Integrand size = 19, antiderivative size = 2262 \[ \int \frac {1}{x^2 \left (a+b \left (c+d x^2\right )^4\right )} \, dx =\text {Too large to display} \] Output:
-1/4*(2*a^(1/4)-2^(1/2)*b^(1/4)*c)/a^(3/4)/(a^(1/2)-2^(1/2)*a^(1/4)*b^(1/4 )*c+b^(1/2)*c^2)/x-1/4*(2*a^(1/4)+2^(1/2)*b^(1/4)*c)/a^(3/4)/(a^(1/2)+2^(1 /2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)/x+1/8*(2*a^(1/4)/b^(1/4)+2^(1/2)*c+(a^( 1/2)*2^(1/2)+4*a^(1/4)*b^(1/4)*c+2^(1/2)*b^(1/2)*c^2)/b^(1/4)/(a^(1/2)+2^( 1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))*d^(1/2)*arctan(((-2^(1/2)*a^(1/ 4)-2*b^(1/4)*c+2*(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1 /2)-2*b^(1/8)*d^(1/2)*x)/(2^(1/2)*a^(1/4)+2*b^(1/4)*c+2*(a^(1/2)+2^(1/2)*a ^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2))/a^(3/4)/b^(1/8)/(a^(1/2)/b^(1/ 2)+2^(1/2)*a^(1/4)*c/b^(1/4)+c^2)/(2^(1/2)*a^(1/4)+2*b^(1/4)*c+2*(a^(1/2)+ 2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2)-1/8*(2*a^(1/4)/b^(1/4) +2^(1/2)*c+(a^(1/2)*2^(1/2)+4*a^(1/4)*b^(1/4)*c+2^(1/2)*b^(1/2)*c^2)/b^(1/ 4)/(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))*d^(1/2)*arctan(( (-2^(1/2)*a^(1/4)-2*b^(1/4)*c+2*(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2) *c^2)^(1/2))^(1/2)+2*b^(1/8)*d^(1/2)*x)/(2^(1/2)*a^(1/4)+2*b^(1/4)*c+2*(a^ (1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2))/a^(3/4)/b^(1/8) /(a^(1/2)/b^(1/2)+2^(1/2)*a^(1/4)*c/b^(1/4)+c^2)/(2^(1/2)*a^(1/4)+2*b^(1/4 )*c+2*(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))^(1/2)+1/8*(2* a^(1/4)/b^(1/4)+2^(1/2)*c-(a^(1/2)*2^(1/2)+4*a^(1/4)*b^(1/4)*c+2^(1/2)*b^( 1/2)*c^2)/b^(1/4)/(a^(1/2)+2^(1/2)*a^(1/4)*b^(1/4)*c+b^(1/2)*c^2)^(1/2))*d ^(1/2)*arctan(b^(1/8)*(2^(1/2)*a^(1/4)+2*b^(1/4)*c-2*(a^(1/2)+2^(1/2)*a...
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.08 \[ \int \frac {1}{x^2 \left (a+b \left (c+d x^2\right )^4\right )} \, dx=-\frac {8+x \text {RootSum}\left [a+b c^4+4 b c^3 d \text {$\#$1}^2+6 b c^2 d^2 \text {$\#$1}^4+4 b c d^3 \text {$\#$1}^6+b d^4 \text {$\#$1}^8\&,\frac {4 c^3 \log (x-\text {$\#$1})+6 c^2 d \log (x-\text {$\#$1}) \text {$\#$1}^2+4 c d^2 \log (x-\text {$\#$1}) \text {$\#$1}^4+d^3 \log (x-\text {$\#$1}) \text {$\#$1}^6}{c^3 \text {$\#$1}+3 c^2 d \text {$\#$1}^3+3 c d^2 \text {$\#$1}^5+d^3 \text {$\#$1}^7}\&\right ]}{8 a x+8 b c^4 x} \] Input:
Integrate[1/(x^2*(a + b*(c + d*x^2)^4)),x]
Output:
-((8 + x*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 & , (4*c^3*Log[x - #1] + 6*c^2*d*Log[x - #1]*#1^2 + 4*c *d^2*Log[x - #1]*#1^4 + d^3*Log[x - #1]*#1^6)/(c^3*#1 + 3*c^2*d*#1^3 + 3*c *d^2*#1^5 + d^3*#1^7) & ])/(8*a*x + 8*b*c^4*x))
Time = 12.39 (sec) , antiderivative size = 3451, normalized size of antiderivative = 1.53, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {7293, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b \left (c+d x^2\right )^4\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {1}{x^2 \left (a+b c^4\right )}+\frac {b d \left (-4 c^3-6 c^2 d x^2-4 c d^2 x^4-d^3 x^6\right )}{\left (a+b c^4\right ) \left (a \left (\frac {b c^4}{a}+1\right )+4 b c^3 d x^2+6 b c^2 d^2 x^4+4 b c d^3 x^6+b d^4 x^8\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^{7/8} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}}}\right ) c^3}{4 (-a)^{3/4} \sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}} \left (b c^4+a\right )}-\frac {b^{7/8} \sqrt {d} \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^3}{4 \sqrt {2} \sqrt {-a} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right ) \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}}}+\frac {b^{7/8} \sqrt {d} \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^3}{4 \sqrt {2} \sqrt {-a} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right ) \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}}}+\frac {b^{7/8} \sqrt {d} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right ) c^3}{4 (-a)^{3/4} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c} \left (b c^4+a\right )}-\frac {b^{7/8} \sqrt {d} \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^3}{8 \sqrt {2} \sqrt {-a} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right ) \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}}+\frac {b^{7/8} \sqrt {d} \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^3}{8 \sqrt {2} \sqrt {-a} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right ) \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}}-\frac {b^{5/8} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}}}\right ) c^2}{4 \sqrt {-a} \sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}} \left (b c^4+a\right )}-\frac {b^{5/8} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}} \sqrt {d} \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 {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right )}+\frac {b^{5/8} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}} \sqrt {d} \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 {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right )}+\frac {b^{5/8} \sqrt {d} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right ) c^2}{4 \sqrt {-a} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c} \left (b c^4+a\right )}+\frac {b^{5/8} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \sqrt {d} \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 {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right )}-\frac {b^{5/8} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \sqrt {d} \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 {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right )}+\frac {b^{3/8} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}}}\right ) c}{4 \sqrt [4]{-a} \sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}} \left (b c^4+a\right )}+\frac {b^{3/8} \sqrt {d} \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}{4 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right ) \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}}}-\frac {b^{3/8} \sqrt {d} \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}{4 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right ) \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}}}+\frac {b^{3/8} \sqrt {d} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right ) c}{4 \sqrt [4]{-a} \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c} \left (b c^4+a\right )}+\frac {b^{3/8} \sqrt {d} \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}{8 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right ) \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}}-\frac {b^{3/8} \sqrt {d} \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}{8 \sqrt {2} \sqrt {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right ) \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c}}-\frac {\sqrt [8]{b} \sqrt {d} \arctan \left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}}}\right )}{4 \sqrt {\sqrt [4]{b} c+\sqrt [4]{-a}} \left (b c^4+a\right )}+\frac {\sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}} \sqrt {d} \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 {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right )}-\frac {\sqrt [8]{b} \sqrt {\sqrt [4]{b} c+\sqrt {\sqrt {b} c^2+\sqrt {-a}}} \sqrt {d} \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 {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right )}+\frac {\sqrt [8]{b} \sqrt {d} \text {arctanh}\left (\frac {\sqrt [8]{b} \sqrt {d} x}{\sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c}}\right )}{4 \sqrt {\sqrt [4]{-a}-\sqrt [4]{b} c} \left (b c^4+a\right )}-\frac {\sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \sqrt {d} \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 {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right )}+\frac {\sqrt [8]{b} \sqrt {\sqrt {\sqrt {b} c^2+\sqrt {-a}}-\sqrt [4]{b} c} \sqrt {d} \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 {\sqrt {b} c^2+\sqrt {-a}} \left (b c^4+a\right )}-\frac {1}{\left (b c^4+a\right ) x}\) |
Input:
Int[1/(x^2*(a + b*(c + d*x^2)^4)),x]
Output:
-(1/((a + b*c^4)*x)) - (b^(1/8)*Sqrt[d]*ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqrt[(- a)^(1/4) + b^(1/4)*c]])/(4*Sqrt[(-a)^(1/4) + b^(1/4)*c]*(a + b*c^4)) + (b^ (3/8)*c*Sqrt[d]*ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqrt[(-a)^(1/4) + b^(1/4)*c]])/ (4*(-a)^(1/4)*Sqrt[(-a)^(1/4) + b^(1/4)*c]*(a + b*c^4)) - (b^(5/8)*c^2*Sqr t[d]*ArcTan[(b^(1/8)*Sqrt[d]*x)/Sqrt[(-a)^(1/4) + b^(1/4)*c]])/(4*Sqrt[-a] *Sqrt[(-a)^(1/4) + b^(1/4)*c]*(a + b*c^4)) + (b^(7/8)*c^3*Sqrt[d]*ArcTan[( b^(1/8)*Sqrt[d]*x)/Sqrt[(-a)^(1/4) + b^(1/4)*c]])/(4*(-a)^(3/4)*Sqrt[(-a)^ (1/4) + b^(1/4)*c]*(a + b*c^4)) + (b^(3/8)*c*Sqrt[d]*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[Sqrt[-a] + Sqrt[b ]*c^2]*(a + b*c^4)*Sqrt[b^(1/4)*c + Sqrt[Sqrt[-a] + Sqrt[b]*c^2]]) - (b^(7 /8)*c^3*Sqrt[d]*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]*Sqrt[Sqrt[-a] + Sqrt[b]*c^2]*(a + b*c^4)*Sqrt[b^(1 /4)*c + Sqrt[Sqrt[-a] + Sqrt[b]*c^2]]) + (b^(1/8)*Sqrt[b^(1/4)*c + Sqrt[Sq rt[-a] + Sqrt[b]*c^2]]*Sqrt[d]*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[Sqrt[-a] + Sqrt[b]*c^2]*(a + b*c^4)) - (b^(5/8)*c^2*Sqrt[b^(1/4)*c + Sqrt[Sqrt[-a] + Sqrt[b]*c^2]]*Sqrt[d]*ArcTan [(Sqrt[-(b^(1/4)*c) + Sqrt[Sqrt[-a] + Sqrt[b]*c^2]] - Sqrt[2]*b^(1/8)*S...
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.07
| method | result | size |
| default | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{8}+4 b c \,d^{3} \textit {\_Z}^{6}+6 b \,c^{2} d^{2} \textit {\_Z}^{4}+4 b \,c^{3} d \,\textit {\_Z}^{2}+b \,c^{4}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6} d^{3}+4 \textit {\_R}^{4} c \,d^{2}+6 \textit {\_R}^{2} c^{2} d +4 c^{3}\right ) \ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{7}+3 c \,d^{2} \textit {\_R}^{5}+3 c^{2} d \,\textit {\_R}^{3}+c^{3} \textit {\_R}}}{8 \left (b \,c^{4}+a \right )}-\frac {1}{\left (b \,c^{4}+a \right ) x}\) | \(149\) |
| risch | \(-\frac {1}{\left (b \,c^{4}+a \right ) x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{6} b^{3} c^{12}+3 a^{7} b^{2} c^{8}+3 a^{8} b \,c^{4}+a^{9}\right ) \textit {\_Z}^{8}+\left (-24 a^{5} b^{2} c^{7} d +40 a^{6} b \,c^{3} d \right ) \textit {\_Z}^{6}+\left (2 a^{3} b^{2} c^{6} d^{2}+42 a^{4} b \,c^{2} d^{2}\right ) \textit {\_Z}^{4}+12 a^{2} b c \,d^{3} \textit {\_Z}^{2}+b \,d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-3 a^{5} b^{4} c^{16}+18 a^{7} b^{2} c^{8}+24 a^{8} b \,c^{4}+9 a^{9}\right ) \textit {\_R}^{8}+\left (a^{3} b^{4} c^{15} d +7 a^{4} b^{3} c^{11} d -181 a^{5} b^{2} c^{7} d +325 a^{6} b \,c^{3} d \right ) \textit {\_R}^{6}+\left (6 a^{2} b^{3} c^{10} d^{2}+28 a^{3} b^{2} c^{6} d^{2}+342 a^{4} b \,c^{2} d^{2}\right ) \textit {\_R}^{4}+\left (b^{3} c^{9} d^{3}+2 a \,b^{2} c^{5} d^{3}+97 a^{2} b c \,d^{3}\right ) \textit {\_R}^{2}+8 b \,d^{4}\right ) x +\left (3 a^{4} b^{4} c^{16}-6 a^{5} b^{3} c^{12}-20 a^{6} b^{2} c^{8}-10 a^{7} b \,c^{4}+a^{8}\right ) \textit {\_R}^{7}+\left (-16 a^{3} b^{3} c^{11} d -32 a^{4} b^{2} c^{7} d -16 a^{5} b \,c^{3} d \right ) \textit {\_R}^{5}+\left (-3 a \,b^{3} c^{10} d^{2}-6 a^{2} b^{2} c^{6} d^{2}-3 a^{3} b \,c^{2} d^{2}\right ) \textit {\_R}^{3}\right )\right )}{8}\) | \(455\) |
Input:
int(1/x^2/(a+b*(d*x^2+c)^4),x,method=_RETURNVERBOSE)
Output:
-1/8/(b*c^4+a)*sum((_R^6*d^3+4*_R^4*c*d^2+6*_R^2*c^2*d+4*c^3)/(_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))-1/(b*c^4+a)/x
Timed out. \[ \int \frac {1}{x^2 \left (a+b \left (c+d x^2\right )^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(a+b*(d*x^2+c)^4),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^2 \left (a+b \left (c+d x^2\right )^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**2/(a+b*(d*x**2+c)**4),x)
Output:
Timed out
\[ \int \frac {1}{x^2 \left (a+b \left (c+d x^2\right )^4\right )} \, dx=\int { \frac {1}{{\left ({\left (d x^{2} + c\right )}^{4} b + a\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*(d*x^2+c)^4),x, algorithm="maxima")
Output:
-b*d*integrate((d^3*x^6 + 4*c*d^2*x^4 + 6*c^2*d*x^2 + 4*c^3)/(b*d^4*x^8 + 4*b*c*d^3*x^6 + 6*b*c^2*d^2*x^4 + 4*b*c^3*d*x^2 + b*c^4 + a), x)/(b*c^4 + a) - 1/((b*c^4 + a)*x)
\[ \int \frac {1}{x^2 \left (a+b \left (c+d x^2\right )^4\right )} \, dx=\int { \frac {1}{{\left ({\left (d x^{2} + c\right )}^{4} b + a\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*(d*x^2+c)^4),x, algorithm="giac")
Output:
sage0*x
Time = 10.14 (sec) , antiderivative size = 2345, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \left (a+b \left (c+d x^2\right )^4\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^2*(a + b*(c + d*x^2)^4)),x)
Output:
symsum(log(root(50331648*a^7*b^2*c^8*z^8 + 16777216*a^6*b^3*c^12*z^8 + 503 31648*a^8*b*c^4*z^8 + 16777216*a^9*z^8 - 6291456*a^5*b^2*c^7*d*z^6 + 10485 760*a^6*b*c^3*d*z^6 + 172032*a^4*b*c^2*d^2*z^4 + 8192*a^3*b^2*c^6*d^2*z^4 + 768*a^2*b*c*d^3*z^2 + b*d^4, z, k)*(root(50331648*a^7*b^2*c^8*z^8 + 1677 7216*a^6*b^3*c^12*z^8 + 50331648*a^8*b*c^4*z^8 + 16777216*a^9*z^8 - 629145 6*a^5*b^2*c^7*d*z^6 + 10485760*a^6*b*c^3*d*z^6 + 172032*a^4*b*c^2*d^2*z^4 + 8192*a^3*b^2*c^6*d^2*z^4 + 768*a^2*b*c*d^3*z^2 + b*d^4, z, k)*(256*a*b^2 0*c^47*d^33 - root(50331648*a^7*b^2*c^8*z^8 + 16777216*a^6*b^3*c^12*z^8 + 50331648*a^8*b*c^4*z^8 + 16777216*a^9*z^8 - 6291456*a^5*b^2*c^7*d*z^6 + 10 485760*a^6*b*c^3*d*z^6 + 172032*a^4*b*c^2*d^2*z^4 + 8192*a^3*b^2*c^6*d^2*z ^4 + 768*a^2*b*c*d^3*z^2 + b*d^4, z, k)*(x*(1024*a^13*b^9*c^3*d^33 + 11264 *a^12*b^10*c^7*d^33 + 56320*a^11*b^11*c^11*d^33 + 168960*a^10*b^12*c^15*d^ 33 + 337920*a^9*b^13*c^19*d^33 + 473088*a^8*b^14*c^23*d^33 + 473088*a^7*b^ 15*c^27*d^33 + 337920*a^6*b^16*c^31*d^33 + 168960*a^5*b^17*c^35*d^33 + 563 20*a^4*b^18*c^39*d^33 + 11264*a^3*b^19*c^43*d^33 + 1024*a^2*b^20*c^47*d^33 ) - root(50331648*a^7*b^2*c^8*z^8 + 16777216*a^6*b^3*c^12*z^8 + 50331648*a ^8*b*c^4*z^8 + 16777216*a^9*z^8 - 6291456*a^5*b^2*c^7*d*z^6 + 10485760*a^6 *b*c^3*d*z^6 + 172032*a^4*b*c^2*d^2*z^4 + 8192*a^3*b^2*c^6*d^2*z^4 + 768*a ^2*b*c*d^3*z^2 + b*d^4, z, k)*(root(50331648*a^7*b^2*c^8*z^8 + 16777216*a^ 6*b^3*c^12*z^8 + 50331648*a^8*b*c^4*z^8 + 16777216*a^9*z^8 - 6291456*a^...
