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

Optimal. Leaf size=318 \[ -\frac{12 a^2 c^2-22 a b^2 c+5 b^4}{2 a^4 x^2 \left (b^2-4 a c\right )}-\frac{\left (3 a^2 c^2-12 a b^2 c+5 b^4\right ) \log \left (a+b x+c x^2\right )}{2 a^6}+\frac{b \left (29 a^2 c^2-27 a b^2 c+5 b^4\right )}{a^5 x \left (b^2-4 a c\right )}+\frac{\log (x) \left (3 a^2 c^2-12 a b^2 c+5 b^4\right )}{a^6}+\frac{b \left (105 a^2 b^2 c^2-70 a^3 c^3-42 a b^4 c+5 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^6 \left (b^2-4 a c\right )^{3/2}}+\frac{b \left (5 b^2-17 a c\right )}{3 a^3 x^3 \left (b^2-4 a c\right )}-\frac{5 b^2-12 a c}{4 a^2 x^4 \left (b^2-4 a c\right )}+\frac{-2 a c+b^2+b c x}{a x^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

-(5*b^2 - 12*a*c)/(4*a^2*(b^2 - 4*a*c)*x^4) + (b*(5*b^2 - 17*a*c))/(3*a^3*(b^2 - 4*a*c)*x^3) - (5*b^4 - 22*a*b
^2*c + 12*a^2*c^2)/(2*a^4*(b^2 - 4*a*c)*x^2) + (b*(5*b^4 - 27*a*b^2*c + 29*a^2*c^2))/(a^5*(b^2 - 4*a*c)*x) + (
b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*x^4*(a + b*x + c*x^2)) + (b*(5*b^6 - 42*a*b^4*c + 105*a^2*b^2*c^2 - 70*a
^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^6*(b^2 - 4*a*c)^(3/2)) + ((5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*
Log[x])/a^6 - ((5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*Log[a + b*x + c*x^2])/(2*a^6)

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Rubi [A]  time = 0.391815, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {1585, 740, 800, 634, 618, 206, 628} \[ -\frac{12 a^2 c^2-22 a b^2 c+5 b^4}{2 a^4 x^2 \left (b^2-4 a c\right )}-\frac{\left (3 a^2 c^2-12 a b^2 c+5 b^4\right ) \log \left (a+b x+c x^2\right )}{2 a^6}+\frac{b \left (29 a^2 c^2-27 a b^2 c+5 b^4\right )}{a^5 x \left (b^2-4 a c\right )}+\frac{\log (x) \left (3 a^2 c^2-12 a b^2 c+5 b^4\right )}{a^6}+\frac{b \left (105 a^2 b^2 c^2-70 a^3 c^3-42 a b^4 c+5 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^6 \left (b^2-4 a c\right )^{3/2}}+\frac{b \left (5 b^2-17 a c\right )}{3 a^3 x^3 \left (b^2-4 a c\right )}-\frac{5 b^2-12 a c}{4 a^2 x^4 \left (b^2-4 a c\right )}+\frac{-2 a c+b^2+b c x}{a x^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a*x^2 + b*x^3 + c*x^4)^2),x]

[Out]

-(5*b^2 - 12*a*c)/(4*a^2*(b^2 - 4*a*c)*x^4) + (b*(5*b^2 - 17*a*c))/(3*a^3*(b^2 - 4*a*c)*x^3) - (5*b^4 - 22*a*b
^2*c + 12*a^2*c^2)/(2*a^4*(b^2 - 4*a*c)*x^2) + (b*(5*b^4 - 27*a*b^2*c + 29*a^2*c^2))/(a^5*(b^2 - 4*a*c)*x) + (
b^2 - 2*a*c + b*c*x)/(a*(b^2 - 4*a*c)*x^4*(a + b*x + c*x^2)) + (b*(5*b^6 - 42*a*b^4*c + 105*a^2*b^2*c^2 - 70*a
^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^6*(b^2 - 4*a*c)^(3/2)) + ((5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*
Log[x])/a^6 - ((5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*Log[a + b*x + c*x^2])/(2*a^6)

