\(\int \frac {1}{(a x^2+b x^3+c x^4)^2} \, dx\) [27]

Optimal result

Integrand size = 18, antiderivative size = 252 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=-\frac {2 \left (2 b^2-5 a c\right )}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (2 b^2-7 a c\right )}{a^3 \left (b^2-4 a c\right ) x^2}-\frac {2 \left (2 b^4-9 a b^2 c+5 a^2 c^2\right )}{a^4 \left (b^2-4 a c\right ) x}+\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) x^3 \left (a+b x+c x^2\right )}-\frac {2 \left (2 b^6-15 a b^4 c+30 a^2 b^2 c^2-10 a^3 c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{3/2}}-\frac {2 b \left (2 b^2-3 a c\right ) \log (x)}{a^5}+\frac {b \left (2 b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{a^5} \]

[Out]

-2/3*(-5*a*c+2*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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1608, 754, 814, 648, 632, 212, 642} \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {b \left (2 b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{a^5}-\frac {2 b \log (x) \left (2 b^2-3 a c\right )}{a^5}+\frac {b \left (2 b^2-7 a c\right )}{a^3 x^2 \left (b^2-4 a c\right )}-\frac {2 \left (2 b^2-5 a c\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}-\frac {2 \left (5 a^2 c^2-9 a b^2 c+2 b^4\right )}{a^4 x \left (b^2-4 a c\right )}-\frac {2 \left (-10 a^3 c^3+30 a^2 b^2 c^2-15 a b^4 c+2 b^6\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{3/2}}+\frac {-2 a c+b^2+b c x}{a x^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^(-2),x]

[Out]

(-2*(2*b^2 - 5*a*c))/(3*a^2*(b^2 - 4*a*c)*x^3) + (b*(2*b^2 - 7*a*c))/(a^3*(b^2 - 4*a*c)*x^2) - (2*(2*b^4 - 9*a
*b^2*c + 5*a^2*c^2))/(a^4*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*x^3*(a + b*x + c*x^2)) - (
2*(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^5*(b^2 - 4*a*c
)^(3/2)) - (2*b*(2*b^2 - 3*a*c)*Log[x])/a^5 + (b*(2*b^2 - 3*a*c)*Log[a + b*x + c*x^2])/a^5

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> 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] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^4 \left (a+b x+c x^2\right )^2} \, dx \\ & = \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 {\int \frac {-2 \left (2 b^2-5 a c\right )-4 b c x}{x^4 \left (a+b x+c x^2\right )} \, dx}{a \left (b^2-4 a c\right )} \\ & = \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 {\int \left (\frac {2 \left (-2 b^2+5 a c\right )}{a x^4}-\frac {2 \left (-2 b^3+7 a b c\right )}{a^2 x^3}-\frac {2 \left (2 b^4-9 a b^2 c+5 a^2 c^2\right )}{a^3 x^2}+\frac {2 b \left (b^2-4 a c\right ) \left (2 b^2-3 a c\right )}{a^4 x}+\frac {2 \left (-2 b^6+13 a b^4 c-21 a^2 b^2 c^2+5 a^3 c^3-b c \left (b^2-4 a c\right ) \left (2 b^2-3 a c\right ) x\right )}{a^4 \left (a+b x+c x^2\right )}\right ) \, dx}{a \left (b^2-4 a c\right )} \\ & = -\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 b \left (2 b^2-3 a c\right ) \log (x)}{a^5}-\frac {2 \int \frac {-2 b^6+13 a b^4 c-21 a^2 b^2 c^2+5 a^3 c^3-b c \left (b^2-4 a c\right ) \left (2 b^2-3 a c\right ) x}{a+b x+c x^2} \, dx}{a^5 \left (b^2-4 a c\right )} \\ & = -\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 b \left (2 b^2-3 a c\right ) \log (x)}{a^5}+\frac {\left (b \left (2 b^2-3 a c\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{a^5}+\frac {\left (2 b^6-15 a b^4 c+30 a^2 b^2 c^2-10 a^3 c^3\right ) \int \frac {1}{a+b x+c x^2} \, dx}{a^5 \left (b^2-4 a c\right )} \\ & = -\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 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}-\frac {\left (2 \left (2 b^6-15 a b^4 c+30 a^2 b^2 c^2-10 a^3 c^3\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^5 \left (b^2-4 a c\right )} \\ & = -\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 ) \tanh ^{-1}\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {-\frac {a^3}{x^3}+\frac {3 a^2 b}{x^2}+\frac {3 a \left (-3 b^2+2 a c\right )}{x}-\frac {3 a \left (b^5-5 a b^3 c+5 a^2 b c^2+b^4 c x-4 a b^2 c^2 x+2 a^2 c^3 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {6 \left (2 b^6-15 a b^4 c+30 a^2 b^2 c^2-10 a^3 c^3\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+6 \left (-2 b^3+3 a b c\right ) \log (x)+3 \left (2 b^3-3 a b c\right ) \log (a+x (b+c x))}{3 a^5} \]

