Integrand size = 18, antiderivative size = 252 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=-\frac {2 \left (2 b^2-5 a c\right )}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (2 b^2-7 a c\right )}{a^3 \left (b^2-4 a c\right ) x^2}-\frac {2 \left (2 b^4-9 a b^2 c+5 a^2 c^2\right )}{a^4 \left (b^2-4 a c\right ) x}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^3 \left (a+b x+c x^2\right )}-\frac {2 \left (2 b^6-15 a b^4 c+30 a^2 b^2 c^2-10 a^3 c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{3/2}}-\frac {2 b \left (2 b^2-3 a c\right ) \log (x)}{a^5}+\frac {b \left (2 b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{a^5} \] Output:
1/3*(10*a*c-4*b^2)/a^2/(-4*a*c+b^2)/x^3+b*(-7*a*c+2*b^2)/a^3/(-4*a*c+b^2)/ x^2-2*(5*a^2*c^2-9*a*b^2*c+2*b^4)/a^4/(-4*a*c+b^2)/x+(b*c*x-2*a*c+b^2)/a/( -4*a*c+b^2)/x^3/(c*x^2+b*x+a)-2*(-10*a^3*c^3+30*a^2*b^2*c^2-15*a*b^4*c+2*b ^6)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/a^5/(-4*a*c+b^2)^(3/2)-2*b*(-3*a *c+2*b^2)*ln(x)/a^5+b*(-3*a*c+2*b^2)*ln(c*x^2+b*x+a)/a^5
Time = 0.35 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {-\frac {a^3}{x^3}+\frac {3 a^2 b}{x^2}+\frac {3 a \left (-3 b^2+2 a c\right )}{x}-\frac {3 a \left (b^5-5 a b^3 c+5 a^2 b c^2+b^4 c x-4 a b^2 c^2 x+2 a^2 c^3 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {6 \left (2 b^6-15 a b^4 c+30 a^2 b^2 c^2-10 a^3 c^3\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+6 \left (-2 b^3+3 a b c\right ) \log (x)+3 \left (2 b^3-3 a b c\right ) \log (a+x (b+c x))}{3 a^5} \] Input:
Integrate[(a*x^2 + b*x^3 + c*x^4)^(-2),x]
Output:
(-(a^3/x^3) + (3*a^2*b)/x^2 + (3*a*(-3*b^2 + 2*a*c))/x - (3*a*(b^5 - 5*a*b ^3*c + 5*a^2*b*c^2 + b^4*c*x - 4*a*b^2*c^2*x + 2*a^2*c^3*x))/((b^2 - 4*a*c )*(a + x*(b + c*x))) - (6*(2*b^6 - 15*a*b^4*c + 30*a^2*b^2*c^2 - 10*a^3*c^ 3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 6*(-2*b^ 3 + 3*a*b*c)*Log[x] + 3*(2*b^3 - 3*a*b*c)*Log[a + x*(b + c*x)])/(3*a^5)
Time = 0.55 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1949, 1165, 27, 1200, 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.
