Integrand size = 25, antiderivative size = 619 \[ \int \frac {x^8 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {x \left (a (2 c d-b e)+\left (b c d-b^2 e+2 a c e\right ) x^4\right )}{4 c \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}-\frac {\left (3 b d-10 a e+\frac {b^2 e}{c}+\frac {3 b^2 c d+4 a c^2 d+b^3 e-12 a b c e}{c \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right ) \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\left (3 b d-10 a e+\frac {b^2 e}{c}-\frac {3 b^2 c d+4 a c^2 d+b^3 e-12 a b c e}{c \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\left (3 b d-10 a e+\frac {b^2 e}{c}+\frac {3 b^2 c d+4 a c^2 d+b^3 e-12 a b c e}{c \sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right ) \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\left (3 b d-10 a e+\frac {b^2 e}{c}-\frac {3 b^2 c d+4 a c^2 d+b^3 e-12 a b c e}{c \sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}} \] Output:
1/4*x*(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)-1/16*(3*b*d-10*a*e+b^2*e/c+(-12*a*b*c*e+4*a*c^2*d+b^3*e+3*b^2*c*d)/c /(-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*(3*b*d- 10*a*e+b^2*e/c-(-12*a*b*c*e+4*a*c^2*d+b^3*e+3*b^2*c*d)/c/(-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*(3*b*d-10*a*e+b^2*e/c+(-1 2*a*b*c*e+4*a*c^2*d+b^3*e+3*b^2*c*d)/c/(-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*(3*b*d-10*a*e+b^2*e/c-(-12*a*b*c*e+4*a*c^2 *d+b^3*e+3*b^2*c*d)/c/(-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 = 0.32 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.28 \[ \int \frac {x^8 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\frac {\frac {4 x \left (-a b e+b (c d-b e) x^4+2 a c \left (d+e x^4\right )\right )}{a+b x^4+c x^8}+\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-2 a c d \log (x-\text {$\#$1})+a b e \log (x-\text {$\#$1})+3 b c d \log (x-\text {$\#$1}) \text {$\#$1}^4+b^2 e \log (x-\text {$\#$1}) \text {$\#$1}^4-10 a c e \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{16 c \left (b^2-4 a c\right )} \] Input:
Integrate[(x^8*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x]
Output:
((4*x*(-(a*b*e) + b*(c*d - b*e)*x^4 + 2*a*c*(d + e*x^4)))/(a + b*x^4 + c*x ^8) + RootSum[a + b*#1^4 + c*#1^8 & , (-2*a*c*d*Log[x - #1] + a*b*e*Log[x - #1] + 3*b*c*d*Log[x - #1]*#1^4 + b^2*e*Log[x - #1]*#1^4 - 10*a*c*e*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(16*c*(b^2 - 4*a*c))
Time = 0.92 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.77, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1822, 25, 1826, 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.
| \(\displaystyle \int \frac {x^8 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx\) |
| \(\Big \downarrow \) 1822 |
| \(\displaystyle -\frac {\int -\frac {x^4 \left ((2 c d-b e) x^4+5 (b d-2 a e)\right )}{c x^8+b x^4+a}dx}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {x^4 \left ((2 c d-b e) x^4+5 (b d-2 a e)\right )}{c x^8+b x^4+a}dx}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 1826 |
| \(\displaystyle \frac {\frac {x (2 c d-b e)}{c}-\frac {\int \frac {a (2 c d-b e)-\left (e b^2+3 c d b-10 a c e\right ) x^4}{c x^8+b x^4+a}dx}{c}}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle \frac {\frac {x (2 c d-b e)}{c}-\frac {-\frac {1}{2} \left (-\frac {-12 a b c e+4 a c^2 d+b^3 e+3 b^2 c d}{\sqrt {b^2-4 a c}}-10 a c e+b^2 e+3 b c d\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx-\frac {1}{2} \left (\frac {-12 a b c e+4 a c^2 d+b^3 e+3 b^2 c d}{\sqrt {b^2-4 a c}}-10 a c e+b^2 e+3 b c d\right ) \int \frac {1}{c