Rule 1585

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

Rule 740

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 800

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 634

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 618

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 206

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

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{1}{x \left (a x^2+b x^3+c x^4\right )^2} \, dx &=\int \frac{1}{x^5 \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^4 \left (a+b x+c x^2\right )}-\frac{\int \frac{-5 b^2+12 a c-5 b c x}{x^5 \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^4 \left (a+b x+c x^2\right )}-\frac{\int \left (\frac{-5 b^2+12 a c}{a x^5}+\frac{5 b^3-17 a b c}{a^2 x^4}+\frac{-5 b^4+22 a b^2 c-12 a^2 c^2}{a^3 x^3}+\frac{5 b^5-27 a b^3 c+29 a^2 b c^2}{a^4 x^2}+\frac{\left (b^2-4 a c\right ) \left (-5 b^4+12 a b^2 c-3 a^2 c^2\right )}{a^5 x}+\frac{b \left (5 b^6-37 a b^4 c+78 a^2 b^2 c^2-41 a^3 c^3\right )+c \left (5 b^6-32 a b^4 c+51 a^2 b^2 c^2-12 a^3 c^3\right ) x}{a^5 \left (a+b x+c x^2\right )}\right ) \, dx}{a \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-12 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (5 b^2-17 a c\right )}{3 a^3 \left (b^2-4 a c\right ) x^3}-\frac{5 b^4-22 a b^2 c+12 a^2 c^2}{2 a^4 \left (b^2-4 a c\right ) x^2}+\frac{b \left (5 b^4-27 a b^2 c+29 a^2 c^2\right )}{a^5 \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^4 \left (a+b x+c x^2\right )}+\frac{\left (5 b^4-12 a b^2 c+3 a^2 c^2\right ) \log (x)}{a^6}-\frac{\int \frac{b \left (5 b^6-37 a b^4 c+78 a^2 b^2 c^2-41 a^3 c^3\right )+c \left (5 b^6-32 a b^4 c+51 a^2 b^2 c^2-12 a^3 c^3\right ) x}{a+b x+c x^2} \, dx}{a^6 \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-12 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (5 b^2-17 a c\right )}{3 a^3 \left (b^2-4 a c\right ) x^3}-\frac{5 b^4-22 a b^2 c+12 a^2 c^2}{2 a^4 \left (b^2-4 a c\right ) x^2}+\frac{b \left (5 b^4-27 a b^2 c+29 a^2 c^2\right )}{a^5 \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^4 \left (a+b x+c x^2\right )}+\frac{\left (5 b^4-12 a b^2 c+3 a^2 c^2\right ) \log (x)}{a^6}-\frac{\left (5 b^4-12 a b^2 c+3 a^2 c^2\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 a^6}-\frac{\left (b \left (5 b^6-42 a b^4 c+105 a^2 b^2 c^2-70 a^3 c^3\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 a^6 \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-12 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (5 b^2-17 a c\right )}{3 a^3 \left (b^2-4 a c\right ) x^3}-\frac{5 b^4-22 a b^2 c+12 a^2 c^2}{2 a^4 \left (b^2-4 a c\right ) x^2}+\frac{b \left (5 b^4-27 a b^2 c+29 a^2 c^2\right )}{a^5 \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^4 \left (a+b x+c x^2\right )}+\frac{\left (5 b^4-12 a b^2 c+3 a^2 c^2\right ) \log (x)}{a^6}-\frac{\left (5 b^4-12 a b^2 c+3 a^2 c^2\right ) \log \left (a+b x+c x^2\right )}{2 a^6}+\frac{\left (b \left (5 b^6-42 a b^4 c+105 a^2 b^2 c^2-70 a^3 c^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^6 \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-12 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (5 b^2-17 a c\right )}{3 a^3 \left (b^2-4 a c\right ) x^3}-\frac{5 b^4-22 a b^2 c+12 a^2 c^2}{2 a^4 \left (b^2-4 a c\right ) x^2}+\frac{b \left (5 b^4-27 a b^2 c+29 a^2 c^2\right )}{a^5 \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^4 \left (a+b x+c x^2\right )}+\frac{b \left (5 b^6-42 a b^4 c+105 a^2 b^2 c^2-70 a^3 c^3\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^6 \left (b^2-4 a c\right )^{3/2}}+\frac{\left (5 b^4-12 a b^2 c+3 a^2 c^2\right ) \log (x)}{a^6}-\frac{\left (5 b^4-12 a b^2 c+3 a^2 c^2\right ) \log \left (a+b x+c x^2\right )}{2 a^6}\\ \end{align*}