| \(\displaystyle \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1949 |
| \(\displaystyle \int \frac {1}{x^4 \left (a+b x+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {-2 a c+b^2+b c x}{a x^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {2 \left (2 b^2+2 c x b-5 a c\right )}{x^4 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {2 b^2+2 c x b-5 a c}{x^4 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {2 \int \left (\frac {2 b^2-5 a c}{a x^4}+\frac {b \left (b^2-4 a c\right ) \left (3 a c-2 b^2\right )}{a^4 x}+\frac {2 b^6-13 a c b^4+21 a^2 c^2 b^2+c \left (b^2-4 a c\right ) \left (2 b^2-3 a c\right ) x b-5 a^3 c^3}{a^4 \left (c x^2+b x+a\right )}+\frac {2 b^4-9 a c b^2+5 a^2 c^2}{a^3 x^2}+\frac {b \left (7 a c-2 b^2\right )}{a^2 x^3}\right )dx}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {b \left (b^2-4 a c\right ) \left (2 b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac {b \log (x) \left (b^2-4 a c\right ) \left (2 b^2-3 a c\right )}{a^4}+\frac {b \left (2 b^2-7 a c\right )}{2 a^2 x^2}-\frac {5 a^2 c^2-9 a b^2 c+2 b^4}{a^3 x}-\frac {\left (-10 a^3 c^3+30 a^2 b^2 c^2-15 a b^4 c+2 b^6\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c}}-\frac {2 b^2-5 a c}{3 a x^3}\right )}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{a x^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
Input:
Int[(a*x^2 + b*x^3 + c*x^4)^(-2),x]
Output:
(b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*x^3*(a + b*x + c*x^2)) + (2*(-1/3*( 2*b^2 - 5*a*c)/(a*x^3) + (b*(2*b^2 - 7*a*c))/(2*a^2*x^2) - (2*b^4 - 9*a*b^ 2*c + 5*a^2*c^2)/(a^3*x) - ((2*b^6 - 15*a*b^4*c + 30*a^2*b^2*c^2 - 10*a^3* c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^4*Sqrt[b^2 - 4*a*c]) - (b* (b^2 - 4*a*c)*(2*b^2 - 3*a*c)*Log[x])/a^4 + (b*(b^2 - 4*a*c)*(2*b^2 - 3*a* c)*Log[a + b*x + c*x^2])/(2*a^4)))/(a*(b^2 - 4*a*c))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_), x_Symbol ] :> Int[x^(p*q)*(a + b*x^(n - q) + c*x^(2*(n - q)))^p, x] /; FreeQ[{a, b, c, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && IntegerQ[p]
Time = 0.10 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(\frac {\frac {\frac {a c \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) x}{4 a c -b^{2}}+\frac {a b \left (5 a^{2} c^{2}-5 a \,b^{2} c +b^{4}\right )}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-12 a^{2} b \,c^{3}+11 a \,b^{3} c^{2}-2 b^{5} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {4 \left (5 a^{3} c^{3}-21 a^{2} b^{2} c^{2}+13 a \,b^{4} c -2 b^{6}-\frac {\left (-12 a^{2} b \,c^{3}+11 a \,b^{3} c^{2}-2 b^{5} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{a^{5}}-\frac {1}{3 a^{2} x^{3}}-\frac {-2 a c +3 b^{2}}{x \,a^{4}}+\frac {b}{a^{3} x^{2}}+\frac {2 b \left (3 a c -2 b^{2}\right ) \ln \left (x \right )}{a^{5}}\) | \(295\) |
