Integrand size = 20, antiderivative size = 636 \[ \int \frac {x^4}{a-b \left (c+d x^2\right )^4} \, dx=\frac {\left (\sqrt [4]{a}+\sqrt [4]{b} c\right )^{3/2} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}\right )}{4 a^{3/4} b^{5/8} d^{5/2}}+\frac {\left (2 \sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}\right ) \arctan \left (\frac {\sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}}\right )}{4 \sqrt {2} \sqrt {a} b^{5/8} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} d^{5/2}}-\frac {\left (2 \sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}\right ) \arctan \left (\frac {\sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}+\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}}\right )}{4 \sqrt {2} \sqrt {a} b^{5/8} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} d^{5/2}}+\frac {\left (\sqrt [4]{a}-\sqrt [4]{b} c\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{4 a^{3/4} b^{5/8} d^{5/2}}+\frac {\left (2 \sqrt [4]{b} c-\sqrt {\sqrt {a}+\sqrt {b} c^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} \sqrt {d} x}{\sqrt {\sqrt {a}+\sqrt {b} c^2}+\sqrt [4]{b} d x^2}\right )}{4 \sqrt {2} \sqrt {a} b^{5/8} \sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} d^{5/2}} \] Output:
1/4*(a^(1/4)+b^(1/4)*c)^(3/2)*arctan(b^(1/8)*d^(1/2)*x/(a^(1/4)+b^(1/4)*c) ^(1/2))/a^(3/4)/b^(5/8)/d^(5/2)+1/8*(2*b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/ 2))*arctan(((-b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2)-2^(1/2)*b^(1/8) *d^(1/2)*x)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2))*2^(1/2)/a^(1/2) /b^(5/8)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2)/d^(5/2)-1/8*(2*b^(1 /4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))*arctan(((-b^(1/4)*c+(a^(1/2)+b^(1/2)*c^ 2)^(1/2))^(1/2)+2^(1/2)*b^(1/8)*d^(1/2)*x)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2 )^(1/2))^(1/2))*2^(1/2)/a^(1/2)/b^(5/8)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^( 1/2))^(1/2)/d^(5/2)+1/4*(a^(1/4)-b^(1/4)*c)^(3/2)*arctanh(b^(1/8)*d^(1/2)* x/(a^(1/4)-b^(1/4)*c)^(1/2))/a^(3/4)/b^(5/8)/d^(5/2)+1/8*(2*b^(1/4)*c-(a^( 1/2)+b^(1/2)*c^2)^(1/2))*arctanh(2^(1/2)*b^(1/8)*(-b^(1/4)*c+(a^(1/2)+b^(1 /2)*c^2)^(1/2))^(1/2)*d^(1/2)*x/((a^(1/2)+b^(1/2)*c^2)^(1/2)+b^(1/4)*d*x^2 ))*2^(1/2)/a^(1/2)/b^(5/8)/(-b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/2)/ d^(5/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.18 \[ \int \frac {x^4}{a-b \left (c+d x^2\right )^4} \, dx=-\frac {\text {RootSum}\left [a-b c^4-4 b c^3 d \text {$\#$1}^2-6 b c^2 d^2 \text {$\#$1}^4-4 b c d^3 \text {$\#$1}^6-b d^4 \text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^3}{c^3+3 c^2 d \text {$\#$1}^2+3 c d^2 \text {$\#$1}^4+d^3 \text {$\#$1}^6}\&\right ]}{8 b d} \] Input:
Integrate[x^4/(a - b*(c + d*x^2)^4),x]
Output:
-1/8*RootSum[a - b*c^4 - 4*b*c^3*d*#1^2 - 6*b*c^2*d^2*#1^4 - 4*b*c*d^3*#1^ 6 - b*d^4*#1^8 & , (Log[x - #1]*#1^3)/(c^3 + 3*c^2*d*#1^2 + 3*c*d^2*#1^4 + d^3*#1^6) & ]/(b*d)
Leaf count is larger than twice the leaf count of optimal. \(2285\) vs. \(2(636)=1272\).