Mathematica [A]  time = 0.413161, size = 272, normalized size = 0.86 \[ \frac{-\frac{12 a \left (-9 a^2 b^2 c^2-5 a^2 b c^3 x+2 a^3 c^3+5 a b^3 c^2 x+6 a b^4 c-b^5 c x-b^6\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+12 \log (x) \left (3 a^2 c^2-12 a b^2 c+5 b^4\right )-6 \left (3 a^2 c^2-12 a b^2 c+5 b^4\right ) \log (a+x (b+c x))+\frac{12 b \left (105 a^2 b^2 c^2-70 a^3 c^3-42 a b^4 c+5 b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{6 a^2 \left (2 a c-3 b^2\right )}{x^2}+\frac{8 a^3 b}{x^3}-\frac{3 a^4}{x^4}-\frac{24 a b \left (3 a c-2 b^2\right )}{x}}{12 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a*x^2 + b*x^3 + c*x^4)^2),x]

[Out]

((-3*a^4)/x^4 + (8*a^3*b)/x^3 + (6*a^2*(-3*b^2 + 2*a*c))/x^2 - (24*a*b*(-2*b^2 + 3*a*c))/x - (12*a*(-b^6 + 6*a
*b^4*c - 9*a^2*b^2*c^2 + 2*a^3*c^3 - b^5*c*x + 5*a*b^3*c^2*x - 5*a^2*b*c^3*x))/((b^2 - 4*a*c)*(a + x*(b + c*x)
)) + (12*b*(5*b^6 - 42*a*b^4*c + 105*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 +
4*a*c)^(3/2) + 12*(5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*Log[x] - 6*(5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*Log[a + x*(b +
c*x)])/(12*a^6)

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Maple [B]  time = 0.018, size = 619, normalized size = 2. \begin{align*} -{\frac{1}{4\,{a}^{2}{x}^{4}}}+{\frac{c}{{x}^{2}{a}^{3}}}-{\frac{3\,{b}^{2}}{2\,{a}^{4}{x}^{2}}}+3\,{\frac{\ln \left ( x \right ){c}^{2}}{{a}^{4}}}-12\,{\frac{\ln \left ( x \right ){b}^{2}c}{{a}^{5}}}+5\,{\frac{\ln \left ( x \right ){b}^{4}}{{a}^{6}}}+{\frac{2\,b}{3\,{a}^{3}{x}^{3}}}-6\,{\frac{bc}{{a}^{4}x}}+4\,{\frac{{b}^{3}}{{a}^{5}x}}-5\,{\frac{b{c}^{3}x}{{a}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+5\,{\frac{{b}^{3}{c}^{2}x}{{a}^{4} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{5}cx}{{a}^{5} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+2\,{\frac{{c}^{3}}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-9\,{\frac{{b}^{2}{c}^{2}}{{a}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+6\,{\frac{{b}^{4}c}{{a}^{4} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{6}}{{a}^{5} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-6\,{\frac{{c}^{3}\ln \left ( c{x}^{2}+bx+a \right ) }{{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{51\,{c}^{2}\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}}{2\,{a}^{4} \left ( 4\,ac-{b}^{2} \right ) }}-16\,{\frac{c\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}}{{a}^{5} \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{5\,\ln \left ( c{x}^{2}+bx+a \right ){b}^{6}}{2\,{a}^{6} \left ( 4\,ac-{b}^{2} \right ) }}-70\,{\frac{b{c}^{3}}{{a}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+105\,{\frac{{b}^{3}{c}^{2}}{{a}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-42\,{\frac{{b}^{5}c}{{a}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+5\,{\frac{{b}^{7}}{{a}^{6} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/a^2/x^4+1/a^3/x^2*c-3/2/a^4/x^2*b^2+3/a^4*ln(x)*c^2-12/a^5*ln(x)*b^2*c+5/a^6*ln(x)*b^4+2/3/a^3*b/x^3-6*b/
a^4/x*c+4*b^3/a^5/x-5/a^3/(c*x^2+b*x+a)*b*c^3/(4*a*c-b^2)*x+5/a^4/(c*x^2+b*x+a)*b^3*c^2/(4*a*c-b^2)*x-1/a^5/(c
*x^2+b*x+a)*b^5*c/(4*a*c-b^2)*x+2/a^2/(c*x^2+b*x+a)/(4*a*c-b^2)*c^3-9/a^3/(c*x^2+b*x+a)/(4*a*c-b^2)*b^2*c^2+6/
a^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b^4*c-1/a^5/(c*x^2+b*x+a)/(4*a*c-b^2)*b^6-6/a^3/(4*a*c-b^2)*c^3*ln(c*x^2+b*x+a)+
51/2/a^4/(4*a*c-b^2)*c^2*ln(c*x^2+b*x+a)*b^2-16/a^5/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b^4+5/2/a^6/(4*a*c-b^2)*ln(c
*x^2+b*x+a)*b^6-70/a^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^3+105/a^4/(4*a*c-b^2)^(3/2)*a
rctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*c^2-42/a^5/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5*c+
5/a^6/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^7

