\(\int \frac {x^9 (d+e x^4)}{(a+b x^4+c x^8)^2} \, dx\) [95]

Optimal result

Integrand size = 25, antiderivative size = 336 \[ \int \frac {x^9 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {x^2 \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x^4\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+\frac {\left (b d-6 a e+\frac {b^2 e}{c}-\frac {b^2 c d+4 a c^2 d+b^3 e-8 a b c e}{c \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b d-6 a e+\frac {b^2 e}{c}+\frac {b^2 c d+4 a c^2 d+b^3 e-8 a b c e}{c \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.09 \[ \int \frac {x^9 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {\frac {2 \sqrt {c} x^2 \left (-a b e+b (c d-b e) x^4+2 a c \left (d+e x^4\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+\frac {\sqrt {2} \left (-b^3 e+b c \left (\sqrt {b^2-4 a c} d+8 a e\right )+b^2 \left (-c d+\sqrt {b^2-4 a c} e\right )-2 a c \left (2 c d+3 \sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (b^3 e+b c \left (\sqrt {b^2-4 a c} d-8 a e\right )+2 a c \left (2 c d-3 \sqrt {b^2-4 a c} e\right )+b^2 \left (c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{8 c^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1814, 1598, 1602, 1480, 218}

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

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

\(\Big \downarrow \) 1598

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {x^4 \left ((2 c d-b e) x^4+3 (b d-2 a e)\right )}{c x^8+b x^4+a}dx^2}{2 \left (b^2-4 a c\right )}-\frac {x^6 \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\right )\)

\(\Big \downarrow \) 1602

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

\(\Big \downarrow \) 1480

\(\displaystyle \frac {1}{2} \left (\frac {\frac {x^2 (2 c d-b e)}{c}-\frac {-\frac {1}{2} \left (-\frac {-8 a b c e+4 a c^2 d+b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}-6 a c e+b^2 e+b c d\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx^2-\frac {1}{2} \left (\frac {-8 a b c e+4 a c^2 d+b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}-6 a c e+b^2 e+b c d\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx^2}{c}}{2 \left (b^2-4 a c\right )}-\frac {x^6 \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (\frac {\frac {x^2 (2 c d-b e)}{c}-\frac {-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (-\frac {-8 a b c e+4 a c^2 d+b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}-6 a c e+b^2 e+b c d\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {-8 a b c e+4 a c^2 d+b^3 e+b^2 c d}{\sqrt {b^2-4 a c}}-6 a c e+b^2 e+b c d\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{c}}{2 \left (b^2-4 a c\right )}-\frac {x^6 \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

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 1602

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 
1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3))   Int[(f*x)^(m - 2)* 
(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p 
+ 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c 
, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | 
| IntegerQ[m])
 

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]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.08

