Integrand size = 22, antiderivative size = 1080 \[ \int \frac {1}{\left (a+b x+c x^2+b x^3+a x^4\right )^2} \, dx =\text {Too large to display} \] Output:
(-12*a^2*b+2*(3*b^2-c^2)*(b+(8*a^2-4*a*c+b^2)^(1/2))-2*a*c*(b+2*(8*a^2-4*a *c+b^2)^(1/2))+4*a*(3*b^2-c*(2*a+c))*x)/a/(2*a-2*b+c)/(2*a+2*b+c)/(8*a^2-4 *a*c+b^2)/(2+(b+(8*a^2-4*a*c+b^2)^(1/2))*x/a+2*x^2)-4*(b^2-2*a*c-b*(8*a^2- 4*a*c+b^2)^(1/2)+a*(b-(8*a^2-4*a*c+b^2)^(1/2))*x)/a/(8*a^2-4*a*c+b^2)^(1/2 )/(4*a^2+2*a*c-b*(b-(8*a^2-4*a*c+b^2)^(1/2)))/(2+(b-(8*a^2-4*a*c+b^2)^(1/2 ))*x/a+2*x^2)/(2+(b+(8*a^2-4*a*c+b^2)^(1/2))*x/a+2*x^2)-4*2^(1/2)*a^2*(c*( 4*b^2-c^2)*(b-(8*a^2-4*a*c+b^2)^(1/2))+4*a^2*c*(2*b-(8*a^2-4*a*c+b^2)^(1/2 ))+12*a^3*(3*b-(8*a^2-4*a*c+b^2)^(1/2))-a*(18*b^3-b*c^2-6*b^2*(8*a^2-4*a*c +b^2)^(1/2)-3*c^2*(8*a^2-4*a*c+b^2)^(1/2)))*arctan(1/2*(b-(8*a^2-4*a*c+b^2 )^(1/2)+4*a*x)*2^(1/2)/(4*a^2+2*a*c-b*(b-(8*a^2-4*a*c+b^2)^(1/2)))^(1/2))/ (8*a^2-4*a*c+b^2)^(3/2)/(4*a^2+2*a*c-b*(b-(8*a^2-4*a*c+b^2)^(1/2)))^(3/2)/ (4*a^2+2*a*c-b*(b+(8*a^2-4*a*c+b^2)^(1/2)))+4*2^(1/2)*a^2*(c*(4*b^2-c^2)*( b+(8*a^2-4*a*c+b^2)^(1/2))+4*a^2*c*(2*b+(8*a^2-4*a*c+b^2)^(1/2))+12*a^3*(3 *b+(8*a^2-4*a*c+b^2)^(1/2))-a*(18*b^3-b*c^2+6*b^2*(8*a^2-4*a*c+b^2)^(1/2)+ 3*c^2*(8*a^2-4*a*c+b^2)^(1/2)))*arctan(1/2*(b+(8*a^2-4*a*c+b^2)^(1/2)+4*a* x)*2^(1/2)/(4*a^2+2*a*c-b*(b+(8*a^2-4*a*c+b^2)^(1/2)))^(1/2))/(8*a^2-4*a*c +b^2)^(3/2)/(4*a^2+2*a*c-b*(b-(8*a^2-4*a*c+b^2)^(1/2)))/(4*a^2+2*a*c-b*(b+ (8*a^2-4*a*c+b^2)^(1/2)))^(3/2)-(3*a-c)*ln(2*a+(b-(8*a^2-4*a*c+b^2)^(1/2)) *x+2*a*x^2)/(8*a^2-4*a*c+b^2)^(3/2)+(3*a-c)*ln(2*a+(b+(8*a^2-4*a*c+b^2)^(1 /2))*x+2*a*x^2)/(8*a^2-4*a*c+b^2)^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.65 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\left (a+b x+c x^2+b x^3+a x^4\right )^2} \, dx=\frac {8 a^3 x+7 b^2 c x-2 c^3 x-b c^2 \left (1+2 x^2\right )+b^3 \left (4+6 x^2\right )+a^2 \left (b \left (2-6 x^2\right )-4 c x \left (-1+x^2\right )\right )+a \left (-2 c^2 x \left (2+x^2\right )-b c \left (5+3 x^2\right )+b^2 \left (-8 x+6 x^3\right )\right )}{\left (8 a^2+b^2-4 a c\right ) \left (4 a^2-4 b^2+4 a c+c^2\right ) \left (x \left (b+c x+b x^2\right )+a \left (1+x^4\right )\right )}+\frac {2 \text {RootSum}\left [a+b \text {$\#$1}+c \text {$\#$1}^2+b \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {12 a^3 \log (x-\text {$\#$1})-9 a b^2 \log (x-\text {$\#$1})+6 a^2 c \log (x-\text {$\#$1})+4 b^2 c \log (x-\text {$\#$1})-2 a c^2 \log (x-\text {$\#$1})-c^3 \log (x-\text {$\#$1})-6 a^2 b \log (x-\text {$\#$1}) \text {$\#$1}+3 b^3 \log (x-\text {$\#$1}) \text {$\#$1}-a b c \log (x-\text {$\#$1}) \text {$\#$1}-b c^2 \log (x-\text {$\#$1}) \text {$\#$1}+3 a b^2 \log (x-\text {$\#$1}) \text {$\#$1}^2-2 a^2 c \log (x-\text {$\#$1}) \text {$\#$1}^2-a c^2 \log (x-\text {$\#$1}) \text {$\#$1}^2}{b+2 c \text {$\#$1}+3 b \text {$\#$1}^2+4 a \text {$\#$1}^3}\&\right ]}{32 a^4-4 b^4+16 a^3 c+b^2 c^2-4 a^2 \left (7 b^2+2 c^2\right )+4 a \left (5 b^2 c-c^3\right )} \] Input:
Integrate[(a + b*x + c*x^2 + b*x^3 + a*x^4)^(-2),x]
Output:
(8*a^3*x + 7*b^2*c*x - 2*c^3*x - b*c^2*(1 + 2*x^2) + b^3*(4 + 6*x^2) + a^2 *(b*(2 - 6*x^2) - 4*c*x*(-1 + x^2)) + a*(-2*c^2*x*(2 + x^2) - b*c*(5 + 3*x ^2) + b^2*(-8*x + 6*x^3)))/((8*a^2 + b^2 - 4*a*c)*(4*a^2 - 4*b^2 + 4*a*c + c^2)*(x*(b + c*x + b*x^2) + a*(1 + x^4))) + (2*RootSum[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , (12*a^3*Log[x - #1] - 9*a*b^2*Log[x - #1] + 6*a^2*c* Log[x - #1] + 4*b^2*c*Log[x - #1] - 2*a*c^2*Log[x - #1] - c^3*Log[x - #1] - 6*a^2*b*Log[x - #1]*#1 + 3*b^3*Log[x - #1]*#1 - a*b*c*Log[x - #1]*#1 - b *c^2*Log[x - #1]*#1 + 3*a*b^2*Log[x - #1]*#1^2 - 2*a^2*c*Log[x - #1]*#1^2 - a*c^2*Log[x - #1]*#1^2)/(b + 2*c*#1 + 3*b*#1^2 + 4*a*#1^3) & ])/(32*a^4 - 4*b^4 + 16*a^3*c + b^2*c^2 - 4*a^2*(7*b^2 + 2*c^2) + 4*a*(5*b^2*c - c^3) )
Time = 4.86 (sec) , antiderivative size = 1135, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2492, 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.