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.72475, size = 3563, normalized size = 11.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(3*a^5*b^4 - 24*a^6*b^2*c + 48*a^7*c^2 - 12*(5*a*b^7*c - 47*a^2*b^5*c^2 + 137*a^3*b^3*c^3 - 116*a^4*b*c
^4)*x^5 - 6*(10*a*b^8 - 99*a^2*b^6*c + 316*a^3*b^4*c^2 - 332*a^4*b^2*c^3 + 48*a^5*c^4)*x^4 - 2*(15*a^2*b^7 - 1
46*a^3*b^5*c + 448*a^4*b^3*c^2 - 416*a^5*b*c^3)*x^3 + (10*a^3*b^6 - 89*a^4*b^4*c + 232*a^5*b^2*c^2 - 144*a^6*c
^3)*x^2 - 6*((5*b^7*c - 42*a*b^5*c^2 + 105*a^2*b^3*c^3 - 70*a^3*b*c^4)*x^6 + (5*b^8 - 42*a*b^6*c + 105*a^2*b^4
*c^2 - 70*a^3*b^2*c^3)*x^5 + (5*a*b^7 - 42*a^2*b^5*c + 105*a^3*b^3*c^2 - 70*a^4*b*c^3)*x^4)*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)) - 5*(a^4*b^5 - 8*a^
5*b^3*c + 16*a^6*b*c^2)*x + 6*((5*b^8*c - 52*a*b^6*c^2 + 179*a^2*b^4*c^3 - 216*a^3*b^2*c^4 + 48*a^4*c^5)*x^6 +
(5*b^9 - 52*a*b^7*c + 179*a^2*b^5*c^2 - 216*a^3*b^3*c^3 + 48*a^4*b*c^4)*x^5 + (5*a*b^8 - 52*a^2*b^6*c + 179*a
^3*b^4*c^2 - 216*a^4*b^2*c^3 + 48*a^5*c^4)*x^4)*log(c*x^2 + b*x + a) - 12*((5*b^8*c - 52*a*b^6*c^2 + 179*a^2*b
^4*c^3 - 216*a^3*b^2*c^4 + 48*a^4*c^5)*x^6 + (5*b^9 - 52*a*b^7*c + 179*a^2*b^5*c^2 - 216*a^3*b^3*c^3 + 48*a^4*
b*c^4)*x^5 + (5*a*b^8 - 52*a^2*b^6*c + 179*a^3*b^4*c^2 - 216*a^4*b^2*c^3 + 48*a^5*c^4)*x^4)*log(x))/((a^6*b^4*
c - 8*a^7*b^2*c^2 + 16*a^8*c^3)*x^6 + (a^6*b^5 - 8*a^7*b^3*c + 16*a^8*b*c^2)*x^5 + (a^7*b^4 - 8*a^8*b^2*c + 16
*a^9*c^2)*x^4), -1/12*(3*a^5*b^4 - 24*a^6*b^2*c + 48*a^7*c^2 - 12*(5*a*b^7*c - 47*a^2*b^5*c^2 + 137*a^3*b^3*c^
3 - 116*a^4*b*c^4)*x^5 - 6*(10*a*b^8 - 99*a^2*b^6*c + 316*a^3*b^4*c^2 - 332*a^4*b^2*c^3 + 48*a^5*c^4)*x^4 - 2*
(15*a^2*b^7 - 146*a^3*b^5*c + 448*a^4*b^3*c^2 - 416*a^5*b*c^3)*x^3 + (10*a^3*b^6 - 89*a^4*b^4*c + 232*a^5*b^2*
c^2 - 144*a^6*c^3)*x^2 - 12*((5*b^7*c - 42*a*b^5*c^2 + 105*a^2*b^3*c^3 - 70*a^3*b*c^4)*x^6 + (5*b^8 - 42*a*b^6
*c + 105*a^2*b^4*c^2 - 70*a^3*b^2*c^3)*x^5 + (5*a*b^7 - 42*a^2*b^5*c + 105*a^3*b^3*c^2 - 70*a^4*b*c^3)*x^4)*sq
rt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 5*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c
^2)*x + 6*((5*b^8*c - 52*a*b^6*c^2 + 179*a^2*b^4*c^3 - 216*a^3*b^2*c^4 + 48*a^4*c^5)*x^6 + (5*b^9 - 52*a*b^7*c
+ 179*a^2*b^5*c^2 - 216*a^3*b^3*c^3 + 48*a^4*b*c^4)*x^5 + (5*a*b^8 - 52*a^2*b^6*c + 179*a^3*b^4*c^2 - 216*a^4
*b^2*c^3 + 48*a^5*c^4)*x^4)*log(c*x^2 + b*x + a) - 12*((5*b^8*c - 52*a*b^6*c^2 + 179*a^2*b^4*c^3 - 216*a^3*b^2
*c^4 + 48*a^4*c^5)*x^6 + (5*b^9 - 52*a*b^7*c + 179*a^2*b^5*c^2 - 216*a^3*b^3*c^3 + 48*a^4*b*c^4)*x^5 + (5*a*b^
8 - 52*a^2*b^6*c + 179*a^3*b^4*c^2 - 216*a^4*b^2*c^3 + 48*a^5*c^4)*x^4)*log(x))/((a^6*b^4*c - 8*a^7*b^2*c^2 +
16*a^8*c^3)*x^6 + (a^6*b^5 - 8*a^7*b^3*c + 16*a^8*b*c^2)*x^5 + (a^7*b^4 - 8*a^8*b^2*c + 16*a^9*c^2)*x^4)]

