Integrand size = 25, antiderivative size = 525 \[ \int \frac {x^4 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=-\frac {x \left (b d-2 a e+(2 c d-b e) x^4\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}+\frac {\left (6 c d-3 b e+\frac {8 b c d-3 b^2 e-4 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right ) \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}+\frac {\left (6 c d-3 b e-\frac {8 b c d-3 b^2 e-4 a c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}+\frac {\left (6 c d-3 b e+\frac {8 b c d-3 b^2 e-4 a c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right ) \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}+\frac {\left (6 c d-3 b e-\frac {8 b c d-3 b^2 e-4 a c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}} \] Output:
-1/4*x*(b*d-2*a*e+(-b*e+2*c*d)*x^4)/(-4*a*c+b^2)/(c*x^8+b*x^4+a)+1/16*(6*c *d-3*b*e+(-4*a*c*e-3*b^2*e+8*b*c*d)/(-4*a*c+b^2)^(1/2))*arctan(2^(1/4)*c^( 1/4)*x/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/c^(1/4)/(-4*a*c+b^2)/(-b-(-4 *a*c+b^2)^(1/2))^(3/4)+1/16*(6*c*d-3*b*e-(-4*a*c*e-3*b^2*e+8*b*c*d)/(-4*a* c+b^2)^(1/2))*arctan(2^(1/4)*c^(1/4)*x/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*2^(3 /4)/c^(1/4)/(-4*a*c+b^2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)+1/16*(6*c*d-3*b*e+( -4*a*c*e-3*b^2*e+8*b*c*d)/(-4*a*c+b^2)^(1/2))*arctanh(2^(1/4)*c^(1/4)*x/(- b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/c^(1/4)/(-4*a*c+b^2)/(-b-(-4*a*c+b^2) ^(1/2))^(3/4)+1/16*(6*c*d-3*b*e-(-4*a*c*e-3*b^2*e+8*b*c*d)/(-4*a*c+b^2)^(1 /2))*arctanh(2^(1/4)*c^(1/4)*x/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/c^(1 /4)/(-4*a*c+b^2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.24 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.27 \[ \int \frac {x^4 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {\frac {4 \left (-b d x+2 a e x-2 c d x^5+b e x^5\right )}{a+b x^4+c x^8}+\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b d \log (x-\text {$\#$1})-2 a e \log (x-\text {$\#$1})-6 c d \log (x-\text {$\#$1}) \text {$\#$1}^4+3 b e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{16 \left (b^2-4 a c\right )} \] Input:
Integrate[(x^4*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x]
Output:
((4*(-(b*d*x) + 2*a*e*x - 2*c*d*x^5 + b*e*x^5))/(a + b*x^4 + c*x^8) + Root Sum[a + b*#1^4 + c*#1^8 & , (b*d*Log[x - #1] - 2*a*e*Log[x - #1] - 6*c*d*L og[x - #1]*#1^4 + 3*b*e*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(16*(b^ 2 - 4*a*c))
Time = 0.69 (sec) , antiderivative size = 423, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1822, 25, 1752, 756, 218, 221}
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.