| risch | \(\frac {\frac {2 c \left (5 a^{2} c^{2}-9 a \,b^{2} c +2 b^{4}\right ) x^{4}}{\left (4 a c -b^{2}\right ) a^{4}}+\frac {b \left (17 a^{2} c^{2}-20 a \,b^{2} c +4 b^{4}\right ) x^{3}}{a^{4} \left (4 a c -b^{2}\right )}+\frac {\left (5 a c -6 b^{2}\right ) x^{2}}{3 a^{3}}+\frac {2 b x}{3 a^{2}}-\frac {1}{3 a}}{x^{3} \left (c \,x^{2}+b x +a \right )}+\frac {6 b \ln \left (x \right ) c}{a^{4}}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 c^{3} a^{8}-48 a^{7} b^{2} c^{2}+12 a^{6} b^{4} c -a^{5} b^{6}\right ) \textit {\_Z}^{2}+\left (192 a^{4} b \,c^{4}-272 a^{3} b^{3} c^{3}+132 a^{2} b^{5} c^{2}-27 a \,b^{7} c +2 b^{9}\right ) \textit {\_Z} +25 a \,c^{6}-6 b^{2} c^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (96 a^{11} c^{3}-80 a^{10} b^{2} c^{2}+22 a^{9} b^{4} c -2 a^{8} b^{6}\right ) \textit {\_R}^{2}+\left (124 a^{7} b \,c^{4}-127 a^{6} b^{3} c^{3}+40 a^{5} b^{5} c^{2}-4 a^{4} b^{7} c \right ) \textit {\_R} +25 a^{4} c^{6}-90 a^{3} b^{2} c^{5}+101 a^{2} b^{4} c^{4}-36 a \,b^{6} c^{3}+4 b^{8} c^{2}\right ) x +\left (-16 a^{11} b \,c^{2}+8 a^{10} b^{3} c -a^{9} b^{5}\right ) \textit {\_R}^{2}+\left (-20 a^{8} c^{4}+89 a^{7} b^{2} c^{3}-73 a^{6} b^{4} c^{2}+21 a^{5} b^{6} c -2 a^{4} b^{8}\right ) \textit {\_R} +60 a^{4} b \,c^{5}-163 a^{3} b^{3} c^{4}+133 a^{2} b^{5} c^{3}-40 a \,b^{7} c^{2}+4 b^{9} c \right )\right )\) | \(524\) |
Input:
int(1/(c*x^4+b*x^3+a*x^2)^2,x,method=_RETURNVERBOSE)
Output:
1/a^5*((a*c*(2*a^2*c^2-4*a*b^2*c+b^4)/(4*a*c-b^2)*x+a*b*(5*a^2*c^2-5*a*b^2 *c+b^4)/(4*a*c-b^2))/(c*x^2+b*x+a)+2/(4*a*c-b^2)*(1/2*(-12*a^2*b*c^3+11*a* b^3*c^2-2*b^5*c)/c*ln(c*x^2+b*x+a)+2*(5*a^3*c^3-21*a^2*b^2*c^2+13*a*b^4*c- 2*b^6-1/2*(-12*a^2*b*c^3+11*a*b^3*c^2-2*b^5*c)*b/c)/(4*a*c-b^2)^(1/2)*arct an((2*c*x+b)/(4*a*c-b^2)^(1/2))))-1/3/a^2/x^3-(-2*a*c+3*b^2)/x/a^4+1/a^3*b /x^2+2*b*(3*a*c-2*b^2)/a^5*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (246) = 492\).
Time = 0.42 (sec) , antiderivative size = 1407, normalized size of antiderivative = 5.58 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="fricas")
Output:
[-1/3*(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 6*(2*a*b^6*c - 17*a^2*b^4*c^2 + 41*a^3*b^2*c^3 - 20*a^4*c^4)*x^4 + 3*(4*a*b^7 - 36*a^2*b^5*c + 97*a^3*b^ 3*c^2 - 68*a^4*b*c^3)*x^3 + (6*a^2*b^6 - 53*a^3*b^4*c + 136*a^4*b^2*c^2 - 80*a^5*c^3)*x^2 - 3*((2*b^6*c - 15*a*b^4*c^2 + 30*a^2*b^2*c^3 - 10*a^3*c^4 )*x^5 + (2*b^7 - 15*a*b^5*c + 30*a^2*b^3*c^2 - 10*a^3*b*c^3)*x^4 + (2*a*b^ 6 - 15*a^2*b^4*c + 30*a^3*b^2*c^2 - 10*a^4*c^3)*x^3)*sqrt(b^2 - 4*a*c)*log ((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^ 2 + b*x + a)) - 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x - 3*((2*b^7*c - 19*a*b^5*c^2 + 56*a^2*b^3*c^3 - 48*a^3*b*c^4)*x^5 + (2*b^8 - 19*a*b^6*c + 56*a^2*b^4*c^2 - 48*a^3*b^2*c^3)*x^4 + (2*a*b^7 - 19*a^2*b^5*c + 56*a^3*b ^3*c^2 - 48*a^4*b*c^3)*x^3)*log(c*x^2 + b*x + a) + 6*((2*b^7*c - 19*a*b^5* c^2 + 56*a^2*b^3*c^3 - 48*a^3*b*c^4)*x^5 + (2*b^8 - 19*a*b^6*c + 56*a^2*b^ 4*c^2 - 48*a^3*b^2*c^3)*x^4 + (2*a*b^7 - 19*a^2*b^5*c + 56*a^3*b^3*c^2 - 4 8*a^4*b*c^3)*x^3)*log(x))/((a^5*b^4*c - 8*a^6*b^2*c^2 + 16*a^7*c^3)*x^5 + (a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^4 + (a^6*b^4 - 8*a^7*b^2*c + 16*a ^8*c^2)*x^3), -1/3*(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 6*(2*a*b^6*c - 17 *a^2*b^4*c^2 + 41*a^3*b^2*c^3 - 20*a^4*c^4)*x^4 + 3*(4*a*b^7 - 36*a^2*b^5* c + 97*a^3*b^3*c^2 - 68*a^4*b*c^3)*x^3 + (6*a^2*b^6 - 53*a^3*b^4*c + 136*a ^4*b^2*c^2 - 80*a^5*c^3)*x^2 + 6*((2*b^6*c - 15*a*b^4*c^2 + 30*a^2*b^2*c^3 - 10*a^3*c^4)*x^5 + (2*b^7 - 15*a*b^5*c + 30*a^2*b^3*c^2 - 10*a^3*b*c^...