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Sympy [B]  time = 47.4723, size = 6181, normalized size = 19.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3
- 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))*log(x + (4608*a**1
9*c**7*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**
3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 - 26432*
a**18*b**2*c**6*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**
6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2
+ 38640*a**17*b**4*c**5*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**
6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*
a**6))**2 - 26124*a**16*b**6*c**4*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*
c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*
b**4)/(2*a**6))**2 + 9603*a**15*b**8*c**3*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42
*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**
2*c + 5*b**4)/(2*a**6))**2 - 6912*a**15*c**9*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 +
42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*
b**2*c + 5*b**4)/(2*a**6)) - 1989*a**14*b**10*c**2*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*
c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 -
12*a*b**2*c + 5*b**4)/(2*a**6))**2 + 37616*a**14*b**2*c**8*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a
**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a*
*2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) + 219*a**13*b**12*c*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105
*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*
a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 - 96472*a**13*b**4*c**7*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c
**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**
6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 10*a**12*b**14*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c*
*3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6
)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 + 112063*a**12*b**6*c**6*(-b*sqrt(-(4*a*c - b**2)**3)*(
70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**
4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 69023*a**11*b**8*c**5*(-b*sqrt(-(4*a*c - b**2)
**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12
*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 20736*a**11*c**11 + 24355*a**10*b**10*c*
*4*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c*
*3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) + 373872*a**10*
b**2*c**10 - 4964*a**9*b**12*c**3*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*
c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*
b**4)/(2*a**6)) - 2277288*a**9*b**4*c**9 + 545*a**8*b**14*c**2*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 10
5*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3
*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) + 6487391*a**8*b**6*c**8 - 25*a**7*b**16*c*(-b*sqrt(-(4*a*c - b**
2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 +
12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 9943570*a**7*b**8*c**7 + 9090837*a**6*
b**10*c**6 - 5264714*a**5*b**12*c**5 + 1984426*a**4*b**14*c**4 - 486146*a**3*b**16*c**3 + 74720*a**2*b**18*c**
2 - 6550*a*b**20*c + 250*b**22)/(90720*a**10*b*c**11 - 844130*a**9*b**3*c**10 + 3174507*a**8*b**5*c**9 - 58850
10*a**7*b**7*c**8 + 6168225*a**6*b**9*c**7 - 3960180*a**5*b**11*c**6 + 1618470*a**4*b**13*c**5 - 423276*a**3*b
**15*c**4 + 68670*a**2*b**17*c**3 - 6300*a*b**19*c**2 + 250*b**21*c)) + (b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c
**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**
6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))*log(x + (4608*a**19*c**7*(b*sqrt(-(4*a*c - b**2)**3)*(70*
a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 - 26432*a**18*b**2*c**6*(b*sqrt(-(4*a*c - b**2)*
*3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*
a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 + 38640*a**17*b**4*c**5*(b*sqrt(-(4*a*c
- b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c*
*2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 - 26124*a**16*b**6*c**4*(b*sqrt(
-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2
*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 + 9603*a**15*b**8*c**3*(
b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 -
48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 - 6912*a**15*c**9
