\(\int \frac {x^5}{(d+e x^4) (a+b x^4+c x^8)^2} \, dx\) [112]

Optimal result

Integrand size = 27, antiderivative size = 619 \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=-\frac {x^2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x^4\right )}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^4+c x^8\right )}+\frac {\sqrt {c} \left (b^3 d e^2-b^2 e \left (7 c d^2-e \left (\sqrt {b^2-4 a c} d+a e\right )\right )-2 c \left (c d^2 \left (\sqrt {b^2-4 a c} d-2 a e\right )+a e^2 \left (5 \sqrt {b^2-4 a c} d+6 a e\right )\right )+b \left (4 c^2 d^3+a \sqrt {b^2-4 a c} e^3+c d e \left (3 \sqrt {b^2-4 a c} d+8 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}-\frac {\sqrt {c} \left (b^3 d e^2+b \left (4 c^2 d^3-a \sqrt {b^2-4 a c} e^3-c d e \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )-b^2 e \left (7 c d^2+e \left (\sqrt {b^2-4 a c} d-a e\right )\right )+2 c \left (a e^2 \left (5 \sqrt {b^2-4 a c} d-6 a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+2 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}-\frac {\sqrt {d} e^{5/2} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{2 \left (c d^2-b d e+a e^2\right )^2} \] Output:

-1/4*x^2*(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)+1/8*c^(1/2)*(b^3*d*e^2-b^2*e*(7*c*d^2-e*((-4*a*c+ 
b^2)^(1/2)*d+a*e))-2*c*(c*d^2*((-4*a*c+b^2)^(1/2)*d-2*a*e)+a*e^2*(5*(-4*a* 
c+b^2)^(1/2)*d+6*a*e))+b*(4*c^2*d^3+a*(-4*a*c+b^2)^(1/2)*e^3+c*d*e*(3*(-4* 
a*c+b^2)^(1/2)*d+8*a*e)))*arctan(2^(1/2)*c^(1/2)*x^2/(b-(-4*a*c+b^2)^(1/2) 
)^(1/2))*2^(1/2)/(-4*a*c+b^2)^(3/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(a*e^2-b* 
d*e+c*d^2)^2-1/8*c^(1/2)*(b^3*d*e^2+b*(4*c^2*d^3-a*(-4*a*c+b^2)^(1/2)*e^3- 
c*d*e*(3*(-4*a*c+b^2)^(1/2)*d-8*a*e))-b^2*e*(7*c*d^2+e*((-4*a*c+b^2)^(1/2) 
*d-a*e))+2*c*(a*e^2*(5*(-4*a*c+b^2)^(1/2)*d-6*a*e)+c*d^2*((-4*a*c+b^2)^(1/ 
2)*d+2*a*e)))*arctan(2^(1/2)*c^(1/2)*x^2/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^( 
1/2)/(-4*a*c+b^2)^(3/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(a*e^2-b*d*e+c*d^2)^2 
-1/2*d^(1/2)*e^(5/2)*arctan(e^(1/2)*x^2/d^(1/2))/(a*e^2-b*d*e+c*d^2)^2
 

Mathematica [A] (verified)

Time = 2.39 (sec) , antiderivative size = 612, normalized size of antiderivative = 0.99 \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {1}{8} \left (\frac {2 x^2 \left (-b^2 e+2 c \left (a e+c d x^4\right )+b c \left (d-e x^4\right )\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \left (a+b x^4+c x^8\right )}+\frac {\sqrt {2} \sqrt {c} \left (b^3 d e^2+b^2 e \left (-7 c d^2+e \left (\sqrt {b^2-4 a c} d+a e\right )\right )-2 c \left (c d^2 \left (\sqrt {b^2-4 a c} d-2 a e\right )+a e^2 \left (5 \sqrt {b^2-4 a c} d+6 a e\right )\right )+b \left (4 c^2 d^3+a \sqrt {b^2-4 a c} e^3+c d e \left (3 \sqrt {b^2-4 a c} d+8 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2+e (-b d+a e)\right )^2}-\frac {\sqrt {2} \sqrt {c} \left (b^3 d e^2+b^2 e \left (-7 c d^2+e \left (-\sqrt {b^2-4 a c} d+a e\right )\right )+2 c \left (a e^2 \left (5 \sqrt {b^2-4 a c} d-6 a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+2 a e\right )\right )+b \left (4 c^2 d^3-a \sqrt {b^2-4 a c} e^3+c d e \left (-3 \sqrt {b^2-4 a c} d+8 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2+e (-b d+a e)\right )^2}-\frac {4 \sqrt {d} e^{5/2} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{\left (c d^2+e (-b d+a e)\right )^2}\right ) \] Input:

Integrate[x^5/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

((2*x^2*(-(b^2*e) + 2*c*(a*e + c*d*x^4) + b*c*(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]*Sqrt[c]*(b^3*d 
*e^2 + b^2*e*(-7*c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d + a*e)) - 2*c*(c*d^2*(Sqrt 
[b^2 - 4*a*c]*d - 2*a*e) + a*e^2*(5*Sqrt[b^2 - 4*a*c]*d + 6*a*e)) + b*(4*c 
^2*d^3 + a*Sqrt[b^2 - 4*a*c]*e^3 + c*d*e*(3*Sqrt[b^2 - 4*a*c]*d + 8*a*e))) 
*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b - Sqrt[b^2 - 4*a*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 
]*Sqrt[c]*(b^3*d*e^2 + b^2*e*(-7*c*d^2 + e*(-(Sqrt[b^2 - 4*a*c]*d) + a*e)) 
 + 2*c*(a*e^2*(5*Sqrt[b^2 - 4*a*c]*d - 6*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d 
 + 2*a*e)) + b*(4*c^2*d^3 - a*Sqrt[b^2 - 4*a*c]*e^3 + c*d*e*(-3*Sqrt[b^2 - 
 4*a*c]*d + 8*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a* 
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) - (4*Sqrt[d]*e^(5/2)*ArcTan[(Sqrt[e]*x^2)/Sqrt[d]])/(c*d^2 + e* 
(-(b*d) + a*e))^2)/8
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 590, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1814, 1652, 1484, 1492, 25, 27, 1480, 218, 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.

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

\(\Big \downarrow \) 1814

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

\(\Big \downarrow \) 1652

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

\(\Big \downarrow \) 1484

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

\(\Big \downarrow \) 1492

\(\displaystyle \frac {1}{2} \left (\frac {-\frac {\int -\frac {a \left (-c (2 c d-b e) x^4+b c d+b^2 e-6 a c e\right )}{c x^8+b x^4+a}dx^2}{2 a \left (b^2-4 a c\right )}-\frac {x^2 \left (2 a c e-b^2 e+c x^4 (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \left (\frac {e^2}{\left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}+\frac {-c e x^4+c d-b e}{\left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}\right )dx^2}{a e^2-b d e+c d^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {a \left (-c (2 c d-b e) x^4+b c d+b^2 e-6 a c e\right )}{c x^8+b x^4+a}dx^2}{2 a \left (b^2-4 a c\right )}-\frac {x^2 \left (2 a c e-b^2 e+c x^4 (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \left (\frac {e^2}{\left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}+\frac {-c e x^4+c d-b e}{\left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}\right )dx^2}{a e^2-b d e+c d^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {-c (2 c d-b e) x^4+b c d+b^2 e-6 a c e}{c x^8+b x^4+a}dx^2}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (2 a c e-b^2 e+c x^4 (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \int \left (\frac {e^2}{\left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}+\frac {-c e x^4+c d-b e}{\left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}\right )dx^2}{a e^2-b d e+c d^2}\right )\)

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {\frac {-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (-\frac {-12 a c e+b^2 e+4 b c d}{\sqrt {b^2-4 a c}}-b e+2 c d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {-12 a c e+b^2 e+4 b c d}{\sqrt {b^2-4 a c}}-b e+2 c d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (2 a c e-b^2 e+c x^4 (2 c d-b e)+b c d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}-\frac {d e \left (-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{\sqrt {d} \left (a e^2-b d e+c d^2\right )}\right )}{a e^2-b d e+c d^2}\right )\)

Input:

Int[x^5/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

((-1/2*(x^2*(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)) + (-((Sqrt[c]*(2*c*d - b*e - (4*b*c*d + b^2*e - 12* 
a*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[b - Sqrt[b^2 - 4*a*c]])) - (Sqrt[c]*(2*c*d - b*e 
+ (4*b*c*d + b^2*e - 12*a*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[b + Sqrt[b^2 - 4*a*c]]))/ 
(2*(b^2 - 4*a*c)))/(c*d^2 - b*d*e + a*e^2) - (d*e*(-((Sqrt[c]*(e - (2*c*d 
- b*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]*(e + (2*c*d - b*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)) + (e^(3/2)*ArcTan[(Sqrt[e]*x^2)/Sqrt[d]])/(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 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 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 1484

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb 
ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre 
eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 
 0] && IntegerQ[q]
 