Time = 7.31 (sec) , antiderivative size = 2285, normalized size of antiderivative = 3.59, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {7291, 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.
| \(\displaystyle \int \frac {x^4}{a-b \left (c+d x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 7291 |
| \(\displaystyle \int \left (\frac {c^2}{d^2 \left (a-b \left (c+d x^2\right )^4\right )}-\frac {2 c \left (c+d x^2\right )}{d^2 \left (a-b \left (c+d x^2\right )^4\right )}+\frac {\left (c+d x^2\right )^2}{d^2 \left (a-b \left (c+d x^2\right )^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right ) c^2}{4 a^{3/4} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} d^{5/2}}-\frac {\arctan \left (\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right ) c^2}{4 \sqrt {2} \sqrt {a} \sqrt [8]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}} d^{5/2}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right ) c^2}{4 \sqrt {2} \sqrt {a} \sqrt [8]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}} d^{5/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right ) c^2}{4 a^{3/4} \sqrt [8]{b} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} d^{5/2}}-\frac {\log \left (\sqrt [4]{b} d x^2-\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right ) c^2}{8 \sqrt {2} \sqrt {a} \sqrt [8]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d^{5/2}}+\frac {\log \left (\sqrt [4]{b} d x^2+\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right ) c^2}{8 \sqrt {2} \sqrt {a} \sqrt [8]{b} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d^{5/2}}+\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right ) c}{2 \sqrt {a} b^{3/8} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} d^{5/2}}+\frac {\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right ) c}{2 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} d^{5/2}}-\frac {\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}} \arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right ) c}{2 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} d^{5/2}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right ) c}{2 \sqrt {a} b^{3/8} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} d^{5/2}}-\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \log \left (\sqrt [4]{b} d x^2-\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right ) c}{4 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} d^{5/2}}+\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \log \left (\sqrt [4]{b} d x^2+\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right ) c}{4 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} d^{5/2}}+\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right )}{4 \sqrt [4]{a} b^{5/8} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} d^{5/2}}+\frac {\arctan \left (\frac {\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right )}{4 \sqrt {2} b^{5/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}} d^{5/2}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {d} x+\sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c}}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}}}\right )}{4 \sqrt {2} b^{5/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {a}}} d^{5/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{4 \sqrt [4]{a} b^{5/8} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} d^{5/2}}+\frac {\log \left (\sqrt [4]{b} d x^2-\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right )}{8 \sqrt {2} b^{5/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d^{5/2}}-\frac {\log \left (\sqrt [4]{b} d x^2+\sqrt {2} \sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} \sqrt {d} x+\sqrt {\sqrt {b} c^2+\sqrt {a}}\right )}{8 \sqrt {2} b^{5/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {a}}-\sqrt [4]{b} c} d^{5/2}}\) |
Input:
Int[x^4/(a - b*(c + d*x^2)^4),x]
Output:
ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqrt[a^(1/4) + b^(1/4)*c]]/(4*a^(1/4)*b^(5/8)*S qrt[a^(1/4) + b^(1/4)*c]*d^(5/2)) + (c*ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqrt[a^( 1/4) + b^(1/4)*c]])/(2*Sqrt[a]*b^(3/8)*Sqrt[a^(1/4) + b^(1/4)*c]*d^(5/2)) + (c^2*ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqrt[a^(1/4) + b^(1/4)*c]])/(4*a^(3/4)*b ^(1/8)*Sqrt[a^(1/4) + b^(1/4)*c]*d^(5/2)) + ArcTan[(Sqrt[-(b^(1/4)*c) + Sq rt[Sqrt[a] + Sqrt[b]*c^2]] - Sqrt[2]*b^(1/8)*Sqrt[d]*x)/Sqrt[b^(1/4)*c + S qrt[Sqrt[a] + Sqrt[b]*c^2]]]/(4*Sqrt[2]*b^(5/8)*Sqrt[Sqrt[a] + Sqrt[b]*c^2 ]*Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]*d^(5/2)) - (c^2*ArcTan[(Sq rt[-(b^(1/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c^2]] - Sqrt[2]*b^(1/8)*Sqrt[d]*x )/Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]])/(4*Sqrt[2]*Sqrt[a]*b^(1/ 8)*Sqrt[Sqrt[a] + Sqrt[b]*c^2]*Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2 ]]*d^(5/2)) + (c*Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]*ArcTan[(Sqr t[-(b^(1/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c^2]] - Sqrt[2]*b^(1/8)*Sqrt[d]*x) /Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]])/(2*Sqrt[2]*Sqrt[a]*b^(3/8 )*Sqrt[Sqrt[a] + Sqrt[b]*c^2]*d^(5/2)) - ArcTan[(Sqrt[-(b^(1/4)*c) + Sqrt[ Sqrt[a] + Sqrt[b]*c^2]] + Sqrt[2]*b^(1/8)*Sqrt[d]*x)/Sqrt[b^(1/4)*c + Sqrt [Sqrt[a] + Sqrt[b]*c^2]]]/(4*Sqrt[2]*b^(5/8)*Sqrt[Sqrt[a] + Sqrt[b]*c^2]*S qrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]*d^(5/2)) + (c^2*ArcTan[(Sqrt[ -(b^(1/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c^2]] + Sqrt[2]*b^(1/8)*Sqrt[d]*x)/S qrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]])/(4*Sqrt[2]*Sqrt[a]*b^(1/...
Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[Polyno mialInSubst[v, u, x]/(a + b*x^n), x] /. x -> u, x] /; FreeQ[{a, b}, x] && I GtQ[n, 0] && PolynomialInQ[v, u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.17
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{8}+4 b c \,d^{3} \textit {\_Z}^{6}+6 b \,c^{2} d^{2} \textit {\_Z}^{4}+4 b \,c^{3} d \,\textit {\_Z}^{2}+b \,c^{4}-a \right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{-d^{3} \textit {\_R}^{7}-3 c \,d^{2} \textit {\_R}^{5}-3 c^{2} d \,\textit {\_R}^{3}-c^{3} \textit {\_R}}}{8 d b}\) | \(107\) |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{8}+4 b c \,d^{3} \textit {\_Z}^{6}+6 b \,c^{2} d^{2} \textit {\_Z}^{4}+4 b \,c^{3} d \,\textit {\_Z}^{2}+b \,c^{4}-a \right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{-d^{3} \textit {\_R}^{7}-3 c \,d^{2} \textit {\_R}^{5}-3 c^{2} d \,\textit {\_R}^{3}-c^{3} \textit {\_R}}}{8 d b}\) | \(107\) |