method	result	size
default	\(\frac {-\frac {\left (2 a c e -b^{2} e +c b d \right ) x^{6}}{2 c \left (4 a c -b^{2}\right )}+\frac {a \left (e b -2 c d \right ) x^{2}}{2 \left (4 a c -b^{2}\right ) c}}{2 c \,x^{8}+2 b \,x^{4}+2 a}+\frac {-\frac {\left (6 a c e \sqrt {-4 a c +b^{2}}-b^{2} e \sqrt {-4 a c +b^{2}}-c b d \sqrt {-4 a c +b^{2}}-8 a b c e +4 a \,c^{2} d +b^{3} e +b^{2} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (6 a c e \sqrt {-4 a c +b^{2}}-b^{2} e \sqrt {-4 a c +b^{2}}-c b d \sqrt {-4 a c +b^{2}}+8 a b c e -4 a \,c^{2} d -b^{3} e -b^{2} c d \right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{4 a c -b^{2}}\)	\(364\)
risch	\(\frac {-\frac {\left (2 a c e -b^{2} e +c b d \right ) x^{6}}{4 c \left (4 a c -b^{2}\right )}+\frac {a \left (e b -2 c d \right ) x^{2}}{4 \left (4 a c -b^{2}\right ) c}}{c \,x^{8}+b \,x^{4}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4096 c^{9} a^{6}-6144 b^{2} c^{8} a^{5}+3840 b^{4} c^{7} a^{4}-1280 b^{6} c^{6} a^{3}+240 b^{8} c^{5} a^{2}-24 b^{10} c^{4} a +b^{12} c^{3}\right ) \textit {\_Z}^{4}+\left (-3840 a^{5} b \,c^{5} e^{2}+3072 c^{6} a^{5} d e +3840 a^{4} b^{3} c^{4} e^{2}-1536 a^{4} b^{2} c^{5} d e -768 b \,c^{6} a^{4} d^{2}-1504 a^{3} b^{5} c^{3} e^{2}-128 b^{4} a^{3} c^{4} d e +512 b^{3} c^{5} a^{3} d^{2}+288 a^{2} b^{7} c^{2} e^{2}+192 b^{6} a^{2} c^{3} d e -96 b^{5} c^{4} a^{2} d^{2}-27 b^{9} a c \,e^{2}-36 b^{8} a \,c^{2} d e +b^{11} e^{2}+2 b^{10} c d e +b^{9} c^{2} d^{2}\right ) \textit {\_Z}^{2}+1296 e^{4} c^{2} a^{5}-360 a^{4} b^{2} c \,e^{4}-2016 a^{4} b \,c^{2} d \,e^{3}+288 a^{4} c^{3} d^{2} e^{2}+25 a^{3} b^{4} e^{4}+496 a^{3} b^{3} c d \,e^{3}+960 a^{3} b^{2} c^{2} d^{2} e^{2}-224 a^{3} b \,c^{3} d^{3} e +16 c^{4} a^{3} d^{4}-30 a^{2} b^{5} d \,e^{3}-198 a^{2} b^{4} c \,d^{2} e^{2}-144 a^{2} b^{3} c^{2} d^{3} e +24 a^{2} b^{2} c^{3} d^{4}+9 a \,b^{6} d^{2} e^{2}+18 a \,b^{5} c \,d^{3} e +9 a \,b^{4} c^{2} d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-384 a^{4} c^{6} e +288 a^{3} b^{2} c^{5} e +192 a^{3} b \,c^{6} d -72 a^{2} b^{4} c^{4} e -144 a^{2} b^{3} c^{5} d +6 a \,b^{6} c^{3} e +36 a \,b^{5} c^{4} d -3 b^{7} c^{3} d \right ) \textit {\_R}^{2}+252 a^{3} b \,c^{2} e^{3}-72 a^{3} c^{3} d \,e^{2}-71 a^{2} b^{3} c \,e^{3}-222 a^{2} b^{2} c^{2} d \,e^{2}+84 a^{2} b \,c^{3} d^{2} e -8 a^{2} c^{4} d^{3}+5 a \,b^{5} e^{3}+54 a \,b^{4} c d \,e^{2}+39 a \,b^{3} c^{2} d^{2} e -10 a \,b^{2} c^{3} d^{3}-3 b^{6} d \,e^{2}-6 b^{5} c \,d^{2} e -3 b^{4} c^{2} d^{3}\right ) x^{2}+\left (256 c^{7} b \,a^{4}-256 c^{6} b^{3} a^{3}+96 c^{5} b^{5} a^{2}-16 c^{4} b^{7} a +b^{9} c^{3}\right ) \textit {\_R}^{3}+\left (288 a^{4} c^{4} e^{2}-368 a^{3} b^{2} c^{3} e^{2}+32 a^{3} b \,c^{4} d e -32 a^{3} c^{5} d^{2}+138 a^{2} b^{4} c^{2} e^{2}+48 a^{2} b^{3} c^{3} d e -20 a \,b^{6} c \,e^{2}-22 a \,b^{5} c^{2} d e -2 a \,b^{4} c^{3} d^{2}+b^{8} e^{2}+2 b^{7} c d e +b^{6} c^{2} d^{2}\right ) \textit {\_R} \right )\right )}{8}\)	\(998\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4420 vs. \(2 (294) = 588\).

Time = 1.88 (sec) , antiderivative size = 4420, normalized size of antiderivative = 13.15 \[ \int \frac {x^9 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

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]

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

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

Output:

1/4*((b*c*d - (b^2 - 2*a*c)*e)*x^6 + (2*a*c*d - a*b*e)*x^2)/((b^2*c^2 - 4* 
a*c^3)*x^8 + (b^3*c - 4*a*b*c^2)*x^4 + a*b^2*c - 4*a^2*c^2) - 1/2*integrat 
e(-((b*c*d + (b^2 - 6*a*c)*e)*x^4 - 2*a*c*d + a*b*e)*x/(c*x^8 + b*x^4 + a) 
, x)/(b^2*c - 4*a*c^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3779 vs. \(2 (294) = 588\).

