\(\int \frac {x^5 (d+e x^4)}{a+b x^4+c x^8} \, dx\) [54]

Optimal result

Integrand size = 25, antiderivative size = 221 \[ \int \frac {x^5 \left (d+e x^4\right )}{a+b x^4+c x^8} \, dx=\frac {e x^2}{2 c}+\frac {\left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} c^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.17 \[ \int \frac {x^5 \left (d+e x^4\right )}{a+b x^4+c x^8} \, dx=\frac {2 \sqrt {c} e x^2-\frac {\sqrt {2} \left (b c d-c \sqrt {b^2-4 a c} d-b^2 e+2 a c e+b \sqrt {b^2-4 a c} e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (-b c d-c \sqrt {b^2-4 a c} d+b^2 e-2 a c e+b \sqrt {b^2-4 a c} e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 c^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1814, 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.

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

\(\Big \downarrow \) 1814

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

\(\Big \downarrow \) 1602

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

((e*x^2)/c - (-(((c*d - b*e - (b*c*d - b^2*e + 2*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[ 
c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])) - ((c*d - b*e + (b*c*d - b^2*e + 2*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[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/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 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 = 0.10 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {e \,x^{2}}{2 c}-\frac {\left (-e b \sqrt {-4 a c +b^{2}}+c d \sqrt {-4 a c +b^{2}}-2 a c e +b^{2} e -c b d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (-e b \sqrt {-4 a c +b^{2}}+c d \sqrt {-4 a c +b^{2}}+2 a c e -b^{2} e +c b d \right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\)	\(219\)
risch	\(\frac {e \,x^{2}}{2 c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{2} c^{3}-8 a \,b^{2} c^{2}+c \,b^{4}\right ) \textit {\_Z}^{4}+\left (12 a^{2} c^{2} e^{2} b -16 c^{3} d e \,a^{2}-7 a c \,e^{2} b^{3}+12 b^{2} c^{2} d e a -4 b \,c^{3} d^{2} a +e^{2} b^{5}-2 d c \,b^{4} e +b^{3} c^{2} d^{2}\right ) \textit {\_Z}^{2}+c^{2} e^{4} a^{3}-2 b \,c^{2} d \,e^{3} a^{2}+2 c^{3} d^{2} e^{2} a^{2}+b^{2} c^{2} d^{2} e^{2} a -2 b \,c^{3} d^{3} e a +c^{4} d^{4} a \right )}{\sum }\textit {\_R} \ln \left (\left (\left (4 a \,c^{3} d -b^{2} c^{2} d \right ) \textit {\_R}^{2}-a^{2} c^{2} e^{3}+a \,b^{2} c \,e^{3}-a \,c^{3} d^{2} e -b^{3} c d \,e^{2}+2 b^{2} c^{2} d^{2} e -b \,c^{3} d^{3}\right ) x^{2}+\left (-4 a b \,c^{2}+b^{3} c \right ) \textit {\_R}^{3}+\left (2 a^{2} c^{2} e^{2}-4 a \,b^{2} c \,e^{2}+6 a b \,c^{2} d e -2 a \,c^{3} d^{2}+b^{4} e^{2}-2 b^{3} c d e +b^{2} c^{2} d^{2}\right ) \textit {\_R} \right )}{4 c}\)	\(376\)

Input:

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

Output:

1/2*e*x^2/c-1/4*(-e*b*(-4*a*c+b^2)^(1/2)+c*d*(-4*a*c+b^2)^(1/2)-2*a*c*e+b^ 
2*e-c*b*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))+1/4*(-e*b*(-4*a*c 
+b^2)^(1/2)+c*d*(-4*a*c+b^2)^(1/2)+2*a*c*e-b^2*e+c*b*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. 2520 vs. \(2 (179) = 358\).