| \(\displaystyle \int \frac {x^4 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx\) |
| \(\Big \downarrow \) 1822 |
| \(\displaystyle -\frac {\int -\frac {-3 (2 c d-b e) x^4+b d-2 a e}{c x^8+b x^4+a}dx}{4 \left (b^2-4 a c\right )}-\frac {x \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-3 (2 c d-b e) x^4+b d-2 a e}{c x^8+b x^4+a}dx}{4 \left (b^2-4 a c\right )}-\frac {x \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle \frac {-\frac {1}{2} \left (-\frac {-4 a c e-3 b^2 e+8 b c d}{\sqrt {b^2-4 a c}}-3 b e+6 c d\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx-\frac {1}{2} \left (\frac {-4 a c e-3 b^2 e+8 b c d}{\sqrt {b^2-4 a c}}-3 b e+6 c d\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{4 \left (b^2-4 a c\right )}-\frac {x \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {-\frac {1}{2} \left (\frac {-4 a c e-3 b^2 e+8 b c d}{\sqrt {b^2-4 a c}}-3 b e+6 c d\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {-b-\sqrt {b^2-4 a c}}}dx}{\sqrt {-\sqrt {b^2-4 a c}-b}}\right )-\frac {1}{2} \left (-\frac {-4 a c e-3 b^2 e+8 b c d}{\sqrt {b^2-4 a c}}-3 b e+6 c d\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {\sqrt {b^2-4 a c}-b}}dx}{\sqrt {\sqrt {b^2-4 a c}-b}}\right )}{4 \left (b^2-4 a c\right )}-\frac {x \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {1}{2} \left (\frac {-4 a c e-3 b^2 e+8 b c d}{\sqrt {b^2-4 a c}}-3 b e+6 c d\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )-\frac {1}{2} \left (-\frac {-4 a c e-3 b^2 e+8 b c d}{\sqrt {b^2-4 a c}}-3 b e+6 c d\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{4 \left (b^2-4 a c\right )}-\frac {x \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {1}{2} \left (\frac {-4 a c e-3 b^2 e+8 b c d}{\sqrt {b^2-4 a c}}-3 b e+6 c d\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )-\frac {1}{2} \left (-\frac {-4 a c e-3 b^2 e+8 b c d}{\sqrt {b^2-4 a c}}-3 b e+6 c d\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{4 \left (b^2-4 a c\right )}-\frac {x \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
Input:
Int[(x^4*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x]
Output:
-1/4*(x*(b*d - 2*a*e + (2*c*d - b*e)*x^4))/((b^2 - 4*a*c)*(a + b*x^4 + c*x ^8)) + (-1/2*((6*c*d - 3*b*e + (8*b*c*d - 3*b^2*e - 4*a*c*e)/Sqrt[b^2 - 4* a*c])*(-(ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/ 4)*c^(1/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTanh[(2^(1/4)*c^(1/4)*x)/ (-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b - Sqrt[b^2 - 4*a*c])^ (3/4)))) - ((6*c*d - 3*b*e - (8*b*c*d - 3*b^2*e - 4*a*c*e)/Sqrt[b^2 - 4*a* c])*(-(ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4) *c^(1/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTanh[(2^(1/4)*c^(1/4)*x)/(- b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b + Sqrt[b^2 - 4*a*c])^(3 /4))))/2)/(4*(b^2 - 4*a*c))
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]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), 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^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Simp[f^(n - 1)*(f*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)*((b*d - 2*a*e - (b*e - 2*c*d)*x^n)/(n*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[f^n/(n*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^(m - n)* (a + b*x^n + c*x^(2*n))^(p + 1)*Simp[(n - m - 1)*(b*d - 2*a*e) + (2*n*p + 2 *n + m + 1)*(b*e - 2*c*d)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] & & EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n - 1] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.27
| method | result | size |
| default | \(\frac {-\frac {\left (e b -2 c d \right ) x^{5}}{4 \left (4 a c -b^{2}\right )}-\frac {\left (2 a e -b d \right ) x}{4 \left (4 a c -b^{2}\right )}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (3 \left (-e b +2 c d \right ) \textit {\_R}^{4}+2 a e -b d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a c -16 b^{2}}\) | \(141\) |
| risch | \(\frac {-\frac {\left (e b -2 c d \right ) x^{5}}{4 \left (4 a c -b^{2}\right )}-\frac {\left (2 a e -b d \right ) x}{4 \left (4 a c -b^{2}\right )}}{c \,x^{8}+b \,x^{4}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-\frac {3 \left (e b -2 c d \right ) \textit {\_R}^{4}}{4 a c -b^{2}}+\frac {2 a e -b d}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{16}\) | \(154\) |
Input:
int(x^4*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
(-1/4*(b*e-2*c*d)/(4*a*c-b^2)*x^5-1/4*(2*a*e-b*d)/(4*a*c-b^2)*x)/(c*x^8+b* x^4+a)+1/16/(4*a*c-b^2)*sum((3*(-b*e+2*c*d)*_R^4+2*a*e-b*d)/(2*_R^7*c+_R^3 *b)*ln(x-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 23255 vs. \(2 (443) = 886\).