[In]

Integrate[(a*x^2 + b*x^3 + c*x^4)^(-2),x]

[Out]

(-(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)

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.17

method	result	size
default	\(-\frac {1}{3 a^{2} x^{3}}-\frac {-2 a c +3 b^{2}}{x \,a^{4}}+\frac {b}{a^{3} x^{2}}+\frac {2 b \left (3 a c -2 b^{2}\right ) \ln \left (x \right )}{a^{5}}+\frac {\frac {\frac {a c \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) x}{4 a c -b^{2}}+\frac {a b \left (5 a^{2} c^{2}-5 a \,b^{2} c +b^{4}\right )}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-12 a^{2} b \,c^{3}+11 a \,b^{3} c^{2}-2 b^{5} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {4 \left (5 c^{3} a^{3}-21 a^{2} b^{2} c^{2}+13 a \,b^{4} c -2 b^{6}-\frac {\left (-12 a^{2} b \,c^{3}+11 a \,b^{3} c^{2}-2 b^{5} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{a^{5}}\)	\(295\)
risch	\(\frac {\frac {2 c \left (5 a^{2} c^{2}-9 a \,b^{2} c +2 b^{4}\right ) x^{4}}{\left (4 a c -b^{2}\right ) a^{4}}+\frac {b \left (17 a^{2} c^{2}-20 a \,b^{2} c +4 b^{4}\right ) x^{3}}{a^{4} \left (4 a c -b^{2}\right )}+\frac {\left (5 a c -6 b^{2}\right ) x^{2}}{3 a^{3}}+\frac {2 b x}{3 a^{2}}-\frac {1}{3 a}}{x^{3} \left (c \,x^{2}+b x +a \right )}+\frac {6 b \ln \left (x \right ) c}{a^{4}}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 c^{3} a^{8}-48 c^{2} a^{7} b^{2}+12 c \,a^{6} b^{4}-a^{5} b^{6}\right ) \textit {\_Z}^{2}+\left (192 c^{4} b \,a^{4}-272 c^{3} a^{3} b^{3}+132 c^{2} a^{2} b^{5}-27 c \,b^{7} a +2 b^{9}\right ) \textit {\_Z} +25 a \,c^{6}-6 b^{2} c^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (96 a^{11} c^{3}-80 a^{10} b^{2} c^{2}+22 a^{9} b^{4} c -2 a^{8} b^{6}\right ) \textit {\_R}^{2}+\left (124 a^{7} b \,c^{4}-127 a^{6} b^{3} c^{3}+40 a^{5} b^{5} c^{2}-4 a^{4} b^{7} c \right ) \textit {\_R} +25 a^{4} c^{6}-90 a^{3} b^{2} c^{5}+101 a^{2} b^{4} c^{4}-36 a \,b^{6} c^{3}+4 b^{8} c^{2}\right ) x +\left (-16 a^{11} b \,c^{2}+8 a^{10} b^{3} c -a^{9} b^{5}\right ) \textit {\_R}^{2}+\left (-20 a^{8} c^{4}+89 a^{7} b^{2} c^{3}-73 a^{6} b^{4} c^{2}+21 a^{5} b^{6} c -2 a^{4} b^{8}\right ) \textit {\_R} +60 a^{4} b \,c^{5}-163 a^{3} b^{3} c^{4}+133 a^{2} b^{5} c^{3}-40 a \,b^{7} c^{2}+4 b^{9} c \right )\right )\)	\(524\)

[In]

int(1/(c*x^4+b*x^3+a*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

-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)+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*c^3*a^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)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (246) = 492\).