Timed out. \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(c*x**4+b*x**3+a*x**2)**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.13 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {2 \, {\left (2 \, b^{6} - 15 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 10 \, a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{2} - 4 \, a^{6} c\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (2 \, b^{3} - 3 \, a b c\right )} \log \left (c x^{2} + b x + a\right )}{a^{5}} - \frac {2 \, {\left (2 \, b^{3} - 3 \, a b c\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {a^{4} b^{2} - 4 \, a^{5} c + 6 \, {\left (2 \, a b^{4} c - 9 \, a^{2} b^{2} c^{2} + 5 \, a^{3} c^{3}\right )} x^{4} + 3 \, {\left (4 \, a b^{5} - 20 \, a^{2} b^{3} c + 17 \, a^{3} b c^{2}\right )} x^{3} + {\left (6 \, a^{2} b^{4} - 29 \, a^{3} b^{2} c + 20 \, a^{4} c^{2}\right )} x^{2} - 2 \, {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x}{3 \, {\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{5} x^{3}} \] Input:
integrate(1/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="giac")
Output:
2*(2*b^6 - 15*a*b^4*c + 30*a^2*b^2*c^2 - 10*a^3*c^3)*arctan((2*c*x + b)/sq rt(-b^2 + 4*a*c))/((a^5*b^2 - 4*a^6*c)*sqrt(-b^2 + 4*a*c)) + (2*b^3 - 3*a* b*c)*log(c*x^2 + b*x + a)/a^5 - 2*(2*b^3 - 3*a*b*c)*log(abs(x))/a^5 - 1/3* (a^4*b^2 - 4*a^5*c + 6*(2*a*b^4*c - 9*a^2*b^2*c^2 + 5*a^3*c^3)*x^4 + 3*(4* a*b^5 - 20*a^2*b^3*c + 17*a^3*b*c^2)*x^3 + (6*a^2*b^4 - 29*a^3*b^2*c + 20* a^4*c^2)*x^2 - 2*(a^3*b^3 - 4*a^4*b*c)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c) *a^5*x^3)
Time = 22.44 (sec) , antiderivative size = 1120, normalized size of antiderivative = 4.44 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/(a*x^2 + b*x^3 + c*x^4)^2,x)
Output:
((x^2*(5*a*c - 6*b^2))/(3*a^3) - 1/(3*a) + (2*b*x)/(3*a^2) + (x^3*(4*b^5 + 17*a^2*b*c^2 - 20*a*b^3*c))/(a^4*(4*a*c - b^2)) + (2*c*x^4*(2*b^4 + 5*a^2 *c^2 - 9*a*b^2*c))/(a^4*(4*a*c - b^2)))/(a*x^3 + b*x^4 + c*x^5) + (log(4*a *b^9 + 4*b^10*x - 4*a*b^6*(-(4*a*c - b^2)^3)^(1/2) - 52*a^2*b^7*c + 308*a^ 5*b*c^4 - 40*a^5*c^5*x - 4*b^7*x*(-(4*a*c - b^2)^3)^(1/2) + 243*a^3*b^5*c^ 2 - 473*a^4*b^3*c^3 + 5*a^4*c^3*(-(4*a*c - b^2)^3)^(1/2) + 24*a^2*b^4*c*(- (4*a*c - b^2)^3)^(1/2) + 266*a^2*b^6*c^2*x - 563*a^3*b^4*c^3*x + 438*a^4*b ^2*c^4*x - 54*a*b^8*c*x - 33*a^3*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 30*a*b ^5*c*x*(-(4*a*c - b^2)^3)^(1/2) + 41*a^3*b*c^3*x*(-(4*a*c - b^2)^3)^(1/2) - 