Integrand size = 20, antiderivative size = 550 \[ \int \frac {x^2}{a-b \left (c+d x^2\right )^4} \, dx=-\frac {\sqrt {\sqrt [4]{a}+\sqrt [4]{b} c} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}+\sqrt [4]{b} c}}\right )}{4 a^{3/4} b^{3/8} d^{3/2}}-\frac {\arctan \left (\frac {\sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}-\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}}\right )}{4 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} d^{3/2}}+\frac {\arctan \left (\frac {\sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}+\sqrt {2} \sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}}}\right )}{4 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} d^{3/2}}+\frac {\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{4 a^{3/4} b^{3/8} d^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{b} \sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} \sqrt {d} x}{\sqrt {\sqrt {a}+\sqrt {b} c^2}+\sqrt [4]{b} d x^2}\right )}{4 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {-\sqrt [4]{b} c+\sqrt {\sqrt {a}+\sqrt {b} c^2}} d^{3/2}} \] Output:
-1/4*(a^(1/4)+b^(1/4)*c)^(1/2)*arctan(b^(1/8)*d^(1/2)*x/(a^(1/4)+b^(1/4)*c )^(1/2))/a^(3/4)/b^(3/8)/d^(3/2)-1/8*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^(3/8)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2) ^(1/2))^(1/2)/d^(3/2)+1/8*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^(3/8)/(b^(1/4)*c+(a^(1/2)+b^(1/2)*c^2)^(1/2))^(1/ 2)/d^(3/2)+1/4*(a^(1/4)-b^(1/4)*c)^(1/2)*arctanh(b^(1/8)*d^(1/2)*x/(a^(1/4 )-b^(1/4)*c)^(1/2))/a^(3/4)/b^(3/8)/d^(3/2)-1/8*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^(3/8)/(-b^(1/4)*c+(a^(1/2)+b^( 1/2)*c^2)^(1/2))^(1/2)/d^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.20 \[ \int \frac {x^2}{a-b \left (c+d x^2\right )^4} \, dx=-\frac {\text {RootSum}\left [a-b c^4-4 b c^3 d \text {$\#$1}^2-6 b c^2 d^2 \text {$\#$1}^4-4 b c d^3 \text {$\#$1}^6-b d^4 \text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{c^3+3 c^2 d \text {$\#$1}^2+3 c d^2 \text {$\#$1}^4+d^3 \text {$\#$1}^6}\&\right ]}{8 b d} \] Input:
Integrate[x^2/(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)/(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. \(1527\) vs. \(2(550)=1100\).
Time = 4.52 (sec) , antiderivative size = 1527, normalized size of antiderivative = 2.78, 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.
| \(\displaystyle \int \frac {x^2}{a-b \left (c+d x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 7291 |
| \(\displaystyle \int \left (\frac {c+d x^2}{d \left (a-b \left (c+d x^2\right )^4\right )}-\frac {c}{d \left (a-b \left (c+d x^2\right )^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right )}{4 a^{3/4} \sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} d^{3/2}}-\frac {\arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{a}}}\right )}{4 \sqrt {a} b^{3/8} \sqrt {\sqrt [4]{b} c+\sqrt [4]{a}} d^{3/2}}-\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 )}{4 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} d^{3/2}}+\frac {c \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} \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^{3/2}}+\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 )}{4 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} d^{3/2}}-\frac {c \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} \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^{3/2}}-\frac {c \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{4 a^{3/4} \sqrt [8]{b} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{a}-\sqrt [4]{b} c}}\right )}{4 \sqrt {a} b^{3/8} \sqrt {\sqrt [4]{a}-\sqrt [4]{b} c} d^{3/2}}+\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 )}{8 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} d^{3/2}}+\frac {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 )}{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^{3/2}}-\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 )}{8 \sqrt {2} \sqrt {a} b^{3/8} \sqrt {\sqrt {b} c^2+\sqrt {a}} d^{3/2}}-\frac {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 )}{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^{3/2}}\) |
Input:
Int[x^2/(a - b*(c + d*x^2)^4),x]
Output:
-1/4*ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqrt[a^(1/4) + b^(1/4)*c]]/(Sqrt[a]*b^(3/8 )*Sqrt[a^(1/4) + b^(1/4)*c]*d^(3/2)) - (c*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^(3/2 )) + (c*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*Sqr t[2]*Sqrt[a]*b^(1/8)*Sqrt[Sqrt[a] + Sqrt[b]*c^2]*Sqrt[b^(1/4)*c + Sqrt[Sqr t[a] + Sqrt[b]*c^2]]*d^(3/2)) - (Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c ^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 ]*Sqrt[a]*b^(3/8)*Sqrt[Sqrt[a] + Sqrt[b]*c^2]*d^(3/2)) - (c*ArcTan[(Sqrt[- (b^(1/4)*c) + Sqrt[Sqrt[a] + Sqrt[b]*c^2]] + Sqrt[2]*b^(1/8)*Sqrt[d]*x)/Sq rt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]])/(4*Sqrt[2]*Sqrt[a]*b^(1/8)*S qrt[Sqrt[a] + Sqrt[b]*c^2]*Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^2]]*d ^(3/2)) + (Sqrt[b^(1/4)*c + Sqrt[Sqrt[a] + Sqrt[b]*c^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]*Sqrt[a]*b^(3/8)*Sqrt [Sqrt[a] + Sqrt[b]*c^2]*d^(3/2)) + ArcTanh[(b^(1/8)*Sqrt[d]*x)/Sqrt[a^(1/4 ) - b^(1/4)*c]]/(4*Sqrt[a]*b^(3/8)*Sqrt[a^(1/4) - b^(1/4)*c]*d^(3/2)) - (c *ArcTanh[(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^(3/2)) + (c*Log[Sqrt[Sqrt[a] + Sqrt[b]*c^...
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 = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.19
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{8}+4 b c \,d^{3} \textit {\_Z}^{6}+6 b \,c^{2} d^{2} \textit {\_Z}^{4}+4 b \,c^{3} d \,\textit {\_Z}^{2}+b \,c^{4}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{-d^{3} \textit {\_R}^{7}-3 c \,d^{2} \textit {\_R}^{5}-3 c^{2} d \,\textit {\_R}^{3}-c^{3} \textit {\_R}}}{8 d b}\) | \(107\) |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{8}+4 b c \,d^{3} \textit {\_Z}^{6}+6 b \,c^{2} d^{2} \textit {\_Z}^{4}+4 b \,c^{3} d \,\textit {\_Z}^{2}+b \,c^{4}-a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{-d^{3} \textit {\_R}^{7}-3 c \,d^{2} \textit {\_R}^{5}-3 c^{2} d \,\textit {\_R}^{3}-c^{3} \textit {\_R}}}{8 d b}\) | \(107\) |
Input:
int(x^2/(a-b*(d*x^2+c)^4),x,method=_RETURNVERBOSE)
Output:
1/8/d/b*sum(_R^2/(-_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))
Result contains complex when optimal does not.