| \(\displaystyle \int \frac {1}{\left (a x^4+a+b x^3+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (-\frac {2 \left (4 a b-c b+2 a (3 a-c) x-(3 a-c) \sqrt {8 a^2-4 c a+b^2}\right ) a^2}{\left (8 a^2-4 c a+b^2\right )^{3/2} \left (2 a x^2+\left (b-\sqrt {8 a^2-4 c a+b^2}\right ) x+2 a\right )}-\frac {2 \left (c \left (b+\sqrt {8 a^2-4 c a+b^2}\right )-a \left (4 b+3 \sqrt {8 a^2-4 c a+b^2}\right )-2 a (3 a-c) x\right ) a^2}{\left (8 a^2-4 c a+b^2\right )^{3/2} \left (2 a x^2+\left (b+\sqrt {8 a^2-4 c a+b^2}\right ) x+2 a\right )}+\frac {2 \left (2 a^2-2 c a+\left (b-\sqrt {8 a^2-4 c a+b^2}\right ) x a+b \left (b-\sqrt {8 a^2-4 c a+b^2}\right )\right ) a^2}{\left (8 a^2-4 c a+b^2\right ) \left (2 a x^2+\left (b-\sqrt {8 a^2-4 c a+b^2}\right ) x+2 a\right )^2}+\frac {2 \left (2 a^2-2 c a+\left (b+\sqrt {8 a^2-4 c a+b^2}\right ) x a+b \left (b+\sqrt {8 a^2-4 c a+b^2}\right )\right ) a^2}{\left (8 a^2-4 c a+b^2\right ) \left (2 a x^2+\left (b+\sqrt {8 a^2-4 c a+b^2}\right ) x+2 a\right )^2}\right )dx}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 \sqrt {2} \left (b^2-\sqrt {8 a^2-4 c a+b^2} b-2 a c\right ) \arctan \left (\frac {b+4 a x-\sqrt {8 a^2-4 c a+b^2}}{\sqrt {2} \sqrt {4 a^2+2 c a-b \left (b-\sqrt {8 a^2-4 c a+b^2}\right )}}\right ) a^3}{\left (8 a^2-4 c a+b^2\right ) \left (4 a^2+2 c a-b \left (b-\sqrt {8 a^2-4 c a+b^2}\right )\right )^{3/2}}+\frac {2 \sqrt {2} \left (b^2+\sqrt {8 a^2-4 c a+b^2} b-2 a c\right ) \arctan \left (\frac {b+4 a x+\sqrt {8 a^2-4 c a+b^2}}{\sqrt {2} \sqrt {4 a^2+2 c a-b \left (b+\sqrt {8 a^2-4 c a+b^2}\right )}}\right ) a^3}{\left (8 a^2-4 c a+b^2\right ) \left (4 a^2+2 c a-b \left (b+\sqrt {8 a^2-4 c a+b^2}\right )\right )^{3/2}}-\frac {\sqrt {2} \left (5 a b-c b-(3 a-c) \sqrt {8 a^2-4 c a+b^2}\right ) \arctan \left (\frac {b+4 a x-\sqrt {8 a^2-4 c a+b^2}}{\sqrt {2} \sqrt {4 a^2+2 c a-b \left (b-\sqrt {8 a^2-4 c a+b^2}\right )}}\right ) a^2}{\left (8 a^2-4 c a+b^2\right )^{3/2} \sqrt {4 a^2+2 c a-b \left (b-\sqrt {8 a^2-4 c a+b^2}\right )}}-\frac {\sqrt {2} \left (c \left (b+\sqrt {8 a^2-4 c a+b^2}\right )-a \left (5 b+3 \sqrt {8 a^2-4 c a+b^2}\right )\right ) \arctan \left (\frac {b+4 a x+\sqrt {8 a^2-4 c a+b^2}}{\sqrt {2} \sqrt {4 a^2+2 c a-b \left (b+\sqrt {8 a^2-4 c a+b^2}\right )}}\right ) a^2}{\left (8 a^2-4 c a+b^2\right )^{3/2} \sqrt {4 a^2+2 c a-b \left (b+\sqrt {8 a^2-4 c a+b^2}\right )}}-\frac {(3 a-c) \log \left (2 a x^2+\left (b-\sqrt {8 a^2-4 c a+b^2}\right ) x+2 a\right ) a^2}{\left (8 a^2-4 c a+b^2\right )^{3/2}}+\frac {(3 a-c) \log \left (2 a x^2+\left (b+\sqrt {8 a^2-4 c a+b^2}\right ) x+2 a\right ) a^2}{\left (8 a^2-4 c a+b^2\right )^{3/2}}-\frac {2 \left (\left (b-\sqrt {8 a^2-4 c a+b^2}\right ) \left (2 a^2+2 c a-b \left (b-\sqrt {8 a^2-4 c a+b^2}\right )\right )-2 a \left (b^2-\sqrt {8 a^2-4 c a+b^2} b-2 a c\right ) x\right ) a^2}{\left (8 a^2-4 c a+b^2\right ) \left (16 a^2-\left (b-\sqrt {8 a^2-4 c a+b^2}\right )^2\right ) \left (2 a x^2+\left (b-\sqrt {8 a^2-4 c a+b^2}\right ) x+2 a\right )}-\frac {2 \left (\left (b+\sqrt {8 a^2-4 c a+b^2}\right ) \left (2 a^2+2 c a-b \left (b+\sqrt {8 a^2-4 c a+b^2}\right )\right )-2 a \left (b^2+\sqrt {8 a^2-4 c a+b^2} b-2 a c\right ) x\right ) a^2}{\left (8 a^2-4 c a+b^2\right ) \left (16 a^2-\left (b+\sqrt {8 a^2-4 c a+b^2}\right )^2\right ) \left (2 a x^2+\left (b+\sqrt {8 a^2-4 c a+b^2}\right ) x+2 a\right )}}{a^2}\) |
Input:
Int[(a + b*x + c*x^2 + b*x^3 + a*x^4)^(-2),x]
Output:
((-2*a^2*((b - Sqrt[8*a^2 + b^2 - 4*a*c])*(2*a^2 + 2*a*c - b*(b - Sqrt[8*a ^2 + b^2 - 4*a*c])) - 2*a*(b^2 - 2*a*c - b*Sqrt[8*a^2 + b^2 - 4*a*c])*x))/ ((8*a^2 + b^2 - 4*a*c)*(16*a^2 - (b - Sqrt[8*a^2 + b^2 - 4*a*c])^2)*(2*a + (b - Sqrt[8*a^2 + b^2 - 4*a*c])*x + 2*a*x^2)) - (2*a^2*((b + Sqrt[8*a^2 + b^2 - 4*a*c])*(2*a^2 + 2*a*c - b*(b + Sqrt[8*a^2 + b^2 - 4*a*c])) - 2*a*( b^2 - 2*a*c + b*Sqrt[8*a^2 + b^2 - 4*a*c])*x))/((8*a^2 + b^2 - 4*a*c)*(16* a^2 - (b + Sqrt[8*a^2 + b^2 - 4*a*c])^2)*(2*a + (b + Sqrt[8*a^2 + b^2 - 4* a*c])*x + 2*a*x^2)) + (2*Sqrt[2]*a^3*(b^2 - 2*a*c - b*Sqrt[8*a^2 + b^2 - 4 *a*c])*ArcTan[(b - Sqrt[8*a^2 + b^2 - 4*a*c] + 4*a*x)/(Sqrt[2]*Sqrt[4*a^2 + 2*a*c - b*(b - Sqrt[8*a^2 + b^2 - 4*a*c])])])/((8*a^2 + b^2 - 4*a*c)*(4* a^2 + 2*a*c - b*(b - Sqrt[8*a^2 + b^2 - 4*a*c]))^(3/2)) - (Sqrt[2]*a^2*(5* a*b - b*c - (3*a - c)*Sqrt[8*a^2 + b^2 - 4*a*c])*ArcTan[(b - Sqrt[8*a^2 + b^2 - 4*a*c] + 4*a*x)/(Sqrt[2]*Sqrt[4*a^2 + 2*a*c - b*(b - Sqrt[8*a^2 + b^ 2 - 4*a*c])])])/((8*a^2 + b^2 - 4*a*c)^(3/2)*Sqrt[4*a^2 + 2*a*c - b*(b - S qrt[8*a^2 + b^2 - 4*a*c])]) + (2*Sqrt[2]*a^3*(b^2 - 2*a*c + b*Sqrt[8*a^2 + b^2 - 4*a*c])*ArcTan[(b + Sqrt[8*a^2 + b^2 - 4*a*c] + 4*a*x)/(Sqrt[2]*Sqr t[4*a^2 + 2*a*c - b*(b + Sqrt[8*a^2 + b^2 - 4*a*c])])])/((8*a^2 + b^2 - 4* a*c)*(4*a^2 + 2*a*c - b*(b + Sqrt[8*a^2 + b^2 - 4*a*c]))^(3/2)) - (Sqrt[2] *a^2*(c*(b + Sqrt[8*a^2 + b^2 - 4*a*c]) - a*(5*b + 3*Sqrt[8*a^2 + b^2 - 4* a*c]))*ArcTan[(b + Sqrt[8*a^2 + b^2 - 4*a*c] + 4*a*x)/(Sqrt[2]*Sqrt[4*a...
Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) ^(p_), x_Symbol] :> Simp[e^p Int[ExpandIntegrand[Px*(b/d + ((d + Sqrt[e*( (b^2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p*(b/d + ((d - Sqrt[e*((b^ 2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && EqQ[a*d^2 - b^2*e, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.94 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(\frac {-\frac {2 a \left (2 a c -3 b^{2}+c^{2}\right ) x^{3}}{\left (8 a^{2}-4 a c +b^{2}\right ) \left (4 a^{2}+4 a c -4 b^{2}+c^{2}\right )}-\frac {\left (6 a^{2}+3 a c -6 b^{2}+2 c^{2}\right ) b \,x^{2}}{\left (4 a^{2}+4 a c -4 b^{2}+c^{2}\right ) \left (8 a^{2}-4 a c +b^{2}\right )}+\frac {\left (8 a^{3}+4 a^{2} c -8 b^{2} a -4 a \,c^{2}+7 c \,b^{2}-2 c^{3}\right ) x}{\left (8 a^{2}-4 a c +b^{2}\right ) \left (4 a^{2}+4 a c -4 b^{2}+c^{2}\right )}+\frac {b \left (2 a^{2}-5 a c +4 b^{2}-c^{2}\right )}{32 a^{4}+16 a^{3} c -28 a^{2} b^{2}-8 a^{2} c^{2}+20 a \,b^{2} c -4 a \,c^{3}-4 b^{4}+b^{2} c^{2}}}{a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{4}+b \,\textit {\_Z}^{3}+\textit {\_Z}^{2} c +b \textit {\_Z} +a \right )}{\sum }\frac {\left (-\frac {a \left (2 a c -3 b^{2}+c^{2}\right ) \textit {\_R}^{2}}{\left (8 a^{2}-4 a c +b^{2}\right ) \left (4 a^{2}+4 a c -4 b^{2}+c^{2}\right )}-\frac {b \left (6 a^{2}+a c -3 b^{2}+c^{2}\right ) \textit {\_R}}{\left (8 a^{2}-4 a c +b^{2}\right ) \left (4 a^{2}+4 a c -4 b^{2}+c^{2}\right )}+\frac {12 a^{3}+6 a^{2} c -9 b^{2} a -2 a \,c^{2}+4 c \,b^{2}-c^{3}}{\left (8 a^{2}-4 a c +b^{2}\right ) \left (4 a^{2}+4 a c -4 b^{2}+c^{2}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3} a +3 \textit {\_R}^{2} b +2 \textit {\_R} c +b}\right )\) | \(525\) |