x^4+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{c}}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\frac {x (2 c d-b e)}{c}-\frac {-\frac {1}{2} \left (\frac {-12 a b c e+4 a c^2 d+b^3 e+3 b^2 c d}{\sqrt {b^2-4 a c}}-10 a c e+b^2 e+3 b c d\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {-b-\sqrt {b^2-4 a c}}}dx}{\sqrt {-\sqrt {b^2-4 a c}-b}}\right )-\frac {1}{2} \left (-\frac {-12 a b c e+4 a c^2 d+b^3 e+3 b^2 c d}{\sqrt {b^2-4 a c}}-10 a c e+b^2 e+3 b c d\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x^2+\sqrt {\sqrt {b^2-4 a c}-b}}dx}{\sqrt {\sqrt {b^2-4 a c}-b}}\right )}{c}}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {x (2 c d-b e)}{c}-\frac {-\frac {1}{2} \left (\frac {-12 a b c e+4 a c^2 d+b^3 e+3 b^2 c d}{\sqrt {b^2-4 a c}}-10 a c e+b^2 e+3 b c d\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )-\frac {1}{2} \left (-\frac {-12 a b c e+4 a c^2 d+b^3 e+3 b^2 c d}{\sqrt {b^2-4 a c}}-10 a c e+b^2 e+3 b c d\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x^2}dx}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{c}}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {x (2 c d-b e)}{c}-\frac {-\frac {1}{2} \left (\frac {-12 a b c e+4 a c^2 d+b^3 e+3 b^2 c d}{\sqrt {b^2-4 a c}}-10 a c e+b^2 e+3 b c d\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )-\frac {1}{2} \left (-\frac {-12 a b c e+4 a c^2 d+b^3 e+3 b^2 c d}{\sqrt {b^2-4 a c}}-10 a c e+b^2 e+3 b c d\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{c}}{4 \left (b^2-4 a c\right )}-\frac {x^5 \left (-2 a e+x^4 (2 c d-b e)+b d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^4+c x^8\right )}\) |
Input:
Int[(x^8*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x]
Output:
-1/4*(x^5*(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)/c - (-1/2*((3*b*c*d + b^2*e - 10*a*c*e + (3*b^ 2*c*d + 4*a*c^2*d + b^3*e - 12*a*b*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)))) - ((3*b*c*d + b^2*e - 10*a*c*e - (3*b^2*c*d + 4*a*c^2*d + b^3*e - 12*a*b*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)/c)/(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]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*( m + n*(2*p + 1) + 1)) Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a* e*(m - n + 1) + (b*e*(m + n*p + 1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x] , x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && Intege rQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.27
| method | result | size |
| default | \(\frac {-\frac {\left (2 a c e -b^{2} e +c b d \right ) x^{5}}{4 c \left (4 a c -b^{2}\right )}+\frac {a \left (e b -2 c d \right ) x}{4 \left (4 a c -b^{2}\right ) c}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (10 a c e -b^{2} e -3 c b d \right ) \textit {\_R}^{4}-a b e +2 a c d \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{16 c \left (4 a c -b^{2}\right )}\) | \(167\) |
| risch | \(\frac {-\frac {\left (2 a c e -b^{2} e +c b d \right ) x^{5}}{4 c \left (4 a c -b^{2}\right )}+\frac {a \left (e b -2 c d \right ) x}{4 \left (4 a c -b^{2}\right ) c}}{c \,x^{8}+b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\frac {\left (10 a c e -b^{2} e -3 c b d \right ) \textit {\_R}^{4}}{4 a c -b^{2}}-\frac {a \left (e b -2 c d \right )}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{16 c}\) | \(180\) |
Input:
int(x^8*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
(-1/4*(2*a*c*e-b^2*e+b*c*d)/c/(4*a*c-b^2)*x^5+1/4*a*(b*e-2*c*d)/(4*a*c-b^2 )/c*x)/(c*x^8+b*x^4+a)+1/16/c/(4*a*c-b^2)*sum(((10*a*c*e-b^2*e-3*b*c*d)*_R ^4-a*b*e+2*a*c*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. 26466 vs. \(2 (537) = 1074\).
