\(\int \frac {d+e x^4}{x^7 (a+b x^4+c x^8)} \, dx\) [57]

Optimal result

Integrand size = 25, antiderivative size = 257 \[ \int \frac {d+e x^4}{x^7 \left (a+b x^4+c x^8\right )} \, dx=-\frac {d}{6 a x^6}+\frac {b d-a e}{2 a^2 x^2}+\frac {\sqrt {c} \left (b d-a e+\frac {b^2 d-2 a c d-a b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (b^2 d-2 a c d-a b e-\sqrt {b^2-4 a c} (b d-a e)\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.53 \[ \int \frac {d+e x^4}{x^7 \left (a+b x^4+c x^8\right )} \, dx=\frac {6 b d x^4-2 a \left (d+3 e x^4\right )+3 x^6 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^2 d \log (x-\text {$\#$1})-a c d \log (x-\text {$\#$1})-a b e \log (x-\text {$\#$1})+b c d \log (x-\text {$\#$1}) \text {$\#$1}^4-a c e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}^2+2 c \text {$\#$1}^6}\&\right ]}{12 a^2 x^6} \] Input:

Integrate[(d + e*x^4)/(x^7*(a + b*x^4 + c*x^8)),x]
 

Output:

(6*b*d*x^4 - 2*a*(d + 3*e*x^4) + 3*x^6*RootSum[a + b*#1^4 + c*#1^8 & , (b^ 
2*d*Log[x - #1] - a*c*d*Log[x - #1] - a*b*e*Log[x - #1] + b*c*d*Log[x - #1 
]*#1^4 - a*c*e*Log[x - #1]*#1^4)/(b*#1^2 + 2*c*#1^6) & ])/(12*a^2*x^6)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1814, 1604, 27, 1604, 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.

\(\displaystyle \int \frac {d+e x^4}{x^7 \left (a+b x^4+c x^8\right )} \, dx\)

\(\Big \downarrow \) 1814

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

\(\Big \downarrow \) 1604

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1604

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 218

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

Input:

Int[(d + e*x^4)/(x^7*(a + b*x^4 + c*x^8)),x]
 

Output:

(-1/3*d/(a*x^6) - (-((b*d - a*e)/(a*x^2)) - ((Sqrt[c]*(b*d - a*e + (b^2*d 
- 2*a*c*d - 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 - a*e - (b^2*d - 2*a*c*d - a*b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq 
rt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a* 
c]]))/a)/a)/2
 

Defintions of rubi rules used

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 1604

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) 
/(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1))   Int[(f*x)^(m + 2)*(a + b*x^2 
 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x 
], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ 
m, -1] && 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 = 0.14 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {2 c \left (-\frac {\left (a \sqrt {-4 a c +b^{2}}\, e -b \sqrt {-4 a c +b^{2}}\, d +a b e +2 a c d -d \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (a \sqrt {-4 a c +b^{2}}\, e -b \sqrt {-4 a c +b^{2}}\, d -a b e -2 a c d +d \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a^{2}}-\frac {d}{6 a \,x^{6}}-\frac {a e -b d}{2 a^{2} x^{2}}\)	\(236\)
risch	\(\frac {-\frac {\left (a e -b d \right ) x^{4}}{2 a^{2}}-\frac {d}{6 a}}{x^{6}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{7} c^{2}-8 a^{6} b^{2} c +a^{5} b^{4}\right ) \textit {\_Z}^{4}+\left (12 a^{4} b \,c^{2} e^{2}+16 a^{4} c^{3} d e -7 a^{3} b^{3} c \,e^{2}-36 a^{3} b^{2} c^{2} d e -20 a^{3} b \,c^{3} d^{2}+a^{2} b^{5} e^{2}+16 a^{2} b^{4} c d e +25 a^{2} b^{3} c^{2} d^{2}-2 a \,b^{6} d e -9 a \,b^{5} c \,d^{2}+b^{7} d^{2}\right ) \textit {\_Z}^{2}+a^{2} c^{3} e^{4}-2 a b \,c^{3} d \,e^{3}+2 a \,c^{4} d^{2} e^{2}+b^{2} c^{3} d^{2} e^{2}-2 b \,c^{4} d^{3} e +c^{5} d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-72 a^{7} c^{2}+38 a^{6} b^{2} c -5 a^{5} b^{4}\right ) \textit {\_R}^{4}+\left (-49 a^{4} b \,c^{2} e^{2}-68 a^{4} c^{3} d e +28 a^{3} b^{3} c \,e^{2}+146 a^{3} b^{2} c^{2} d e +83 a^{3} b \,c^{3} d^{2}-4 a^{2} b^{5} e^{2}-64 a^{2} b^{4} c d e -101 a^{2} b^{3} c^{2} d^{2}+8 a \,b^{6} d e +36 a \,b^{5} c \,d^{2}-4 b^{7} d^{2}\right ) \textit {\_R}^{2}-4 a^{2} c^{3} e^{4}+8 a b \,c^{3} d \,e^{3}-8 a \,c^{4} d^{2} e^{2}-4 b^{2} c^{3} d^{2} e^{2}+8 b \,c^{4} d^{3} e -4 c^{5} d^{4}\right ) x^{2}+\left (-4 a^{6} c^{2} e +5 a^{5} b^{2} c e +8 a^{5} b \,c^{2} d -a^{4} b^{4} e -6 a^{4} b^{3} c d +a^{3} b^{5} d \right ) \textit {\_R}^{3}+\left (-a^{3} c^{3} d \,e^{2}+a^{2} b \,c^{3} d^{2} e -a^{2} c^{4} d^{3}\right ) \textit {\_R} \right )\right )}{4}\)	\(582\)

Input:

int((e*x^4+d)/x^7/(c*x^8+b*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4924 vs. \(2 (211) = 422\).