| default | \(\text {Expression too large to display}\) | \(3166\) |
Input:
int(1/(a*x^4+b*x^3+c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
(-2*a*(2*a*c-3*b^2+c^2)/(8*a^2-4*a*c+b^2)/(4*a^2+4*a*c-4*b^2+c^2)*x^3-(6*a ^2+3*a*c-6*b^2+2*c^2)*b/(4*a^2+4*a*c-4*b^2+c^2)/(8*a^2-4*a*c+b^2)*x^2+(8*a ^3+4*a^2*c-8*a*b^2-4*a*c^2+7*b^2*c-2*c^3)/(8*a^2-4*a*c+b^2)/(4*a^2+4*a*c-4 *b^2+c^2)*x+b*(2*a^2-5*a*c+4*b^2-c^2)/(32*a^4+16*a^3*c-28*a^2*b^2-8*a^2*c^ 2+20*a*b^2*c-4*a*c^3-4*b^4+b^2*c^2))/(a*x^4+b*x^3+c*x^2+b*x+a)+2*sum((-a*( 2*a*c-3*b^2+c^2)/(8*a^2-4*a*c+b^2)/(4*a^2+4*a*c-4*b^2+c^2)*_R^2-b*(6*a^2+a *c-3*b^2+c^2)/(8*a^2-4*a*c+b^2)/(4*a^2+4*a*c-4*b^2+c^2)*_R+(12*a^3+6*a^2*c -9*a*b^2-2*a*c^2+4*b^2*c-c^3)/(8*a^2-4*a*c+b^2)/(4*a^2+4*a*c-4*b^2+c^2))/( 4*_R^3*a+3*_R^2*b+2*_R*c+b)*ln(x-_R),_R=RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a ))
Timed out. \[ \int \frac {1}{\left (a+b x+c x^2+b x^3+a x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(a*x^4+b*x^3+c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (a+b x+c x^2+b x^3+a x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(a*x**4+b*x**3+c*x**2+b*x+a)**2,x)
Output:
Timed out
\[ \int \frac {1}{\left (a+b x+c x^2+b x^3+a x^4\right )^2} \, dx=\int { \frac {1}{{\left (a x^{4} + b x^{3} + c x^{2} + b x + a\right )}^{2}} \,d x } \] Input:
integrate(1/(a*x^4+b*x^3+c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
(2*(3*a*b^2 - 2*a^2*c - a*c^2)*x^3 + 2*a^2*b + 4*b^3 - 5*a*b*c - b*c^2 - ( 6*a^2*b - 6*b^3 + 3*a*b*c + 2*b*c^2)*x^2 + (8*a^3 - 8*a*b^2 - 4*a*c^2 - 2* c^3 + (4*a^2 + 7*b^2)*c)*x)/(32*a^5 - 28*a^3*b^2 - 4*a*b^4 - 4*a^2*c^3 + ( 32*a^5 - 28*a^3*b^2 - 4*a*b^4 - 4*a^2*c^3 - (8*a^3 - a*b^2)*c^2 + 4*(4*a^4 + 5*a^2*b^2)*c)*x^4 + (32*a^4*b - 28*a^2*b^3 - 4*b^5 - 4*a*b*c^3 - (8*a^2 *b - b^3)*c^2 + 4*(4*a^3*b + 5*a*b^3)*c)*x^3 - (8*a^3 - a*b^2)*c^2 - (4*a* c^4 + (8*a^2 - b^2)*c^3 - 4*(4*a^3 + 5*a*b^2)*c^2 - 4*(8*a^4 - 7*a^2*b^2 - b^4)*c)*x^2 + 4*(4*a^4 + 5*a^2*b^2)*c + (32*a^4*b - 28*a^2*b^3 - 4*b^5 - 4*a*b*c^3 - (8*a^2*b - b^3)*c^2 + 4*(4*a^3*b + 5*a*b^3)*c)*x) - 2*integrat e(-(12*a^3 - 9*a*b^2 - 2*a*c^2 - c^3 + (3*a*b^2 - 2*a^2*c - a*c^2)*x^2 + 2 *(3*a^2 + 2*b^2)*c - (6*a^2*b - 3*b^3 + a*b*c + b*c^2)*x)/(a*x^4 + b*x^3 + c*x^2 + b*x + a), x)/(32*a^4 - 28*a^2*b^2 - 4*b^4 - 4*a*c^3 - (8*a^2 - b^ 2)*c^2 + 4*(4*a^3 + 5*a*b^2)*c)