Time = 73.64 (sec) , antiderivative size = 103351, normalized size of antiderivative = 187.91 \[ \int \frac {x^2}{a-b \left (c+d x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate(x^2/(a-b*(d*x^2+c)^4),x, algorithm="fricas")
Output:
Too large to include
Time = 2.98 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.27 \[ \int \frac {x^2}{a-b \left (c+d x^2\right )^4} \, dx=- \operatorname {RootSum} {\left (16777216 t^{8} a^{6} b^{3} d^{12} - 8192 t^{4} a^{3} b^{2} c^{2} d^{6} + 256 t^{2} a^{2} b c d^{3} - a + b c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 4194304 t^{7} a^{5} b^{3} c^{2} d^{10} - 32768 t^{5} a^{4} b^{2} c d^{7} - 512 t^{3} a^{3} b d^{4} + 1024 t^{3} a^{2} b^{2} c^{4} d^{4} - 40 t a b c^{3} d}{a + 4 b c^{4}} \right )} \right )\right )} \] Input:
integrate(x**2/(a-b*(d*x**2+c)**4),x)
Output:
-RootSum(16777216*_t**8*a**6*b**3*d**12 - 8192*_t**4*a**3*b**2*c**2*d**6 + 256*_t**2*a**2*b*c*d**3 - a + b*c**4, Lambda(_t, _t*log(x + (-4194304*_t* *7*a**5*b**3*c**2*d**10 - 32768*_t**5*a**4*b**2*c*d**7 - 512*_t**3*a**3*b* d**4 + 1024*_t**3*a**2*b**2*c**4*d**4 - 40*_t*a*b*c**3*d)/(a + 4*b*c**4))) )
\[ \int \frac {x^2}{a-b \left (c+d x^2\right )^4} \, dx=\int { -\frac {x^{2}}{{\left (d x^{2} + c\right )}^{4} b - a} \,d x } \] Input:
integrate(x^2/(a-b*(d*x^2+c)^4),x, algorithm="maxima")
Output:
-integrate(x^2/((d*x^2 + c)^4*b - a), x)
\[ \int \frac {x^2}{a-b \left (c+d x^2\right )^4} \, dx=\int { -\frac {x^{2}}{{\left (d x^{2} + c\right )}^{4} b - a} \,d x } \] Input:
integrate(x^2/(a-b*(d*x^2+c)^4),x, algorithm="giac")
Output:
integrate(-x^2/((d*x^2 + c)^4*b - a), x)
Time = 9.92 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.17 \[ \int \frac {x^2}{a-b \left (c+d x^2\right )^4} \, dx=\sum _{k=1}^8\ln \left (-a\,b^5\,d^{20}+b^6\,c^4\,d^{20}-\mathrm {root}\left (16777216\,a^6\,b^3\,d^{12}\,z^8-8192\,a^3\,b^2\,c^2\,d^6\,z^4+256\,a^2\,b\,c\,d^3\,z^2+b\,c^4-a,z,k\right )\,b^7\,c^6\,d^{22}\,x\,8+{\mathrm {root}\left (16777216\,a^6\,b^3\,d^{12}\,z^8-8192\,a^3\,b^2\,c^2\,d^6\,z^4+256\,a^2\,b\,c\,d^3\,z^2+b\,c^4-a,z,k\right )}^2\,a^2\,b^6\,c\,d^{23}\,128-{\mathrm {root}\left (16777216\,a^6\,b^3\,d^{12}\,z^8-8192\,a^3\,b^2\,c^2\,d^6\,z^4+256\,a^2\,b\,c\,d^3\,z^2+b\,c^4-a,z,k\right )}^2\,a\,b^7\,c^5\,d^{23}\,128-{\mathrm {root}\left (16777216\,a^6\,b^3\,d^{12}\,z^8-8192\,a^3\,b^2\,c^2\,d^6\,z^4+256\,a^2\,b\,c\,d^3\,z^2+b\,c^4-a,z,k\right )}^5\,a^4\,b^7\,d^{28}\,x\,32768-{\mathrm {root}\left (16777216\,a^6\,b^3\,d^{12}\,z^8-8192\,a^3\,b^2\,c^2\,d^6\,z^4+256\,a^2\,b\,c\,d^3\,z^2+b\,c^4-a,z,k\right )}^7\,a^5\,b^8\,c\,d^{31}\,x\,2097152+{\mathrm {root}\left (16777216\,a^6\,b^3\,d^{12}\,z^8-8192\,a^3\,b^2\,c^2\,d^6\,z^4+256\,a^2\,b\,c\,d^3\,z^2+b\,c^4-a,z,k\right )}^3\,a^2\,b^7\,c^3\,d^{25}\,x\,1024+{\mathrm {root}\left (16777216\,a^6\,b^3\,d^{12}\,z^8-8192\,a^3\,b^2\,c^2\,d^6\,z^4+256\,a^2\,b\,c\,d^3\,z^2+b\,c^4-a,z,k\right )}^5\,a^3\,b^8\,c^4\,d^{28}\,x\,32768-\mathrm {root}\left (16777216\,a^6\,b^3\,d^{12}\,z^8-8192\,a^3\,b^2\,c^2\,d^6\,z^4+256\,a^2\,b\,c\,d^3\,z^2+b\,c^4-a,z,k\right )\,a\,b^6\,c^2\,d^{22}\,x\,24\right )\,\mathrm {root}\left (16777216\,a^6\,b^3\,d^{12}\,z^8-8192\,a^3\,b^2\,c^2\,d^6\,z^4+256\,a^2\,b\,c\,d^3\,z^2+b\,c^4-a,z,k\right ) \] Input:
int(x^2/(a - b*(c + d*x^2)^4),x)
Output:
symsum(log(b^6*c^4*d^20 - a*b^5*d^20 - 8*root(16777216*a^6*b^3*d^12*z^8 - 8192*a^3*b^2*c^2*d^6*z^4 + 256*a^2*b*c*d^3*z^2 + b*c^4 - a, z, k)*b^7*c^6* d^22*x + 128*root(16777216*a^6*b^3*d^12*z^8 - 8192*a^3*b^2*c^2*d^6*z^4 + 2 56*a^2*b*c*d^3*z^2 + b*c^4 - a, z, k)^2*a^2*b^6*c*d^23 - 128*root(16777216 *a^6*b^3*d^12*z^8 - 8192*a^3*b^2*c^2*d^6*z^4 + 256*a^2*b*c*d^3*z^2 + b*c^4 - a, z, k)^2*a*b^7*c^5*d^23 - 32768*root(16777216*a^6*b^3*d^12*z^8 - 8192 *a^3*b^2*c^2*d^6*z^4 + 256*a^2*b*c*d^3*z^2 + b*c^4 - a, z, k)^5*a^4*b^7*d^ 28*x - 2097152*root(16777216*a^6*b^3*d^12*z^8 - 8192*a^3*b^2*c^2*d^6*z^4 + 256*a^2*b*c*d^3*z^2 + b*c^4 - a, z, k)^7*a^5*b^8*c*d^31*x + 1024*root(167 77216*a^6*b^3*d^12*z^8 - 8192*a^3*b^2*c^2*d^6*z^4 + 256*a^2*b*c*d^3*z^2 + b*c^4 - a, z, k)^3*a^2*b^7*c^3*d^25*x + 32768*root(16777216*a^6*b^3*d^12*z ^8 - 8192*a^3*b^2*c^2*d^6*z^4 + 256*a^2*b*c*d^3*z^2 + b*c^4 - a, z, k)^5*a ^3*b^8*c^4*d^28*x - 24*root(16777216*a^6*b^3*d^12*z^8 - 8192*a^3*b^2*c^2*d ^6*z^4 + 256*a^2*b*c*d^3*z^2 + b*c^4 - a, z, k)*a*b^6*c^2*d^22*x)*root(167 77216*a^6*b^3*d^12*z^8 - 8192*a^3*b^2*c^2*d^6*z^4 + 256*a^2*b*c*d^3*z^2 + b*c^4 - a, z, k), k, 1, 8)
\[ \int \frac {x^2}{a-b \left (c+d x^2\right )^4} \, dx=\int \frac {x^{2}}{a -b \left (d \,x^{2}+c \right )^{4}}d x \] Input:
int(x^2/(a-b*(d*x^2+c)^4),x)
Output:
int(x^2/(a-b*(d*x^2+c)^4),x)