\[ \int \frac {1}{x^2 \left (a+b \left (c+d x^2\right )^4\right )} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(a+b*(d*x^2+c)^4),x)
Output:
( - int(x**6/(a**2 + 2*a*b*c**4 + 4*a*b*c**3*d*x**2 + 6*a*b*c**2*d**2*x**4 + 4*a*b*c*d**3*x**6 + a*b*d**4*x**8 + b**2*c**8 + 4*b**2*c**7*d*x**2 + 6* b**2*c**6*d**2*x**4 + 4*b**2*c**5*d**3*x**6 + b**2*c**4*d**4*x**8),x)*a*b* d**4*x - int(x**6/(a**2 + 2*a*b*c**4 + 4*a*b*c**3*d*x**2 + 6*a*b*c**2*d**2 *x**4 + 4*a*b*c*d**3*x**6 + a*b*d**4*x**8 + b**2*c**8 + 4*b**2*c**7*d*x**2 + 6*b**2*c**6*d**2*x**4 + 4*b**2*c**5*d**3*x**6 + b**2*c**4*d**4*x**8),x) *b**2*c**4*d**4*x - 4*int(x**4/(a**2 + 2*a*b*c**4 + 4*a*b*c**3*d*x**2 + 6* a*b*c**2*d**2*x**4 + 4*a*b*c*d**3*x**6 + a*b*d**4*x**8 + b**2*c**8 + 4*b** 2*c**7*d*x**2 + 6*b**2*c**6*d**2*x**4 + 4*b**2*c**5*d**3*x**6 + b**2*c**4* d**4*x**8),x)*a*b*c*d**3*x - 4*int(x**4/(a**2 + 2*a*b*c**4 + 4*a*b*c**3*d* x**2 + 6*a*b*c**2*d**2*x**4 + 4*a*b*c*d**3*x**6 + a*b*d**4*x**8 + b**2*c** 8 + 4*b**2*c**7*d*x**2 + 6*b**2*c**6*d**2*x**4 + 4*b**2*c**5*d**3*x**6 + b **2*c**4*d**4*x**8),x)*b**2*c**5*d**3*x - 6*int(x**2/(a**2 + 2*a*b*c**4 + 4*a*b*c**3*d*x**2 + 6*a*b*c**2*d**2*x**4 + 4*a*b*c*d**3*x**6 + a*b*d**4*x* *8 + b**2*c**8 + 4*b**2*c**7*d*x**2 + 6*b**2*c**6*d**2*x**4 + 4*b**2*c**5* d**3*x**6 + b**2*c**4*d**4*x**8),x)*a*b*c**2*d**2*x - 6*int(x**2/(a**2 + 2 *a*b*c**4 + 4*a*b*c**3*d*x**2 + 6*a*b*c**2*d**2*x**4 + 4*a*b*c*d**3*x**6 + a*b*d**4*x**8 + b**2*c**8 + 4*b**2*c**7*d*x**2 + 6*b**2*c**6*d**2*x**4 + 4*b**2*c**5*d**3*x**6 + b**2*c**4*d**4*x**8),x)*b**2*c**6*d**2*x - 4*int(1 /(a**2 + 2*a*b*c**4 + 4*a*b*c**3*d*x**2 + 6*a*b*c**2*d**2*x**4 + 4*a*b*...