Integrand size = 26, antiderivative size = 188 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac {a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}+\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac {3 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}} \]
1/32*x*(5*e*x^2+6*d*x+7*c)/a^2/(-b*x^4+a)+1/8*(a*f+b*x*(e*x^2+d*x+c))/a/b/ (-b*x^4+a)^2+3/16*d*arctanh(x^2*b^(1/2)/a^(1/2))/a^(5/2)/b^(1/2)+1/64*arct an(b^(1/4)*x/a^(1/4))*(-5*e*a^(1/2)+21*c*b^(1/2))/a^(11/4)/b^(3/4)+1/64*ar ctanh(b^(1/4)*x/a^(1/4))*(5*e*a^(1/2)+21*c*b^(1/2))/a^(11/4)/b^(3/4)
Time = 0.21 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.35 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\frac {\frac {4 a x (7 c+x (6 d+5 e x))}{a-b x^4}+\frac {16 a^2 (a f+b x (c+x (d+e x)))}{b \left (a-b x^4\right )^2}+\frac {2 \sqrt [4]{a} \left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}-\frac {\left (21 \sqrt [4]{a} \sqrt {b} c+12 \sqrt {a} \sqrt [4]{b} d+5 a^{3/4} e\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{b^{3/4}}+\frac {\left (21 \sqrt [4]{a} \sqrt {b} c-12 \sqrt {a} \sqrt [4]{b} d+5 a^{3/4} e\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{b^{3/4}}+\frac {12 \sqrt {a} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt {b}}}{128 a^3} \]
((4*a*x*(7*c + x*(6*d + 5*e*x)))/(a - b*x^4) + (16*a^2*(a*f + b*x*(c + x*( d + e*x))))/(b*(a - b*x^4)^2) + (2*a^(1/4)*(21*Sqrt[b]*c - 5*Sqrt[a]*e)*Ar cTan[(b^(1/4)*x)/a^(1/4)])/b^(3/4) - ((21*a^(1/4)*Sqrt[b]*c + 12*Sqrt[a]*b ^(1/4)*d + 5*a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x])/b^(3/4) + ((21*a^(1/4)*S qrt[b]*c - 12*Sqrt[a]*b^(1/4)*d + 5*a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/b ^(3/4) + (12*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(128*a^3)
Time = 0.42 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2393, 25, 2394, 25, 2415, 2009}
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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle \frac {a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}-\frac {\int -\frac {5 e x^2+6 d x+7 c}{\left (a-b x^4\right )^2}dx}{8 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 e x^2+6 d x+7 c}{\left (a-b x^4\right )^2}dx}{8 a}+\frac {a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {x \left (7 c+6 d x+5 e x^2\right )}{4 a \left (a-b x^4\right )}-\frac {\int -\frac {5 e x^2+12 d x+21 c}{a-b x^4}dx}{4 a}}{8 a}+\frac {a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {5 e x^2+12 d x+21 c}{a-b x^4}dx}{4 a}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{4 a \left (a-b x^4\right )}}{8 a}+\frac {a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {\int \left (\frac {12 d x}{a-b x^4}+\frac {5 e x^2+21 c}{a-b x^4}\right )dx}{4 a}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{4 a \left (a-b x^4\right )}}{8 a}+\frac {a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (21 \sqrt {b} c-5 \sqrt {a} e\right )}{2 a^{3/4} b^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} e+21 \sqrt {b} c\right )}{2 a^{3/4} b^{3/4}}+\frac {6 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}}{4 a}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{4 a \left (a-b x^4\right )}}{8 a}+\frac {a f+b x \left (c+d x+e x^2\right )}{8 a b \left (a-b x^4\right )^2}\) |
(a*f + b*x*(c + d*x + e*x^2))/(8*a*b*(a - b*x^4)^2) + ((x*(7*c + 6*d*x + 5 *e*x^2))/(4*a*(a - b*x^4)) + (((21*Sqrt[b]*c - 5*Sqrt[a]*e)*ArcTan[(b^(1/4 )*x)/a^(1/4)])/(2*a^(3/4)*b^(3/4)) + ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*ArcTanh [(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(3/4)) + (6*d*ArcTanh[(Sqrt[b]*x^2)/Sq rt[a]])/(Sqrt[a]*Sqrt[b]))/(4*a))/(8*a)
3.2.50.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q , x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) In t[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^( p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n , 0] && LtQ[p, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b *x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[ExpandToSum[n *(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x ] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff [Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1 }]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.50 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {-\frac {5 b e \,x^{7}}{32 a^{2}}-\frac {3 b d \,x^{6}}{16 a^{2}}-\frac {7 b c \,x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}+\frac {11 c x}{32 a}+\frac {f}{8 b}}{\left (-b \,x^{4}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (5 \textit {\_R}^{2} e +12 \textit {\_R} d +21 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b}\) | \(120\) |
| default | \(c \left (\frac {x}{8 a \left (-b \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (-b \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{128 a^{2}}}{a}\right )+d \left (\frac {x^{2}}{8 a \left (-b \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (-b \,x^{4}+a \right )}+\frac {3 \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{32 a \sqrt {a b}}}{a}\right )+e \left (\frac {x^{3}}{8 a \left (-b \,x^{4}+a \right )^{2}}+\frac {\frac {5 x^{3}}{32 a \left (-b \,x^{4}+a \right )}-\frac {5 \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{128 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a}\right )+f \left (\frac {x^{4}}{8 a \left (-b \,x^{4}+a \right )^{2}}+\frac {x^{4}}{8 a^{2} \left (-b \,x^{4}+a \right )}\right )\) | \(312\) |
(-5/32*b*e/a^2*x^7-3/16*b*d/a^2*x^6-7/32*b*c/a^2*x^5+9/32/a*e*x^3+5/16*d/a *x^2+11/32*c/a*x+1/8*f/b)/(-b*x^4+a)^2-1/128/a^2/b*sum((5*_R^2*e+12*_R*d+2 1*c)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b-a))
Result contains complex when optimal does not.