Input:
int(x^4/(a-b*(d*x^2+c)^4),x,method=_RETURNVERBOSE)
Output:
1/8/d/b*sum(_R^4/(-_R^7*d^3-3*_R^5*c*d^2-3*_R^3*c^2*d-_R*c^3)*ln(x-_R),_R= RootOf(_Z^8*b*d^4+4*_Z^6*b*c*d^3+6*_Z^4*b*c^2*d^2+4*_Z^2*b*c^3*d+b*c^4-a))
Timed out. \[ \int \frac {x^4}{a-b \left (c+d x^2\right )^4} \, dx=\text {Timed out} \] Input:
integrate(x^4/(a-b*(d*x^2+c)^4),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {x^4}{a-b \left (c+d x^2\right )^4} \, dx=\text {Timed out} \] Input:
integrate(x**4/(a-b*(d*x**2+c)**4),x)
Output:
Timed out
\[ \int \frac {x^4}{a-b \left (c+d x^2\right )^4} \, dx=\int { -\frac {x^{4}}{{\left (d x^{2} + c\right )}^{4} b - a} \,d x } \] Input:
integrate(x^4/(a-b*(d*x^2+c)^4),x, algorithm="maxima")
Output:
-integrate(x^4/((d*x^2 + c)^4*b - a), x)
Timed out. \[ \int \frac {x^4}{a-b \left (c+d x^2\right )^4} \, dx=\text {Timed out} \] Input:
integrate(x^4/(a-b*(d*x^2+c)^4),x, algorithm="giac")
Output:
Timed out
Time = 10.77 (sec) , antiderivative size = 2511, normalized size of antiderivative = 3.95 \[ \int \frac {x^4}{a-b \left (c+d x^2\right )^4} \, dx=\text {Too large to display} \] Input:
int(x^4/(a - b*(c + d*x^2)^4),x)
Output:
symsum(log(8*a*b^5*c^7*d^13 - 4*b^6*c^11*d^13 - 262144*root(16777216*a^6*b ^5*d^20*z^8 + 3145728*a^5*b^4*c*d^15*z^6 + 172032*a^4*b^3*c^2*d^10*z^4 - 8 192*a^3*b^4*c^6*d^10*z^4 + 2560*a^3*b^2*c^3*d^5*z^2 + 1536*a^2*b^3*c^7*d^5 *z^2 - 3*a*b^2*c^8 + 3*a^2*b*c^4 + b^3*c^12 - a^3, z, k)^6*a^5*b^7*d^28 - 4*a^2*b^4*c^3*d^13 - 960*root(16777216*a^6*b^5*d^20*z^8 + 3145728*a^5*b^4* c*d^15*z^6 + 172032*a^4*b^3*c^2*d^10*z^4 - 8192*a^3*b^4*c^6*d^10*z^4 + 256 0*a^3*b^2*c^3*d^5*z^2 + 1536*a^2*b^3*c^7*d^5*z^2 - 3*a*b^2*c^8 + 3*a^2*b*c ^4 + b^3*c^12 - a^3, z, k)^2*a^3*b^5*c^2*d^18 + 640*root(16777216*a^6*b^5* d^20*z^8 + 3145728*a^5*b^4*c*d^15*z^6 + 172032*a^4*b^3*c^2*d^10*z^4 - 8192 *a^3*b^4*c^6*d^10*z^4 + 2560*a^3*b^2*c^3*d^5*z^2 + 1536*a^2*b^3*c^7*d^5*z^ 2 - 3*a*b^2*c^8 + 3*a^2*b*c^4 + b^3*c^12 - a^3, z, k)^2*a^2*b^6*c^6*d^18 + 32768*root(16777216*a^6*b^5*d^20*z^8 + 3145728*a^5*b^4*c*d^15*z^6 + 17203 2*a^4*b^3*c^2*d^10*z^4 - 8192*a^3*b^4*c^6*d^10*z^4 + 2560*a^3*b^2*c^3*d^5* z^2 + 1536*a^2*b^3*c^7*d^5*z^2 - 3*a*b^2*c^8 + 3*a^2*b*c^4 + b^3*c^12 - a^ 3, z, k)^4*a^3*b^7*c^5*d^23 + 262144*root(16777216*a^6*b^5*d^20*z^8 + 3145 728*a^5*b^4*c*d^15*z^6 + 172032*a^4*b^3*c^2*d^10*z^4 - 8192*a^3*b^4*c^6*d^ 10*z^4 + 2560*a^3*b^2*c^3*d^5*z^2 + 1536*a^2*b^3*c^7*d^5*z^2 - 3*a*b^2*c^8 + 3*a^2*b*c^4 + b^3*c^12 - a^3, z, k)^6*a^4*b^8*c^4*d^28 - 8*root(1677721 6*a^6*b^5*d^20*z^8 + 3145728*a^5*b^4*c*d^15*z^6 + 172032*a^4*b^3*c^2*d^10* z^4 - 8192*a^3*b^4*c^6*d^10*z^4 + 2560*a^3*b^2*c^3*d^5*z^2 + 1536*a^2*b...
\[ \int \frac {x^4}{a-b \left (c+d x^2\right )^4} \, dx=\int \frac {x^{4}}{a -b \left (d \,x^{2}+c \right )^{4}}d x \] Input:
int(x^4/(a-b*(d*x^2+c)^4),x)
Output:
int(x^4/(a-b*(d*x^2+c)^4),x)