Time = 0.56 (sec) , antiderivative size = 1407, normalized size of antiderivative = 5.58 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="fricas")

[Out]

[-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)*lo
g(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 + 5
6*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(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^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)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 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 - 48*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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(c*x**4+b*x**3+a*x**2)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {2 \, {\left (2 \, b^{6} - 15 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 10 \, a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{2} - 4 \, a^{6} c\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (2 \, b^{3} - 3 \, a b c\right )} \log \left (c x^{2} + b x + a\right )}{a^{5}} - \frac {2 \, {\left (2 \, b^{3} - 3 \, a b c\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {a^{4} b^{2} - 4 \, a^{5} c + 6 \, {\left (2 \, a b^{4} c - 9 \, a^{2} b^{2} c^{2} + 5 \, a^{3} c^{3}\right )} x^{4} + 3 \, {\left (4 \, a b^{5} - 20 \, a^{2} b^{3} c + 17 \, a^{3} b c^{2}\right )} x^{3} + {\left (6 \, a^{2} b^{4} - 29 \, a^{3} b^{2} c + 20 \, a^{4} c^{2}\right )} x^{2} - 2 \, {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x}{3 \, {\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{5} x^{3}} \]

[In]

integrate(1/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="giac")

[Out]

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)/sqrt(-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)

Mupad [B] (verification not implemented)

Time = 9.22 (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=\frac {\frac {x^2\,\left (5\,a\,c-6\,b^2\right )}{3\,a^3}-\frac {1}{3\,a}+\frac {2\,b\,x}{3\,a^2}+\frac {x^3\,\left (17\,a^2\,b\,c^2-20\,a\,b^3\,c+4\,b^5\right )}{a^4\,\left (4\,a\,c-b^2\right )}+\frac {2\,c\,x^4\,\left (5\,a^2\,c^2-9\,a\,b^2\,c+2\,b^4\right )}{a^4\,\left (4\,a\,c-b^2\right )}}{c\,x^5+b\,x^4+a\,x^3}+\frac {\ln \left (4\,a\,b^9+4\,b^{10}\,x-4\,a\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-52\,a^2\,b^7\,c+308\,a^5\,b\,c^4-40\,a^5\,c^5\,x-4\,b^7\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+243\,a^3\,b^5\,c^2-473\,a^4\,b^3\,c^3+5\,a^4\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a^2\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+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\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+30\,a\,b^5\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+41\,a^3\,b\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-66\,a^2\,b^3\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (a^2\,\left (132\,b^5\,c^2-30\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )-a^3\,\left (272\,b^3\,c^3-10\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )+2\,b^9-2\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-a\,\left (27\,b^7\,c-15\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )+192\,a^4\,b\,c^4\right )}{-64\,a^8\,c^3+48\,a^7\,b^2\,c^2-12\,a^6\,b^4\,c+a^5\,b^6}+\frac {\ln \left (4\,a\,b^9+4\,b^{10}\,x+4\,a\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-52\,a^2\,b^7\,c+308\,a^5\,b\,c^4-40\,a^5\,c^5\,x+4\,b^7\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+243\,a^3\,b^5\,c^2-473\,a^4\,b^3\,c^3-5\,a^4\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-24\,a^2\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+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\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-30\,a\,b^5\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-41\,a^3\,b\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+66\,a^2\,b^3\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (a^2\,\left (132\,b^5\,c^2+30\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )-a^3\,\left (272\,b^3\,c^3+10\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )+2\,b^9+2\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-a\,\left (27\,b^7\,c+15\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )+192\,a^4\,b\,c^4\right )}{-64\,a^8\,c^3+48\,a^7\,b^2\,c^2-12\,a^6\,b^4\,c+a^5\,b^6}+\frac {2\,b\,\ln \left (x\right )\,\left (3\,a\,c-2\,b^2\right )}{a^5} \]

[In]

int(1/(a*x^2 + b*x^3 + c*x^4)^2,x)

[Out]

((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 - 1
5*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 + 3
3*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
) + (2*b*log(x)*(3*a*c - 2*b^2))/a^5