Time = 6.66 (sec) , antiderivative size = 118761, normalized size of antiderivative = 631.71 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.32 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=-\frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} - 9 \, a b e x^{3} - 10 \, a b d x^{2} - 11 \, a b c x - 4 \, a^{2} f}{32 \, {\left (a^{2} b^{3} x^{8} - 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} + \frac {\frac {12 \, d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {12 \, d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (21 \, \sqrt {b} c + 5 \, \sqrt {a} e\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{128 \, a^{2}} \]
-1/32*(5*b^2*e*x^7 + 6*b^2*d*x^6 + 7*b^2*c*x^5 - 9*a*b*e*x^3 - 10*a*b*d*x^ 2 - 11*a*b*c*x - 4*a^2*f)/(a^2*b^3*x^8 - 2*a^3*b^2*x^4 + a^4*b) + 1/128*(1 2*d*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 12*d*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*sqrt(b)) + 2*(21*sqrt(b)*c - 5*sqrt(a)*e)*arctan(sqrt(b) *x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - (21*sq rt(b)*c + 5*sqrt(a)*e)*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/a^2
Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (148) = 296\).
Time = 0.28 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.87 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=-\frac {\sqrt {2} {\left (21 \, b^{2} c - 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c + 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} + \frac {\sqrt {2} {\left (21 \, b^{2} c - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} - 9 \, a b e x^{3} - 10 \, a b d x^{2} - 11 \, a b c x - 4 \, a^{2} f}{32 \, {\left (b x^{4} - a\right )}^{2} a^{2} b} \]
-1/128*sqrt(2)*(21*b^2*c - 12*sqrt(2)*(-a*b^3)^(1/4)*b*d + 5*sqrt(-a*b)*b* e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*b^3) ^(3/4)*a^2) - 1/128*sqrt(2)*(21*b^2*c + 12*sqrt(2)*(-a*b^3)^(1/4)*b*d - 5* sqrt(-a*b)*b*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/ 4))/((-a*b^3)^(3/4)*a^2) - 1/256*sqrt(2)*(21*b^2*c - 5*sqrt(-a*b)*b*e)*log (x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a^2) + 1/256*s qrt(2)*(21*b^2*c - 5*sqrt(-a*b)*b*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sq rt(-a/b))/((-a*b^3)^(3/4)*a^2) - 1/32*(5*b^2*e*x^7 + 6*b^2*d*x^6 + 7*b^2*c *x^5 - 9*a*b*e*x^3 - 10*a*b*d*x^2 - 11*a*b*c*x - 4*a^2*f)/((b*x^4 - a)^2*a ^2*b)
Time = 9.52 (sec) , antiderivative size = 832, normalized size of antiderivative = 4.43 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx=\left (\sum _{k=1}^4\ln \left (-\frac {b\,\left (125\,a\,e^3+3024\,b\,c\,d^2-2205\,b\,c^2\,e+1728\,b\,d^3\,x+{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,c\,344064+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\,a^3\,b\,e^2\,x\,3200-2520\,b\,c\,d\,e\,x+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\,a^2\,b^2\,c^2\,x\,56448-{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,d\,x\,196608-\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\,a^3\,b\,d\,e\,15360\right )}{a^6\,32768}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\right )+\frac {\frac {f}{8\,b}+\frac {5\,d\,x^2}{16\,a}+\frac {9\,e\,x^3}{32\,a}+\frac {11\,c\,x}{32\,a}-\frac {7\,b\,c\,x^5}{32\,a^2}-\frac {3\,b\,d\,x^6}{16\,a^2}-\frac {5\,b\,e\,x^7}{32\,a^2}}{a^2-2\,a\,b\,x^4+b^2\,x^8} \]
symsum(log(-(b*(125*a*e^3 + 3024*b*c*d^2 - 2205*b*c^2*e + 1728*b*d^3*x + 3 44064*root(268435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6* b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c *d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k)^2*a^5*b^2*c + 3200*root(268435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c* e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d *e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e ^4 - 194481*b^2*c^4, z, k)*a^3*b*e^2*x - 2520*b*c*d*e*x + 56448*root(26843 5456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 27 09504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a *b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k)*a^2*b^2*c ^2*x - 196608*root(268435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718 592*a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 604 80*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481* b^2*c^4, z, k)^2*a^5*b^2*d*x - 15360*root(268435456*a^11*b^3*z^4 - 6881280 *a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153 600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k)*a^3*b*d*e))/(32768*a^6))*root(268435 456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 270 9504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050...