Time = 57.92 (sec) , antiderivative size = 23255, normalized size of antiderivative = 44.30 \[ \int \frac {x^4 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(x^4*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {x^4 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**4*(e*x**4+d)/(c*x**8+b*x**4+a)**2,x)
Output:
Timed out
\[ \int \frac {x^4 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\int { \frac {{\left (e x^{4} + d\right )} x^{4}}{{\left (c x^{8} + b x^{4} + a\right )}^{2}} \,d x } \] Input:
integrate(x^4*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="maxima")
Output:
-1/4*((2*c*d - b*e)*x^5 + (b*d - 2*a*e)*x)/((b^2*c - 4*a*c^2)*x^8 + (b^3 - 4*a*b*c)*x^4 + a*b^2 - 4*a^2*c) + 1/4*integrate(-(3*(2*c*d - b*e)*x^4 - b *d + 2*a*e)/(c*x^8 + b*x^4 + a), x)/(b^2 - 4*a*c)
Timed out. \[ \int \frac {x^4 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^4*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
Output:
Timed out
Time = 28.34 (sec) , antiderivative size = 99213, normalized size of antiderivative = 188.98 \[ \int \frac {x^4 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:
int((x^4*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x)
Output:
atan(((((((-(b^19*c*d^4 + 81*a^3*b^17*e^4 - 3*a*b^17*c^2*d^4 + 12386304*a^ 9*b*c^10*d^4 - 1184*a^4*b^15*c*e^4 - 983040*a^11*b*c^8*e^4 + 4*a^4*c*e^4*( -(4*a*c - b^2)^15)^(1/2) + b^4*c*d^4*(-(4*a*c - b^2)^15)^(1/2) - 14155776* a^10*c^10*d^3*e + 1572864*a^11*c^9*d*e^3 - 96*a^2*b^15*c^3*d^4 + 2752*a^3* b^13*c^4*d^4 - 55296*a^4*b^11*c^5*d^4 + 585216*a^5*b^9*c^6*d^4 - 3350528*a ^6*b^7*c^7*d^4 + 10665984*a^7*b^5*c^8*d^4 - 17891328*a^8*b^3*c^9*d^4 + 324 *a^2*c^3*d^4*(-(4*a*c - b^2)^15)^(1/2) - 81*a^3*b^2*e^4*(-(4*a*c - b^2)^15 )^(1/2) + 960*a^5*b^13*c^2*e^4 + 84480*a^6*b^11*c^3*e^4 - 719360*a^7*b^9*c ^4*e^4 + 2727936*a^8*b^7*c^5*e^4 - 5259264*a^9*b^5*c^6*e^4 + 4587520*a^10* b^3*c^7*e^4 - 20*a*b^18*c*d^3*e + 576*a^3*b^15*c^2*d^2*e^2 - 58752*a^4*b^1 3*c^3*d^2*e^2 + 678912*a^5*b^11*c^4*d^2*e^2 - 3456000*a^6*b^9*c^5*d^2*e^2 + 8110080*a^7*b^7*c^6*d^2*e^2 - 4030464*a^8*b^5*c^7*d^2*e^2 - 16515072*a^9 *b^3*c^8*d^2*e^2 - 216*a^3*c^2*d^2*e^2*(-(4*a*c - b^2)^15)^(1/2) - 1256*a^ 3*b^16*c*d*e^3 + 27*a*b^2*c^2*d^4*(-(4*a*c - b^2)^15)^(1/2) + 168*a^2*b^16 *c^2*d^3*e + 150*a^2*b^17*c*d^2*e^2 - 3072*a^3*b^14*c^3*d^3*e + 85504*a^4* b^12*c^4*d^3*e + 26880*a^4*b^14*c^2*d*e^3 - 976896*a^5*b^10*c^5*d^3*e - 22 2720*a^5*b^12*c^3*d*e^3 + 5468160*a^6*b^8*c^6*d^3*e + 815104*a^6*b^10*c^4* d*e^3 - 15859712*a^7*b^6*c^7*d^3*e - 552960*a^7*b^8*c^5*d*e^3 + 20840448*a ^8*b^4*c^8*d^3*e - 5308416*a^8*b^6*c^6*d*e^3 - 2359296*a^9*b^2*c^9*d^3*e + 16384000*a^9*b^4*c^7*d*e^3 + 22413312*a^10*b*c^9*d^2*e^2 - 15728640*a^...
\[ \int \frac {x^4 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {x^{4} \left (e \,x^{4}+d \right )}{\left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:
int(x^4*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
Output:
int(x^4*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)