^2 \left (c d^2 \left (\sqrt {b^2-4 a c} d-3 a e\right )+a e^2 \left (-2 \sqrt {b^2-4 a c} d+a e\right )\right )-a b \left (8 c^2 d^3-a \sqrt {b^2-4 a c} e^3+c d e \left (\sqrt {b^2-4 a c} d+4 a e\right )\right )+2 a c \left (a e^2 \left (\sqrt {b^2-4 a c} d+2 a e\right )+c d^2 \left (-3 \sqrt {b^2-4 a c} d+10 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2+e (-b d+a e)\right )^2}-\frac {4 d^{5/2} \sqrt {e} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{\left (c d^2+e (-b d+a e)\right )^2}\right ) \] Input:

Integrate[x^13/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

((2*x^2*(-2*a^2*e + b^2*d*x^4 - 2*a*c*d*x^4 + a*b*(d - e*x^4)))/((b^2 - 4* 
a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + b*x^4 + c*x^8)) + (Sqrt[2]*(-(b^4*d^2 
*e) + b^3*(-(c*d^3) + d*e*(Sqrt[b^2 - 4*a*c]*d + 2*a*e)) + b^2*(-(a*e^2*(2 
*Sqrt[b^2 - 4*a*c]*d + a*e)) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 3*a*e)) + a*b* 
(8*c^2*d^3 + a*Sqrt[b^2 - 4*a*c]*e^3 + c*d*e*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a 
*e)) - 2*a*c*(a*e^2*(-(Sqrt[b^2 - 4*a*c]*d) + 2*a*e) + c*d^2*(3*Sqrt[b^2 - 
 4*a*c]*d + 10*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b - Sqrt[b^2 - 4*a 
*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 + e 
*(-(b*d) + a*e))^2) + (Sqrt[2]*(b^4*d^2*e + b^3*(c*d^3 + d*e*(Sqrt[b^2 - 4 
*a*c]*d - 2*a*e)) + b^2*(c*d^2*(Sqrt[b^2 - 4*a*c]*d - 3*a*e) + a*e^2*(-2*S 
qrt[b^2 - 4*a*c]*d + a*e)) - a*b*(8*c^2*d^3 - a*Sqrt[b^2 - 4*a*c]*e^3 + c* 
d*e*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)) + 2*a*c*(a*e^2*(Sqrt[b^2 - 4*a*c]*d + 2 
*a*e) + c*d^2*(-3*Sqrt[b^2 - 4*a*c]*d + 10*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]* 
x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + S 
qrt[b^2 - 4*a*c]]*(c*d^2 + e*(-(b*d) + a*e))^2) - (4*d^(5/2)*Sqrt[e]*ArcTa 
n[(Sqrt[e]*x^2)/Sqrt[d]])/(c*d^2 + e*(-(b*d) + a*e))^2)/8
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 611, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1814, 1650, 1598, 25, 1480, 218, 1610, 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.

\(\displaystyle \int \frac {x^{13}}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx\)

\(\Big \downarrow \) 1814

\(\displaystyle \frac {1}{2} \int \frac {x^{12}}{\left (e x^4+d\right ) \left (c x^8+b x^4+a\right )^2}dx^2\)

\(\Big \downarrow \) 1650

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

\(\Big \downarrow \) 1598

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^13/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

(-(((x^2*(a*(b*d - 2*a*e) + (b^2*d - 2*a*c*d - a*b*e)*x^4))/(2*(b^2 - 4*a* 
c)*(a + b*x^4 + c*x^8)) - (-(((b^2*d - 2*a*c*d - a*b*e - (b^3*d - a*b^2*e 
- 4*a^2*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b - Sqrt 
[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) - ((b^2*d 
- 2*a*c*d - a*b*e + (b^3*d - a*b^2*e - 4*a^2*c*e)/Sqrt[b^2 - 4*a*c])*ArcTa 
n[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqr 
t[b + Sqrt[b^2 - 4*a*c]]))/(2*(b^2 - 4*a*c)))/(c*d^2 - b*d*e + a*e^2)) + ( 
d^2*((Sqrt[c]*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c 
]*x^2)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]* 
(c*d^2 - b*d*e + a*e^2)) + (Sqrt[c]*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])* 
ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b 
 + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[d]*Sqrt[e]*ArcTan[( 
Sqrt[e]*x^2)/Sqrt[d]])/(c*d^2 - b*d*e + a*e^2)))/(c*d^2 - b*d*e + a*e^2))/ 
2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), 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^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1598