Time = 0.36 (sec) , antiderivative size = 2520, normalized size of antiderivative = 11.40 \[ \int \frac {x^5 \left (d+e x^4\right )}{a+b x^4+c x^8} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(2*e*x^2 - sqrt(1/2)*c*sqrt(-(b*c^2*d^2 - 2*(b^2*c - 2*a*c^2)*d*e + (b 
^3 - 3*a*b*c)*e^2 + (b^2*c^3 - 4*a*c^4)*sqrt((c^4*d^4 - 4*b*c^3*d^3*e + 2* 
(3*b^2*c^2 - a*c^3)*d^2*e^2 - 4*(b^3*c - a*b*c^2)*d*e^3 + (b^4 - 2*a*b^2*c 
 + a^2*c^2)*e^4)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-(c^3*d^4 
- 3*b*c^2*d^3*e + 3*b^2*c*d^2*e^2 - (b^3 + a*b*c)*d*e^3 + (a*b^2 - a^2*c)* 
e^4)*x^2 + 1/2*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*d^2*e - 2*(b^3*c - 4*a*b*c^2 
)*d*e^2 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^3 + (2*(b^2*c^4 - 4*a*c^5)*d - ( 
b^3*c^3 - 4*a*b*c^4)*e)*sqrt((c^4*d^4 - 4*b*c^3*d^3*e + 2*(3*b^2*c^2 - a*c 
^3)*d^2*e^2 - 4*(b^3*c - a*b*c^2)*d*e^3 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^4) 
/(b^2*c^6 - 4*a*c^7)))*sqrt(-(b*c^2*d^2 - 2*(b^2*c - 2*a*c^2)*d*e + (b^3 - 
 3*a*b*c)*e^2 + (b^2*c^3 - 4*a*c^4)*sqrt((c^4*d^4 - 4*b*c^3*d^3*e + 2*(3*b 
^2*c^2 - a*c^3)*d^2*e^2 - 4*(b^3*c - a*b*c^2)*d*e^3 + (b^4 - 2*a*b^2*c + a 
^2*c^2)*e^4)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))) + sqrt(1/2)*c*sqr 
t(-(b*c^2*d^2 - 2*(b^2*c - 2*a*c^2)*d*e + (b^3 - 3*a*b*c)*e^2 + (b^2*c^3 - 
 4*a*c^4)*sqrt((c^4*d^4 - 4*b*c^3*d^3*e + 2*(3*b^2*c^2 - a*c^3)*d^2*e^2 - 
4*(b^3*c - a*b*c^2)*d*e^3 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^4)/(b^2*c^6 - 4* 
a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-(c^3*d^4 - 3*b*c^2*d^3*e + 3*b^2*c*d^2* 
e^2 - (b^3 + a*b*c)*d*e^3 + (a*b^2 - a^2*c)*e^4)*x^2 - 1/2*sqrt(1/2)*((b^2 
*c^2 - 4*a*c^3)*d^2*e - 2*(b^3*c - 4*a*b*c^2)*d*e^2 + (b^4 - 5*a*b^2*c + 4 
*a^2*c^2)*e^3 + (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*sqr...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2966 vs. \(2 (179) = 358\).

Time = 1.27 (sec) , antiderivative size = 2966, normalized size of antiderivative = 13.42 \[ \int \frac {x^5 \left (d+e x^4\right )}{a+b x^4+c x^8} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*e*x^2/c + 1/8*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c - 8*sqrt 
(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt( 
b^2 - 4*a*c)*c)*b^3*c^2 - 2*b^4*c^2 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a 
*c)*c)*a^2*c^3 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + sqrt( 
2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 + 16*a*b^2*c^3 - 4*sqrt(2)*sqrt 
(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^4 - 32*a^2*c^4 + 2*(b^2 - 4*a*c)*b^2*c^2 - 
 8*(b^2 - 4*a*c)*a*c^3)*d*x^4*abs(c) - (sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a* 
c)*c)*b^5 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c - 2*sqrt(2)* 
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c - 2*b^5*c + 16*sqrt(2)*sqrt(b*c + sq 
rt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 8*sqrt(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 + 16*a*b^3*c^2 
- 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 32*a^2*b*c^3 + 2*(b^ 
2 - 4*a*c)*b^3*c - 8*(b^2 - 4*a*c)*a*b*c^2)*e*x^4*abs(c) + (2*b^3*c^4 - 8* 
a*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^ 
2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 
2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 - sqrt 
(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^4 - 2*(b^2 - 4*a 
*c)*b*c^4)*d*x^4 - (2*b^4*c^3 - 8*a*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sq 
rt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c 
 + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*...
 