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

rule 1652

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

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 = 3.02 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {\frac {-\frac {c \left (a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}\right ) x^{6}}{2 \left (4 a c -b^{2}\right )}+\frac {\left (2 a^{2} c \,e^{3}-a \,b^{2} e^{3}-a b c d \,e^{2}+2 a \,c^{2} d^{2} e +b^{3} d \,e^{2}-2 b^{2} c \,d^{2} e +b \,c^{2} d^{3}\right ) x^{2}}{8 a c -2 b^{2}}}{c \,x^{8}+b \,x^{4}+a}+\frac {2 c \left (-\frac {\left (-a b \,e^{3} \sqrt {-4 a c +b^{2}}+10 a c d \,e^{2} \sqrt {-4 a c +b^{2}}-b^{2} d \,e^{2} \sqrt {-4 a c +b^{2}}-3 b c \,d^{2} e \sqrt {-4 a c +b^{2}}+2 c^{2} d^{3} \sqrt {-4 a c +b^{2}}+12 a^{2} c \,e^{3}-a \,b^{2} e^{3}-8 a b c d \,e^{2}-4 a \,c^{2} d^{2} e -b^{3} d \,e^{2}+7 b^{2} c \,d^{2} e -4 b \,c^{2} d^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-a b \,e^{3} \sqrt {-4 a c +b^{2}}+10 a c d \,e^{2} \sqrt {-4 a c +b^{2}}-b^{2} d \,e^{2} \sqrt {-4 a c +b^{2}}-3 b c \,d^{2} e \sqrt {-4 a c +b^{2}}+2 c^{2} d^{3} \sqrt {-4 a c +b^{2}}-12 a^{2} c \,e^{3}+a \,b^{2} e^{3}+8 a b c d \,e^{2}+4 a \,c^{2} d^{2} e +b^{3} d \,e^{2}-7 b^{2} c \,d^{2} e +4 b \,c^{2} d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}-\frac {d \,e^{3} \arctan \left (\frac {e \,x^{2}}{\sqrt {d e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {d e}}\)	\(660\)
risch	\(\text {Expression too large to display}\)	\(16937\)

Input:

int(x^5/(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*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^6+1/2*(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^2)/(c*x^8+b*x^4 
+a)+2/(4*a*c-b^2)*c*(-1/8*(-a*b*e^3*(-4*a*c+b^2)^(1/2)+10*a*c*d*e^2*(-4*a* 
c+b^2)^(1/2)-b^2*d*e^2*(-4*a*c+b^2)^(1/2)-3*b*c*d^2*e*(-4*a*c+b^2)^(1/2)+2 
*c^2*d^3*(-4*a*c+b^2)^(1/2)+12*a^2*c*e^3-a*b^2*e^3-8*a*b*c*d*e^2-4*a*c^2*d 
^2*e-b^3*d*e^2+7*b^2*c*d^2*e-4*b*c^2*d^3)/(-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*b*e^3*(-4*a*c+b^2)^(1/2)+10*a*c*d*e^2*(-4*a*c+b^2)^(1/ 
2)-b^2*d*e^2*(-4*a*c+b^2)^(1/2)-3*b*c*d^2*e*(-4*a*c+b^2)^(1/2)+2*c^2*d^3*( 
-4*a*c+b^2)^(1/2)-12*a^2*c*e^3+a*b^2*e^3+8*a*b*c*d*e^2+4*a*c^2*d^2*e+b^3*d 
*e^2-7*b^2*c*d^2*e+4*b*c^2*d^3)/(-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*e^3/(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. \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x^5/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**5/(e*x**4+d)/(c*x**8+b*x**4+a)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^5/(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. 19913 vs. \(2 (550) = 1100\).

Time = 7.40 (sec) , antiderivative size = 19913, normalized size of antiderivative = 32.17 \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
 

Output:

-1/2*d*e^3*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*(2*b^3*c^7 - 
8*a*b*c^8 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3* 
c^5 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^6 
+ 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^6 - sq 
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^7 - 2*(b^2 - 4 
*a*c)*b*c^7)*d^7*x^4 - 7*(2*b^4*c^6 - 8*a*b^2*c^7 - sqrt(2)*sqrt(b^2 - 4*a 
*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^4 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)* 
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sq 
rt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
 + sqrt(b^2 - 4*a*c)*c)*b^2*c^6 - 2*(b^2 - 4*a*c)*b^2*c^6)*d^6*e*x^4 + 7*( 
2*b^5*c^5 - 4*a*b^3*c^6 - 16*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 + 2*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 + sq 
rt(b^2 - 4*a*c)*c)*b^4*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b 
^2 - 4*a*c)*c)*a^2*b*c^5 + 4*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 - 2*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 - 4*(b^2 - 4*a*c)*a*b*c^6)*d^5*e^2* 
x^4 - (2*b^6*c^4 + 54*a*b^4*c^5 - 248*a^2*b^2*c^6 - sqrt(2)*sqrt(b^2 - ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Hanged} \] Input:

int(x^5/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {x^{5}}{\left (e \,x^{4}+d \right ) \left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:

int(x^5/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
 

Output:

int(x^5/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)