\(\int \frac {x^9}{(d+e x^4) (a+b x^4+c x^8)^2} \, dx\) [111]

Optimal result

Integrand size = 27, antiderivative size = 531 \[ \int \frac {x^9}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {x^2 \left (a (2 c d-b e)+c (b d-2 a e) x^4\right )}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^4+c x^8\right )}-\frac {\sqrt {c} \left (3 b^2 d^2 e-2 a e \left (3 c d^2-a e^2\right )-b \left (c d^3+3 a d e^2\right )-\frac {3 b^3 d^2 e+4 a^2 b e^3-4 a c d \left (c d^2-3 a e^2\right )-b^2 \left (c d^3+9 a d e^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}+\frac {\sqrt {c} \left (b c d^3-3 b^2 d^2 e+6 a c d^2 e+3 a b d e^2-2 a^2 e^3-\frac {3 b^3 d^2 e+4 a^2 b e^3-4 a c d \left (c d^2-3 a e^2\right )-b^2 \left (c d^3+9 a d e^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}+\frac {d^{3/2} e^{3/2} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{2 \left (c d^2-b d e+a e^2\right )^2} \] Output:

1/4*x^2*(a*(-b*e+2*c*d)+c*(-2*a*e+b*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)*(3*b^2*d^2*e-2*a*e*(-a*e^2+3*c*d^2)-b*(3*a* 
d*e^2+c*d^3)-(3*b^3*d^2*e+4*a^2*b*e^3-4*a*c*d*(-3*a*e^2+c*d^2)-b^2*(9*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)/(-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*c^(1/2)*(b*c*d^3-3*b^2*d^2*e+6*a*c*d^2*e+3*a*b*d*e^2-2*a^ 
2*e^3-(3*b^3*d^2*e+4*a^2*b*e^3-4*a*c*d*(-3*a*e^2+c*d^2)-b^2*(9*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)/(-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^(3/2)*e^(3/2)*arctan(e^(1/2)*x^2/d^(1/2))/(a*e^2-b*d*e+c*d^2)^ 
2
 

Mathematica [A] (verified)

Time = 2.03 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.12 \[ \int \frac {x^9}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\frac {1}{8} \left (\frac {-2 b c d x^6+2 a x^2 \left (-2 c d+b e+2 c e x^4\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \left (a+b x^4+c x^8\right )}-\frac {\sqrt {2} \sqrt {c} \left (-3 b^3 d^2 e+2 a \left (2 c^2 d^3+a \sqrt {b^2-4 a c} e^3-3 c d e \left (\sqrt {b^2-4 a c} d+2 a e\right )\right )+b^2 \left (c d^3+3 d e \left (\sqrt {b^2-4 a c} d+3 a e\right )\right )-b \left (c \sqrt {b^2-4 a c} d^3+a e^2 \left (3 \sqrt {b^2-4 a c} d+4 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2+e (-b d+a e)\right )^2}+\frac {\sqrt {2} \sqrt {c} \left (4 a c^2 d^3+b c \sqrt {b^2-4 a c} d^3-3 b^3 d^2 e-2 a^2 \sqrt {b^2-4 a c} e^3+a b e^2 \left (3 \sqrt {b^2-4 a c} d-4 a e\right )+6 a c d e \left (\sqrt {b^2-4 a c} d-2 a e\right )+b^2 d \left (c d^2-3 \sqrt {b^2-4 a c} d e+9 a e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2+e (-b d+a e)\right )^2}+\frac {4 d^{3/2} e^{3/2} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{\left (c d^2+e (-b d+a e)\right )^2}\right ) \] Input:

Integrate[x^9/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

((-2*b*c*d*x^6 + 2*a*x^2*(-2*c*d + b*e + 2*c*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]*(-3*b^3*d^2*e 
 + 2*a*(2*c^2*d^3 + a*Sqrt[b^2 - 4*a*c]*e^3 - 3*c*d*e*(Sqrt[b^2 - 4*a*c]*d 
 + 2*a*e)) + b^2*(c*d^3 + 3*d*e*(Sqrt[b^2 - 4*a*c]*d + 3*a*e)) - b*(c*Sqrt 
[b^2 - 4*a*c]*d^3 + a*e^2*(3*Sqrt[b^2 - 4*a*c]*d + 4*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]*(4*a* 
c^2*d^3 + b*c*Sqrt[b^2 - 4*a*c]*d^3 - 3*b^3*d^2*e - 2*a^2*Sqrt[b^2 - 4*a*c 
]*e^3 + a*b*e^2*(3*Sqrt[b^2 - 4*a*c]*d - 4*a*e) + 6*a*c*d*e*(Sqrt[b^2 - 4* 
a*c]*d - 2*a*e) + b^2*d*(c*d^2 - 3*Sqrt[b^2 - 4*a*c]*d*e + 9*a*e^2))*ArcTa 
n[(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*d^(3/2)*e^ 
(3/2)*ArcTan[(Sqrt[e]*x^2)/Sqrt[d]])/(c*d^2 + e*(-(b*d) + a*e))^2)/8
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1814, 1650, 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.

\(\displaystyle \int \frac {x^9}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx\)

\(\Big \downarrow \) 1814

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

\(\Big \downarrow \) 1650

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

\(\Big \downarrow \) 1484

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

\(\Big \downarrow \) 1492

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

\(\displaystyle \frac {1}{2} \left (\frac {d^2 \int \left (\frac {e^2}{\left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}+\frac {-c e x^4+c d-b e}{\left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}\right )dx^2}{a e^2-b d e+c d^2}-\frac {\frac {-\frac {1}{2} c \left (-\frac {-4 a b e-12 a c d+5 b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx^2-\frac {1}{2} c \left (\frac {-4 a b e-12 a c d+5 b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx^2}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (c x^4 (b d-2 a e)+a (2 c d-b e)\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (\frac {d^2 \int \left (\frac {e^2}{\left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}+\frac {-c e x^4+c d-b e}{\left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}\right )dx^2}{a e^2-b d e+c d^2}-\frac {\frac {-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (-\frac {-4 a b e-12 a c d+5 b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {-4 a b e-12 a c d+5 b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (c x^4 (b d-2 a e)+a (2 c d-b e)\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {d^2 \left (-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{\sqrt {d} \left (a e^2-b d e+c d^2\right )}\right )}{a e^2-b d e+c d^2}-\frac {\frac {-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (-\frac {-4 a b e-12 a c d+5 b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {-4 a b e-12 a c d+5 b^2 d}{\sqrt {b^2-4 a c}}-2 a e+b d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 \left (b^2-4 a c\right )}-\frac {x^2 \left (c x^4 (b d-2 a e)+a (2 c d-b e)\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}}{a e^2-b d e+c d^2}\right )\)

Input:

Int[x^9/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x]
 

Output:

(-((-1/2*(x^2*(a*(2*c*d - b*e) + c*(b*d - 2*a*e)*x^4))/((b^2 - 4*a*c)*(a + 
 b*x^4 + c*x^8)) + (-((Sqrt[c]*(b*d - 2*a*e - (5*b^2*d - 12*a*c*d - 4*a*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]])) - (Sqrt[c]*(b*d - 2*a*e + (5 
*b^2*d - 12*a*c*d - 4*a*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]]))/(2 
*(b^2 - 4*a*c)))/(c*d^2 - b*d*e + a*e^2)) + (d^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))) - 
 (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 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 = 3.08 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.17