*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3
- 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 1989*a**14*b**10
*c**2*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*
c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 + 37616*a*
*14*b**2*c**8*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(
64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) + 219
*a**13*b**12*c*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*
(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 -
96472*a**13*b**4*c**7*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/
(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**
6)) - 10*a**12*b**14*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2
*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)
)**2 + 112063*a**12*b**6*c**6*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5
*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)
/(2*a**6)) - 69023*a**11*b**8*c**5*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*
c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*
b**4)/(2*a**6)) - 20736*a**11*c**11 + 24355*a**10*b**10*c**4*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a
**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a*
*2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) + 373872*a**10*b**2*c**10 - 4964*a**9*b**12*c**3*(b*sqrt(-(4*a*c - b
**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 2277288*a**9*b**4*c**9 + 545*a**8*b*
*14*c**2*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a*
*3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) + 6487391*
a**8*b**6*c**8 - 25*a**7*b**16*c*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c
- 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b*
*4)/(2*a**6)) - 9943570*a**7*b**8*c**7 + 9090837*a**6*b**10*c**6 - 5264714*a**5*b**12*c**5 + 1984426*a**4*b**1
4*c**4 - 486146*a**3*b**16*c**3 + 74720*a**2*b**18*c**2 - 6550*a*b**20*c + 250*b**22)/(90720*a**10*b*c**11 - 8
44130*a**9*b**3*c**10 + 3174507*a**8*b**5*c**9 - 5885010*a**7*b**7*c**8 + 6168225*a**6*b**9*c**7 - 3960180*a**
5*b**11*c**6 + 1618470*a**4*b**13*c**5 - 423276*a**3*b**15*c**4 + 68670*a**2*b**17*c**3 - 6300*a*b**19*c**2 +
250*b**21*c)) - (12*a**5*c - 3*a**4*b**2 + x**5*(348*a**2*b*c**3 - 324*a*b**3*c**2 + 60*b**5*c) + x**4*(-72*a*
*3*c**3 + 480*a**2*b**2*c**2 - 354*a*b**4*c + 60*b**6) + x**3*(208*a**3*b*c**2 - 172*a**2*b**3*c + 30*a*b**5)
+ x**2*(-36*a**4*c**2 + 49*a**3*b**2*c - 10*a**2*b**4) + x*(-20*a**4*b*c + 5*a**3*b**3))/(x**6*(48*a**6*c**2 -
12*a**5*b**2*c) + x**5*(48*a**6*b*c - 12*a**5*b**3) + x**4*(48*a**7*c - 12*a**6*b**2)) + (3*a**2*c**2 - 12*a*
b**2*c + 5*b**4)*log(x + (-20736*a**11*c**11 + 373872*a**10*b**2*c**10 - 2277288*a**9*b**4*c**9 - 6912*a**9*c*
*9*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) + 6487391*a**8*b**6*c**8 + 37616*a**8*b**2*c**8*(3*a**2*c**2 - 12*a*b*
*2*c + 5*b**4) - 9943570*a**7*b**8*c**7 - 96472*a**7*b**4*c**7*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) + 4608*a**
7*c**7*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)**2 + 9090837*a**6*b**10*c**6 + 112063*a**6*b**6*c**6*(3*a**2*c**2
- 12*a*b**2*c + 5*b**4) - 26432*a**6*b**2*c**6*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)**2 - 5264714*a**5*b**12*c*
*5 - 69023*a**5*b**8*c**5*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) + 38640*a**5*b**4*c**5*(3*a**2*c**2 - 12*a*b**2
*c + 5*b**4)**2 + 1984426*a**4*b**14*c**4 + 24355*a**4*b**10*c**4*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) - 26124
*a**4*b**6*c**4*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)**2 - 486146*a**3*b**16*c**3 - 4964*a**3*b**12*c**3*(3*a**
2*c**2 - 12*a*b**2*c + 5*b**4) + 9603*a**3*b**8*c**3*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)**2 + 74720*a**2*b**1
8*c**2 + 545*a**2*b**14*c**2*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) - 1989*a**2*b**10*c**2*(3*a**2*c**2 - 12*a*b
**2*c + 5*b**4)**2 - 6550*a*b**20*c - 25*a*b**16*c*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) + 219*a*b**12*c*(3*a**
2*c**2 - 12*a*b**2*c + 5*b**4)**2 + 250*b**22 - 10*b**14*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)**2)/(90720*a**10
*b*c**11 - 844130*a**9*b**3*c**10 + 3174507*a**8*b**5*c**9 - 5885010*a**7*b**7*c**8 + 6168225*a**6*b**9*c**7 -
3960180*a**5*b**11*c**6 + 1618470*a**4*b**13*c**5 - 423276*a**3*b**15*c**4 + 68670*a**2*b**17*c**3 - 6300*a*b
**19*c**2 + 250*b**21*c))/a**6