Time = 85.48 (sec) , antiderivative size = 26466, normalized size of antiderivative = 42.76 \[ \int \frac {x^8 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(x^8*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {x^8 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**8*(e*x**4+d)/(c*x**8+b*x**4+a)**2,x)
Output:
Timed out
\[ \int \frac {x^8 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\int { \frac {{\left (e x^{4} + d\right )} x^{8}}{{\left (c x^{8} + b x^{4} + a\right )}^{2}} \,d x } \] Input:
integrate(x^8*(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^5 + (2*a*c*d - a*b*e)*x)/((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) - 1/4*integrate( -((3*b*c*d + (b^2 - 10*a*c)*e)*x^4 - 2*a*c*d + a*b*e)/(c*x^8 + b*x^4 + a), x)/(b^2*c - 4*a*c^2)
\[ \int \frac {x^8 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\int { \frac {{\left (e x^{4} + d\right )} x^{8}}{{\left (c x^{8} + b x^{4} + a\right )}^{2}} \,d x } \] Input:
integrate(x^8*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x, algorithm="giac")
Output:
integrate((e*x^4 + d)*x^8/(c*x^8 + b*x^4 + a)^2, x)
Time = 29.87 (sec) , antiderivative size = 113499, normalized size of antiderivative = 183.36 \[ \int \frac {x^8 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\text {Too large to display} \] Input:
int((x^8*(d + e*x^4))/(a + b*x^4 + c*x^8)^2,x)
Output:
((x*(a*b*e - 2*a*c*d))/(4*c*(4*a*c - b^2)) - (x^5*(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(((((397*a^4*b^7*c*e^5 - 32*a^5*c^7*d^5 - 9*a^3*b^9*e^5 - 130000*a^7*b*c^4*e^5 + 5*a^2*b^10*d*e^4 + 60000*a^7*c^5*d*e^4 + 405*a^2*b^6*c^4*d^5 + 918*a^3*b^4*c^5*d^5 + 96*a^4 *b^2*c^6*d^5 - 6549*a^5*b^5*c^2*e^5 + 47800*a^6*b^3*c^3*e^5 + 1600*a^6*c^6 *d^3*e^2 + 270*a^2*b^8*c^2*d^3*e^2 - 6660*a^3*b^6*c^3*d^3*e^2 - 2450*a^3*b ^7*c^2*d^2*e^3 + 35940*a^4*b^4*c^4*d^3*e^2 + 31230*a^4*b^5*c^3*d^2*e^3 + 4 7520*a^5*b^2*c^5*d^3*e^2 - 118160*a^5*b^3*c^4*d^2*e^3 - 330*a^3*b^8*c*d*e^ 4 - 720*a^5*b*c^6*d^4*e + 540*a^2*b^7*c^3*d^4*e + 60*a^2*b^9*c*d^2*e^3 - 5 805*a^3*b^5*c^4*d^4*e - 10840*a^4*b^3*c^5*d^4*e + 7205*a^4*b^6*c^2*d*e^4 - 64610*a^5*b^4*c^3*d*e^4 - 90400*a^6*b*c^5*d^2*e^3 + 199200*a^6*b^2*c^4*d* e^4)/(64*(b^8*c + 256*a^4*c^5 - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^ 2*c^4)) - (((-(b^21*e^4 + 81*b^17*c^4*d^4 + b^6*e^4*(-(4*a*c - b^2)^15)^(1 /2) - 1184*a*b^15*c^5*d^4 - 983040*a^8*b*c^12*d^4 - 4*a*c^5*d^4*(-(4*a*c - b^2)^15)^(1/2) + 73728000*a^10*b*c^10*e^4 + 2621440*a^9*c^12*d^3*e - 6553 6000*a^10*c^11*d*e^3 + 108*b^18*c^3*d^3*e + 960*a^2*b^13*c^6*d^4 + 84480*a ^3*b^11*c^7*d^4 - 719360*a^4*b^9*c^8*d^4 + 2727936*a^5*b^7*c^9*d^4 - 52592 64*a^6*b^5*c^10*d^4 + 4587520*a^7*b^3*c^11*d^4 + 2085*a^2*b^17*c^2*e^4 - 3 6320*a^3*b^15*c^3*e^4 + 404160*a^4*b^13*c^4*e^4 - 3001344*a^5*b^11*c^5*e^4 + 15064576*a^6*b^9*c^6*e^4 - 50503680*a^7*b^7*c^7*e^4 + 108380160*a^8*...
\[ \int \frac {x^8 \left (d+e x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx=\int \frac {x^{8} \left (e \,x^{4}+d \right )}{\left (c \,x^{8}+b \,x^{4}+a \right )^{2}}d x \] Input:
int(x^8*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)
Output:
int(x^8*(e*x^4+d)/(c*x^8+b*x^4+a)^2,x)