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

rule 1610

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

rule 1650

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

rule 1814

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e 
_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Sub 
st[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + b*x^(n/k) + c*x^(2*(n/k)))^ 
p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 
 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
 

rule 2009

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

Maple [A] (verified)

Time = 3.19 (sec) , antiderivative size = 807, normalized size of antiderivative = 1.27

method	result	size
default	\(\frac {\frac {-\frac {\left (a^{2} b \,e^{3}+2 a^{2} c d \,e^{2}-2 a \,b^{2} d \,e^{2}-a b c \,d^{2} e +2 a \,c^{2} d^{3}+b^{3} d^{2} e -b^{2} c \,d^{3}\right ) x^{6}}{2 \left (4 a c -b^{2}\right )}-\frac {a \left (2 a^{2} e^{3}-3 a b d \,e^{2}+2 a c \,d^{2} e +b^{2} d^{2} e -b c \,d^{3}\right ) x^{2}}{2 \left (4 a c -b^{2}\right )}}{c \,x^{8}+b \,x^{4}+a}+\frac {2 c \left (-\frac {\left (-a^{2} b \,e^{3} \sqrt {-4 a c +b^{2}}-2 a^{2} c d \,e^{2} \sqrt {-4 a c +b^{2}}+2 a \,b^{2} d \,e^{2} \sqrt {-4 a c +b^{2}}+a b c \,d^{2} e \sqrt {-4 a c +b^{2}}+6 a \,c^{2} d^{3} \sqrt {-4 a c +b^{2}}-b^{3} d^{2} e \sqrt {-4 a c +b^{2}}-b^{2} c \,d^{3} \sqrt {-4 a c +b^{2}}+4 a^{3} e^{3} c +a^{2} b^{2} e^{3}-4 a^{2} b d \,e^{2} c +20 a^{2} d^{2} e \,c^{2}-2 a \,b^{3} d \,e^{2}-3 a \,b^{2} d^{2} e c -8 a b \,c^{2} d^{3}+b^{4} d^{2} e +b^{3} c \,d^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-a^{2} b \,e^{3} \sqrt {-4 a c +b^{2}}-2 a^{2} c d \,e^{2} \sqrt {-4 a c +b^{2}}+2 a \,b^{2} d \,e^{2} \sqrt {-4 a c +b^{2}}+a b c \,d^{2} e \sqrt {-4 a c +b^{2}}+6 a \,c^{2} d^{3} \sqrt {-4 a c +b^{2}}-b^{3} d^{2} e \sqrt {-4 a c +b^{2}}-b^{2} c \,d^{3} \sqrt {-4 a c +b^{2}}-4 a^{3} e^{3} c -a^{2} b^{2} e^{3}+4 a^{2} b d \,e^{2} c -20 a^{2} d^{2} e \,c^{2}+2 a \,b^{3} d \,e^{2}+3 a \,b^{2} d^{2} e c +8 a b \,c^{2} d^{3}-b^{4} d^{2} e -b^{3} c \,d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {d^{3} e \arctan \left (\frac {e \,x^{2}}{\sqrt {d e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {d e}}\)	\(807\)
risch	\(\text {Expression too large to display}\)	\(18793\)

Input:

int(x^13/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2/(a*e^2-b*d*e+c*d^2)^2*((-1/2*(a^2*b*e^3+2*a^2*c*d*e^2-2*a*b^2*d*e^2-a* 
b*c*d^2*e+2*a*c^2*d^3+b^3*d^2*e-b^2*c*d^3)/(4*a*c-b^2)*x^6-1/2*a*(2*a^2*e^ 
3-3*a*b*d*e^2+2*a*c*d^2*e+b^2*d^2*e-b*c*d^3)/(4*a*c-b^2)*x^2)/(c*x^8+b*x^4 
+a)+2/(4*a*c-b^2)*c*(-1/8*(-a^2*b*e^3*(-4*a*c+b^2)^(1/2)-2*a^2*c*d*e^2*(-4 
*a*c+b^2)^(1/2)+2*a*b^2*d*e^2*(-4*a*c+b^2)^(1/2)+a*b*c*d^2*e*(-4*a*c+b^2)^ 
(1/2)+6*a*c^2*d^3*(-4*a*c+b^2)^(1/2)-b^3*d^2*e*(-4*a*c+b^2)^(1/2)-b^2*c*d^ 
3*(-4*a*c+b^2)^(1/2)+4*a^3*e^3*c+a^2*b^2*e^3-4*a^2*b*d*e^2*c+20*a^2*d^2*e* 
c^2-2*a*b^3*d*e^2-3*a*b^2*d^2*e*c-8*a*b*c^2*d^3+b^4*d^2*e+b^3*c*d^3)/c/(-4 
*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^2*2^ 
(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+1/8*(-a^2*b*e^3*(-4*a*c+b^2)^(1/2 
)-2*a^2*c*d*e^2*(-4*a*c+b^2)^(1/2)+2*a*b^2*d*e^2*(-4*a*c+b^2)^(1/2)+a*b*c* 
d^2*e*(-4*a*c+b^2)^(1/2)+6*a*c^2*d^3*(-4*a*c+b^2)^(1/2)-b^3*d^2*e*(-4*a*c+ 
b^2)^(1/2)-b^2*c*d^3*(-4*a*c+b^2)^(1/2)-4*a^3*e^3*c-a^2*b^2*e^3+4*a^2*b*d* 
e^2*c-20*a^2*d^2*e*c^2+2*a*b^3*d*e^2+3*a*b^2*d^2*e*c+8*a*b*c^2*d^3-b^4*d^2 
*e-b^3*c*d^3)/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2 
)*arctan(c*x^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))))-1/2*d^3*e/(a*e^ 
2-b*d*e+c*d^2)^2/(d*e)^(1/2)*arctan(e*x^2/(d*e)^(1/2))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^{13}}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x^13/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{13}}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**13/(e*x**4+d)/(c*x**8+b*x**4+a)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^{13}}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^13/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20106 vs. \(2 (581) = 1162\).

Time = 7.24 (sec) , antiderivative size = 20106, normalized size of antiderivative = 31.76 \[ \int \frac {x^{13}}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^13/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
 

Output:

-1/2*d^3*e*arctan(e*x^2/sqrt(d*e))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 
 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(d*e)) - 1/16*((2*b^5*c^5 - 20 
*a*b^3*c^6 + 48*a^2*b*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 
- 4*a*c)*c)*b^5*c^3 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 
*a*c)*c)*a*b^3*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a 
*c)*c)*b^4*c^4 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c) 
*c)*a^2*b*c^5 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)* 
c)*a*b^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b 
^3*c^5 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c 
^6 - 2*(b^2 - 4*a*c)*b^3*c^5 + 12*(b^2 - 4*a*c)*a*b*c^6)*d^7*x^4 - (2*b^6* 
c^4 - 30*a*b^4*c^5 + 88*a^2*b^2*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + 
 sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 15*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq 
rt(b^2 - 4*a*c)*c)*a*b^4*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt 
(b^2 - 4*a*c)*c)*b^5*c^3 - 44*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^ 
2 - 4*a*c)*c)*a^2*b^2*c^4 - 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b 
^2 - 4*a*c)*c)*a*b^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 
 4*a*c)*c)*b^4*c^4 + 11*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4* 
a*c)*c)*a*b^2*c^5 - 2*(b^2 - 4*a*c)*b^4*c^4 + 22*(b^2 - 4*a*c)*a*b^2*c^5)* 
d^6*e*x^4 - (2*b^7*c^3 - 12*a^2*b^3*c^5 - 80*a^3*b*c^6 - sqrt(2)*sqrt(b^2 
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^7*c + 2*sqrt(2)*sqrt(b^2 - 4...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{13}}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Hanged} \] Input:

int(x^13/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {x^{13}}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {x^{13}}{\left (e \,x^{4}+d \right ) \left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:

int(x^13/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
 

Output:

int(x^13/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)