\[ \int \frac {1}{\left (a+b x+c x^2+b x^3+a x^4\right )^2} \, dx=\int { \frac {1}{{\left (a x^{4} + b x^{3} + c x^{2} + b x + a\right )}^{2}} \,d x } \] Input:
integrate(1/(a*x^4+b*x^3+c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
integrate((a*x^4 + b*x^3 + c*x^2 + b*x + a)^(-2), x)
Time = 23.47 (sec) , antiderivative size = 12699, normalized size of antiderivative = 11.76 \[ \int \frac {1}{\left (a+b x+c x^2+b x^3+a x^4\right )^2} \, dx=\text {Too large to display} \] Input:
int(1/(a + b*x + a*x^4 + b*x^3 + c*x^2)^2,x)
Output:
symsum(log((576*a^8*c + 216*a^5*b^4 - 40*a^5*c^4 - 80*a^6*c^3 + 288*a^7*c^ 2 - 72*a^4*b^4*c - 720*a^6*b^2*c + 16*a^4*b^2*c^3 + 216*a^5*b^2*c^2)/(4096 *a^9*c + 8192*a^10 + 16*b^10 + 352*a^2*b^8 + 2320*a^4*b^6 + 2432*a^6*b^4 - 13312*a^8*b^2 - 64*a^3*c^7 - 128*a^4*c^6 + 768*a^5*c^5 + 1536*a^6*c^4 - 3 072*a^7*c^3 - 6144*a^8*c^2 + b^6*c^4 - 8*b^8*c^2 - 12*a*b^4*c^5 + 104*a*b^ 6*c^3 - 3424*a^3*b^6*c - 11712*a^5*b^4*c + 11264*a^7*b^2*c + 48*a^2*b^2*c^ 6 - 456*a^2*b^4*c^4 + 984*a^2*b^6*c^2 + 704*a^3*b^2*c^5 - 1120*a^3*b^4*c^3 - 1216*a^4*b^2*c^4 + 9408*a^4*b^4*c^2 - 5632*a^5*b^2*c^3 + 7424*a^6*b^2*c ^2 - 224*a*b^8*c) - root(1728*a*b^16*c*z^4 + 85131264*a^12*b^4*c^2*z^4 + 6 4880640*a^9*b^6*c^3*z^4 - 58195968*a^13*b^2*c^3*z^4 - 45121536*a^10*b^4*c^ 4*z^4 - 41680896*a^10*b^6*c^2*z^4 - 32870400*a^8*b^8*c^2*z^4 - 32194560*a^ 8*b^6*c^4*z^4 + 22806528*a^11*b^2*c^5*z^4 + 22333440*a^7*b^8*c^3*z^4 + 220 20096*a^14*b^2*c^2*z^4 + 14843904*a^9*b^4*c^5*z^4 - 6828288*a^6*b^10*c^2*z ^4 + 5627904*a^8*b^4*c^6*z^4 - 5076480*a^6*b^8*c^4*z^4 - 4128768*a^9*b^2*c ^7*z^4 - 3538944*a^10*b^2*c^6*z^4 - 2998272*a^7*b^4*c^7*z^4 + 2551680*a^5* b^10*c^3*z^4 + 2359296*a^11*b^4*c^3*z^4 + 2326528*a^6*b^6*c^6*z^4 + 208896 0*a^7*b^6*c^5*z^4 + 1572864*a^12*b^2*c^4*z^4 + 835584*a^8*b^2*c^8*z^4 - 58 8048*a^4*b^12*c^2*z^4 - 536064*a^5*b^8*c^5*z^4 + 319488*a^7*b^2*c^9*z^4 - 294912*a^5*b^6*c^7*z^4 + 218880*a^4*b^8*c^6*z^4 - 112320*a^4*b^10*c^4*z^4 + 89088*a^5*b^4*c^9*z^4 + 84384*a^3*b^12*c^3*z^4 - 73824*a^3*b^10*c^5*z...