Time = 5.08 (sec) , antiderivative size = 3779, normalized size of antiderivative = 11.25 \[ \int \frac {x^9 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(b*c*d*x^6 - b^2*e*x^6 + 2*a*c*e*x^6 + 2*a*c*d*x^2 - a*b*e*x^2)/((c*x^ 
8 + b*x^4 + a)*(b^2*c - 4*a*c^2)) + 1/16*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4 
*a*c)*c)*b^3*c - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 2*sqr 
t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - 2*b^3*c^2 + sqrt(2)*sqrt(b* 
c + sqrt(b^2 - 4*a*c)*c)*b*c^3 + 8*a*b*c^3 + 2*(b^2 - 4*a*c)*b*c^2)*d*x^4* 
abs(b^2*c - 4*a*c^2) + (sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 10*s 
qrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c + sqrt 
(b^2 - 4*a*c)*c)*b^3*c - 2*b^4*c + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c) 
*c)*a^2*c^2 + 12*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2) 
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 20*a*b^2*c^2 - 6*sqrt(2)*sqrt(b 
*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 48*a^2*c^3 + 2*(b^2 - 4*a*c)*b^2*c - 12* 
(b^2 - 4*a*c)*a*c^2)*e*x^4*abs(b^2*c - 4*a*c^2) + (2*b^4*c^4 - 8*a*b^2*c^5 
 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 4*s 
qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 + 2*sqr 
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 - sqrt(2)*s 
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^4 - 2*(b^2 - 4*a*c) 
*b^2*c^4)*d*x^4 + (2*b^5*c^3 - 20*a*b^3*c^4 + 48*a^2*b*c^5 - sqrt(2)*sqrt( 
b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c + 10*sqrt(2)*sqrt(b^2 - 
 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 2*sqrt(2)*sqrt(b^2 - 4 
*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 - 24*sqrt(2)*sqrt(b^2 - 4...
 

Mupad [B] (verification not implemented)

Time = 29.17 (sec) , antiderivative size = 34824, normalized size of antiderivative = 103.64 \[ \int \frac {x^9 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^2*(a*b*e - 2*a*c*d))/(4*c*(4*a*c - b^2)) - (x^6*(2*a*c*e - b^2*e + b*c 
*d))/(4*c*(4*a*c - b^2)))/(a + b*x^4 + c*x^8) + atan(((((a*b^12*e^4 + 64*a 
^5*c^8*d^4 + 5184*a^7*c^6*e^4 + a*b^8*c^4*d^4 - 32*a^2*b^10*c*e^4 + 4*a^2* 
b^6*c^5*d^4 + 20*a^3*b^4*c^6*d^4 + 32*a^4*b^2*c^7*d^4 + 404*a^3*b^8*c^2*e^ 
4 - 2512*a^4*b^6*c^3*e^4 + 7780*a^5*b^4*c^4*e^4 - 10656*a^6*b^2*c^5*e^4 - 
1152*a^6*c^7*d^2*e^2 + 4*a*b^11*c*d*e^3 - 84*a^2*b^8*c^3*d^2*e^2 + 264*a^3 
*b^6*c^4*d^2*e^2 + 120*a^4*b^4*c^5*d^2*e^2 + 960*a^5*b^2*c^6*d^2*e^2 + 4*a 
*b^9*c^3*d^3*e - 128*a^5*b*c^7*d^3*e + 1152*a^6*b*c^6*d*e^3 + 6*a*b^10*c^2 
*d^2*e^2 - 20*a^2*b^7*c^4*d^3*e - 92*a^2*b^9*c^2*d*e^3 - 40*a^3*b^5*c^5*d^ 
3*e + 728*a^3*b^7*c^3*d*e^3 - 256*a^4*b^3*c^6*d^3*e - 2104*a^4*b^5*c^4*d*e 
^3 + 832*a^5*b^3*c^5*d*e^3)/(256*a^4*c^6 + b^8*c^2 - 16*a*b^6*c^3 + 96*a^2 
*b^4*c^4 - 256*a^3*b^2*c^5) + ((((x^2*(294912*a^2*b^12*c^7*d - 2949120*a^3 
*b^10*c^8*d + 15728640*a^4*b^8*c^9*d - 47185920*a^5*b^6*c^10*d + 75497472* 
a^6*b^4*c^11*d - 50331648*a^7*b^2*c^12*d + 24576*a^2*b^13*c^6*e - 589824*a 
^3*b^11*c^7*e + 5898240*a^4*b^9*c^8*e - 31457280*a^5*b^7*c^9*e + 94371840* 
a^6*b^5*c^10*e - 150994944*a^7*b^3*c^11*e - 12288*a*b^14*c^6*d + 100663296 
*a^8*b*c^12*e))/(8*(1024*a^5*c^7 - b^10*c^2 + 20*a*b^8*c^3 - 160*a^2*b^6*c 
^4 + 640*a^3*b^4*c^5 - 1280*a^4*b^2*c^6)) + ((-(b^11*e^2 + b^9*c^2*d^2 + b 
^2*e^2*(-(4*a*c - b^2)^9)^(1/2) + c^2*d^2*(-(4*a*c - b^2)^9)^(1/2) - 768*a 
^4*b*c^6*d^2 - 3840*a^5*b*c^5*e^2 + 2*b^10*c*d*e - 96*a^2*b^5*c^4*d^2 +...
 

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)