method	result	size
default	\(-\frac {\frac {-\frac {c \left (2 a^{2} e^{3}-3 a b d \,e^{2}+2 a c \,d^{2} e +b^{2} d^{2} e -b c \,d^{3}\right ) x^{6}}{2 \left (4 a c -b^{2}\right )}-\frac {a \left (a b \,e^{3}-2 a c d \,e^{2}-b^{2} d \,e^{2}+3 b c \,d^{2} e -2 c^{2} d^{3}\right ) x^{2}}{2 \left (4 a c -b^{2}\right )}}{c \,x^{8}+b \,x^{4}+a}+\frac {2 c \left (-\frac {\left (-2 a^{2} e^{3} \sqrt {-4 a c +b^{2}}+3 a b d \,e^{2} \sqrt {-4 a c +b^{2}}+6 a c \,d^{2} e \sqrt {-4 a c +b^{2}}-3 b^{2} d^{2} e \sqrt {-4 a c +b^{2}}+b c \,d^{3} \sqrt {-4 a c +b^{2}}+4 a^{2} b \,e^{3}+12 a^{2} c d \,e^{2}-9 a \,b^{2} d \,e^{2}-4 a \,c^{2} d^{3}+3 b^{3} d^{2} e -b^{2} c \,d^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-2 a^{2} e^{3} \sqrt {-4 a c +b^{2}}+3 a b d \,e^{2} \sqrt {-4 a c +b^{2}}+6 a c \,d^{2} e \sqrt {-4 a c +b^{2}}-3 b^{2} d^{2} e \sqrt {-4 a c +b^{2}}+b c \,d^{3} \sqrt {-4 a c +b^{2}}-4 a^{2} b \,e^{3}-12 a^{2} c d \,e^{2}+9 a \,b^{2} d \,e^{2}+4 a \,c^{2} d^{3}-3 b^{3} d^{2} e +b^{2} c \,d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {d^{2} e^{2} \arctan \left (\frac {e \,x^{2}}{\sqrt {d e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {d e}}\)	\(620\)
risch	\(\text {Expression too large to display}\)	\(17247\)

Input:

int(x^9/(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*(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^6-1/2*a*(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^2)/(c*x^8+b*x^4+a)+2/(4*a*c-b^2)*c*(-1/8*( 
-2*a^2*e^3*(-4*a*c+b^2)^(1/2)+3*a*b*d*e^2*(-4*a*c+b^2)^(1/2)+6*a*c*d^2*e*( 
-4*a*c+b^2)^(1/2)-3*b^2*d^2*e*(-4*a*c+b^2)^(1/2)+b*c*d^3*(-4*a*c+b^2)^(1/2 
)+4*a^2*b*e^3+12*a^2*c*d*e^2-9*a*b^2*d*e^2-4*a*c^2*d^3+3*b^3*d^2*e-b^2*c*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*(-2*a^2*e^3*(-4*a*c+b^ 
2)^(1/2)+3*a*b*d*e^2*(-4*a*c+b^2)^(1/2)+6*a*c*d^2*e*(-4*a*c+b^2)^(1/2)-3*b 
^2*d^2*e*(-4*a*c+b^2)^(1/2)+b*c*d^3*(-4*a*c+b^2)^(1/2)-4*a^2*b*e^3-12*a^2* 
c*d*e^2+9*a*b^2*d*e^2+4*a*c^2*d^3-3*b^3*d^2*e+b^2*c*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^2*e^2/(a*e^2-b*d*e+c*d^2)^2/(d*e)^(1/2)*ar 
ctan(e*x^2/(d*e)^(1/2))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^9}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x^9/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^9}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**9/(e*x**4+d)/(c*x**8+b*x**4+a)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^9}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^9/(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. 18563 vs. \(2 (479) = 958\).

Time = 6.98 (sec) , antiderivative size = 18563, normalized size of antiderivative = 34.96 \[ \int \frac {x^9}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^9/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
 

Output:

1/2*d^2*e^2*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^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)*sqrt(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^7*x^4 - (10*b^5*c^5 - 52*a*b^3*c^6 + 48*a^2*b*c^7 - 
5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^3 + 26*s 
qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^4 + 10*sq 
rt(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 - 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 - 10*(b^2 - 4*a*c)*b 
^3*c^5 + 12*(b^2 - 4*a*c)*a*b*c^6)*d^6*e*x^4 + 7*(2*b^6*c^4 - 10*a*b^4*c^5 
 + 8*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 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(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 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2 
*c^4 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^9}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\text {Hanged} \] Input:

int(x^9/((d + e*x^4)*(a + b*x^4 + c*x^8)^2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {x^9}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {x^{9}}{\left (e \,x^{4}+d \right ) \left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:

int(x^9/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
 

Output:

int(x^9/(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)