66*a^2*b^3*c^2*x*(-(4*a*c - b^2)^3)^(1/2))*(a^2*(132*b^5*c^2 - 30*b^2*c^ 2*(-(4*a*c - b^2)^3)^(1/2)) - a^3*(272*b^3*c^3 - 10*c^3*(-(4*a*c - b^2)^3) ^(1/2)) + 2*b^9 - 2*b^6*(-(4*a*c - b^2)^3)^(1/2) - a*(27*b^7*c - 15*b^4*c* (-(4*a*c - b^2)^3)^(1/2)) + 192*a^4*b*c^4))/(a^5*b^6 - 64*a^8*c^3 - 12*a^6 *b^4*c + 48*a^7*b^2*c^2) + (log(4*a*b^9 + 4*b^10*x + 4*a*b^6*(-(4*a*c - b^ 2)^3)^(1/2) - 52*a^2*b^7*c + 308*a^5*b*c^4 - 40*a^5*c^5*x + 4*b^7*x*(-(4*a *c - b^2)^3)^(1/2) + 243*a^3*b^5*c^2 - 473*a^4*b^3*c^3 - 5*a^4*c^3*(-(4*a* c - b^2)^3)^(1/2) - 24*a^2*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 266*a^2*b^6*c^ 2*x - 563*a^3*b^4*c^3*x + 438*a^4*b^2*c^4*x - 54*a*b^8*c*x + 33*a^3*b^2*c^ 2*(-(4*a*c - b^2)^3)^(1/2) - 30*a*b^5*c*x*(-(4*a*c - b^2)^3)^(1/2) - 41*a^ 3*b*c^3*x*(-(4*a*c - b^2)^3)^(1/2) + 66*a^2*b^3*c^2*x*(-(4*a*c - b^2)^3...
Time = 0.16 (sec) , antiderivative size = 1276, normalized size of antiderivative = 5.06 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/(c*x^4+b*x^3+a*x^2)^2,x)
Output:
(60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b*c**3*x* *3 - 180*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**3 *c**2*x**3 + 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a* *3*b**2*c**3*x**4 + 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* *2))*a**3*b*c**4*x**5 + 90*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**5*c*x**3 - 180*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 *a*c - b**2))*a**2*b**4*c**2*x**4 - 180*sqrt(4*a*c - b**2)*atan((b + 2*c*x )/sqrt(4*a*c - b**2))*a**2*b**3*c**3*x**5 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**7*x**3 + 90*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**6*c*x**4 + 90*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**5*c**2*x**5 - 12*sqrt(4*a*c - b**2)*atan ((b + 2*c*x)/sqrt(4*a*c - b**2))*b**8*x**4 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**7*c*x**5 - 144*log(a + b*x + c*x**2)*a**4 *b**2*c**3*x**3 + 168*log(a + b*x + c*x**2)*a**3*b**4*c**2*x**3 - 144*log( a + b*x + c*x**2)*a**3*b**3*c**3*x**4 - 144*log(a + b*x + c*x**2)*a**3*b** 2*c**4*x**5 - 57*log(a + b*x + c*x**2)*a**2*b**6*c*x**3 + 168*log(a + b*x + c*x**2)*a**2*b**5*c**2*x**4 + 168*log(a + b*x + c*x**2)*a**2*b**4*c**3*x **5 + 6*log(a + b*x + c*x**2)*a*b**8*x**3 - 57*log(a + b*x + c*x**2)*a*b** 7*c*x**4 - 57*log(a + b*x + c*x**2)*a*b**6*c**2*x**5 + 6*log(a + b*x + c*x **2)*b**9*x**4 + 6*log(a + b*x + c*x**2)*b**8*c*x**5 + 288*log(x)*a**4*...