\[ \int \frac {1}{\left (a+b x+c x^2+b x^3+a x^4\right )^2} \, dx=\text {too large to display} \] Input:
int(1/(a*x^4+b*x^3+c*x^2+b*x+a)^2,x)
Output:
( - 4*int(x**3/(a**2*x**8 + 2*a**2*x**4 + a**2 + 2*a*b*x**7 + 2*a*b*x**5 + 2*a*b*x**3 + 2*a*b*x + 2*a*c*x**6 + 2*a*c*x**2 + b**2*x**6 + 2*b**2*x**4 + b**2*x**2 + 2*b*c*x**5 + 2*b*c*x**3 + c**2*x**4),x)*a**2*x**4 - 4*int(x* *3/(a**2*x**8 + 2*a**2*x**4 + a**2 + 2*a*b*x**7 + 2*a*b*x**5 + 2*a*b*x**3 + 2*a*b*x + 2*a*c*x**6 + 2*a*c*x**2 + b**2*x**6 + 2*b**2*x**4 + b**2*x**2 + 2*b*c*x**5 + 2*b*c*x**3 + c**2*x**4),x)*a**2 - 4*int(x**3/(a**2*x**8 + 2 *a**2*x**4 + a**2 + 2*a*b*x**7 + 2*a*b*x**5 + 2*a*b*x**3 + 2*a*b*x + 2*a*c *x**6 + 2*a*c*x**2 + b**2*x**6 + 2*b**2*x**4 + b**2*x**2 + 2*b*c*x**5 + 2* b*c*x**3 + c**2*x**4),x)*a*b*x**3 - 4*int(x**3/(a**2*x**8 + 2*a**2*x**4 + a**2 + 2*a*b*x**7 + 2*a*b*x**5 + 2*a*b*x**3 + 2*a*b*x + 2*a*c*x**6 + 2*a*c *x**2 + b**2*x**6 + 2*b**2*x**4 + b**2*x**2 + 2*b*c*x**5 + 2*b*c*x**3 + c* *2*x**4),x)*a*b*x - 4*int(x**3/(a**2*x**8 + 2*a**2*x**4 + a**2 + 2*a*b*x** 7 + 2*a*b*x**5 + 2*a*b*x**3 + 2*a*b*x + 2*a*c*x**6 + 2*a*c*x**2 + b**2*x** 6 + 2*b**2*x**4 + b**2*x**2 + 2*b*c*x**5 + 2*b*c*x**3 + c**2*x**4),x)*a*c* x**2 - 3*int(x**2/(a**2*x**8 + 2*a**2*x**4 + a**2 + 2*a*b*x**7 + 2*a*b*x** 5 + 2*a*b*x**3 + 2*a*b*x + 2*a*c*x**6 + 2*a*c*x**2 + b**2*x**6 + 2*b**2*x* *4 + b**2*x**2 + 2*b*c*x**5 + 2*b*c*x**3 + c**2*x**4),x)*a*b*x**4 - 3*int( x**2/(a**2*x**8 + 2*a**2*x**4 + a**2 + 2*a*b*x**7 + 2*a*b*x**5 + 2*a*b*x** 3 + 2*a*b*x + 2*a*c*x**6 + 2*a*c*x**2 + b**2*x**6 + 2*b**2*x**4 + b**2*x** 2 + 2*b*c*x**5 + 2*b*c*x**3 + c**2*x**4),x)*a*b - 3*int(x**2/(a**2*x**8...