Mupad [B] (verification not implemented)

Time = 27.96 (sec) , antiderivative size = 17577, normalized size of antiderivative = 79.53 \[ \int \frac {x^5 \left (d+e x^4\right )}{a+b x^4+c x^8} \, dx=\text {Too large to display} \] Input:

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

Output:

(e*x^2)/(2*c) - atan(((((16*(a*b^8*e^4 + 4*a^3*c^6*d^4 + 4*a^5*c^4*e^4 + a 
*b^4*c^4*d^4 - 8*a^2*b^6*c*e^4 - 4*a^2*b^2*c^5*d^4 + 20*a^3*b^4*c^2*e^4 - 
16*a^4*b^2*c^3*e^4 - 8*a^4*c^5*d^2*e^2 - 4*a*b^7*c*d*e^3 - 36*a^2*b^4*c^3* 
d^2*e^2 + 56*a^3*b^2*c^4*d^2*e^2 - 4*a*b^5*c^3*d^3*e - 24*a^3*b*c^5*d^3*e 
+ 24*a^4*b*c^4*d*e^3 + 6*a*b^6*c^2*d^2*e^2 + 20*a^2*b^3*c^4*d^3*e + 28*a^2 
*b^5*c^2*d*e^3 - 56*a^3*b^3*c^3*d*e^3))/c^2 + ((((16*(-(b^5*e^2 + b^3*c^2* 
d^2 - b^2*e^2*(-(4*a*c - b^2)^3)^(1/2) - c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) 
+ 12*a^2*b*c^2*e^2 - 2*b^4*c*d*e - 4*a*b*c^3*d^2 - 7*a*b^3*c*e^2 + a*c*e^2 
*(-(4*a*c - b^2)^3)^(1/2) - 16*a^2*c^3*d*e + 12*a*b^2*c^2*d*e + 2*b*c*d*e* 
(-(4*a*c - b^2)^3)^(1/2))/(32*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) 
*(256*a*b^6*c^6 - 2048*a^2*b^4*c^7 + 4096*a^3*b^2*c^8))/c^2 - (4*x^2*(256* 
a*b^5*c^6*d - 2048*a^2*b^3*c^7*d + 4096*a^3*b*c^8*d))/c^2)*(-(b^5*e^2 + b^ 
3*c^2*d^2 - b^2*e^2*(-(4*a*c - b^2)^3)^(1/2) - c^2*d^2*(-(4*a*c - b^2)^3)^ 
(1/2) + 12*a^2*b*c^2*e^2 - 2*b^4*c*d*e - 4*a*b*c^3*d^2 - 7*a*b^3*c*e^2 + a 
*c*e^2*(-(4*a*c - b^2)^3)^(1/2) - 16*a^2*c^3*d*e + 12*a*b^2*c^2*d*e + 2*b* 
c*d*e*(-(4*a*c - b^2)^3)^(1/2))/(32*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4))) 
^(1/2) + (16*(32*a*b^5*c^5*d^2 + 256*a^3*b*c^7*d^2 + 32*a*b^7*c^3*e^2 - 25 
6*a^4*b*c^6*e^2 - 192*a^2*b^3*c^6*d^2 - 256*a^2*b^5*c^4*e^2 + 576*a^3*b^3* 
c^5*e^2 - 64*a*b^6*c^4*d*e + 448*a^2*b^4*c^5*d*e - 768*a^3*b^2*c^6*d*e))/c 
^2)*(-(b^5*e^2 + b^3*c^2*d^2 - b^2*e^2*(-(4*a*c - b^2)^3)^(1/2) - c^2*d...
 

Reduce [F]

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

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

Output:

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