Time = 3.58 (sec) , antiderivative size = 4924, normalized size of antiderivative = 19.16 \[ \int \frac {d+e x^4}{x^7 \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:

integrate((e*x^4+d)/x^7/(c*x^8+b*x^4+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x^4}{x^7 \left (a+b x^4+c x^8\right )} \, dx=\text {Timed out} \] Input:

integrate((e*x**4+d)/x**7/(c*x**8+b*x**4+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {d+e x^4}{x^7 \left (a+b x^4+c x^8\right )} \, dx=\int { \frac {e x^{4} + d}{{\left (c x^{8} + b x^{4} + a\right )} x^{7}} \,d x } \] Input:

integrate((e*x^4+d)/x^7/(c*x^8+b*x^4+a),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4428 vs. \(2 (211) = 422\).

Time = 1.39 (sec) , antiderivative size = 4428, normalized size of antiderivative = 17.23 \[ \int \frac {d+e x^4}{x^7 \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:

integrate((e*x^4+d)/x^7/(c*x^8+b*x^4+a),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 27.55 (sec) , antiderivative size = 24353, normalized size of antiderivative = 94.76 \[ \int \frac {d+e x^4}{x^7 \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:

int((d + e*x^4)/(x^7*(a + b*x^4 + c*x^8)),x)
 

Output:

- atan(((x^2*(8*a^17*c^11*d^5 - 8*a^19*c^9*d*e^4 + 4*a^15*b^4*c^9*d^5 - 16 
*a^16*b^2*c^10*d^5 + 12*a^16*b^4*c^8*d^3*e^2 - 36*a^17*b^2*c^9*d^3*e^2 - 1 
2*a^17*b^3*c^8*d^2*e^3 + 16*a^17*b*c^10*d^4*e - 4*a^15*b^5*c^8*d^4*e + 8*a 
^16*b^3*c^9*d^4*e + 32*a^18*b*c^9*d^2*e^3 + 4*a^18*b^2*c^8*d*e^4) - (-(b^7 
*d^2 + a^2*b^5*e^2 + b^4*d^2*(-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3*d^2 - 
 7*a^3*b^3*c*e^2 + 12*a^4*b*c^2*e^2 - a^3*c*e^2*(-(4*a*c - b^2)^3)^(1/2) - 
 2*a*b^6*d*e + 25*a^2*b^3*c^2*d^2 + a^2*b^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + 
 a^2*c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*d^2 + 16*a^4*c^3*d*e - 2 
*a*b^3*d*e*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b^4*c*d*e - 3*a*b^2*c*d^2*(-( 
4*a*c - b^2)^3)^(1/2) - 36*a^3*b^2*c^2*d*e + 4*a^2*b*c*d*e*(-(4*a*c - b^2) 
^3)^(1/2))/(32*(a^5*b^4 + 16*a^7*c^2 - 8*a^6*b^2*c)))^(1/2)*((x^2*(512*a^2 
2*c^8*e^3 + 1280*a^20*b*c^9*d^3 - 512*a^21*c^9*d^2*e + 64*a^16*b^9*c^5*d^3 
 - 640*a^17*b^7*c^6*d^3 + 2176*a^18*b^5*c^7*d^3 - 2880*a^19*b^3*c^8*d^3 - 
64*a^19*b^6*c^5*e^3 + 448*a^20*b^4*c^6*e^3 - 896*a^21*b^2*c^7*e^3 - 2304*a 
^21*b*c^8*d*e^2 - 192*a^17*b^8*c^5*d^2*e + 1728*a^18*b^6*c^6*d^2*e + 192*a 
^18*b^7*c^5*d*e^2 - 4928*a^19*b^4*c^7*d^2*e - 1536*a^19*b^5*c^6*d*e^2 + 44 
80*a^20*b^2*c^8*d^2*e + 3648*a^20*b^3*c^7*d*e^2) + (-(b^7*d^2 + a^2*b^5*e^ 
2 + b^4*d^2*(-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3*d^2 - 7*a^3*b^3*c*e^2 
+ 12*a^4*b*c^2*e^2 - a^3*c*e^2*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b^6*d*e + 25 
*a^2*b^3*c^2*d^2 + a^2*b^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + a^2*c^2*d^2*(...
 

Reduce [F]

\[ \int \frac {d+e x^4}{x^7 \left (a+b x^4+c x^8\right )} \, dx=\int \frac {e \,x^{4}+d}{x^{7} \left (c \,x^{8}+b \,x^{4}+a \right )}d x \] Input:

int((e*x^4+d)/x^7/(c*x^8+b*x^4+a),x)
 

Output:

int((e*x^4+d)/x^7/(c*x^8+b*x^4+a),x)