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Giac [A]  time = 1.12268, size = 468, normalized size = 1.47 \begin{align*} -\frac{{\left (5 \, b^{7} - 42 \, a b^{5} c + 105 \, a^{2} b^{3} c^{2} - 70 \, a^{3} b c^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (5 \, b^{4} - 12 \, a b^{2} c + 3 \, a^{2} c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{6}} + \frac{{\left (5 \, b^{4} - 12 \, a b^{2} c + 3 \, a^{2} c^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} - \frac{3 \, a^{5} b^{2} - 12 \, a^{6} c - 12 \,{\left (5 \, a b^{5} c - 27 \, a^{2} b^{3} c^{2} + 29 \, a^{3} b c^{3}\right )} x^{5} - 6 \,{\left (10 \, a b^{6} - 59 \, a^{2} b^{4} c + 80 \, a^{3} b^{2} c^{2} - 12 \, a^{4} c^{3}\right )} x^{4} - 2 \,{\left (15 \, a^{2} b^{5} - 86 \, a^{3} b^{3} c + 104 \, a^{4} b c^{2}\right )} x^{3} +{\left (10 \, a^{3} b^{4} - 49 \, a^{4} b^{2} c + 36 \, a^{5} c^{2}\right )} x^{2} - 5 \,{\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x}{12 \,{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} a^{6} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(5*b^7 - 42*a*b^5*c + 105*a^2*b^3*c^2 - 70*a^3*b*c^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^6*b^2 - 4*a^
7*c)*sqrt(-b^2 + 4*a*c)) - 1/2*(5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*log(c*x^2 + b*x + a)/a^6 + (5*b^4 - 12*a*b^2*c
+ 3*a^2*c^2)*log(abs(x))/a^6 - 1/12*(3*a^5*b^2 - 12*a^6*c - 12*(5*a*b^5*c - 27*a^2*b^3*c^2 + 29*a^3*b*c^3)*x^
5 - 6*(10*a*b^6 - 59*a^2*b^4*c + 80*a^3*b^2*c^2 - 12*a^4*c^3)*x^4 - 2*(15*a^2*b^5 - 86*a^3*b^3*c + 104*a^4*b*c
^2)*x^3 + (10*a^3*b^4 - 49*a^4*b^2*c + 36*a^5*c^2)*x^2 - 5*(a^4*b^3 - 4*a^5*b*c)*x)/((c*x^2 + b*x + a)*(b^2 -
4*a*c)*a^6*x^4)