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

Optimal. Leaf size=202 \[ \frac{b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{3/2}}-\frac{3 b^2-8 a c}{2 a^2 x^2 \left (b^2-4 a c\right )}-\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}+\frac{b \left (3 b^2-11 a c\right )}{a^3 x \left (b^2-4 a c\right )}+\frac{\log (x) \left (3 b^2-2 a c\right )}{a^4}+\frac{-2 a c+b^2+b c x}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

-(3*b^2 - 8*a*c)/(2*a^2*(b^2 - 4*a*c)*x^2) + (b*(3*b^2 - 11*a*c))/(a^3*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*c*x
)/(a*(b^2 - 4*a*c)*x^2*(a + b*x + c*x^2)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2
- 4*a*c]])/(a^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*Log[x])/a^4 - ((3*b^2 - 2*a*c)*Log[a + b*x + c*x^2])/(
2*a^4)

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Rubi [A]  time = 0.250297, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1585, 740, 800, 634, 618, 206, 628} \[ \frac{b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{3/2}}-\frac{3 b^2-8 a c}{2 a^2 x^2 \left (b^2-4 a c\right )}-\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}+\frac{b \left (3 b^2-11 a c\right )}{a^3 x \left (b^2-4 a c\right )}+\frac{\log (x) \left (3 b^2-2 a c\right )}{a^4}+\frac{-2 a c+b^2+b c x}{a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(a*x^2 + b*x^3 + c*x^4)^2,x]

[Out]

-(3*b^2 - 8*a*c)/(2*a^2*(b^2 - 4*a*c)*x^2) + (b*(3*b^2 - 11*a*c))/(a^3*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*c*x
)/(a*(b^2 - 4*a*c)*x^2*(a + b*x + c*x^2)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2
- 4*a*c]])/(a^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*Log[x])/a^4 - ((3*b^2 - 2*a*c)*Log[a + b*x + c*x^2])/(
2*a^4)

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{x}{\left (a x^2+b x^3+c x^4\right )^2} \, dx &=\int \frac{1}{x^3 \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^2 \left (a+b x+c x^2\right )}-\frac{\int \frac{-3 b^2+8 a c-3 b c x}{x^3 \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^2 \left (a+b x+c x^2\right )}-\frac{\int \left (\frac{-3 b^2+8 a c}{a x^3}+\frac{3 b^3-11 a b c}{a^2 x^2}+\frac{\left (b^2-4 a c\right ) \left (-3 b^2+2 a c\right )}{a^3 x}+\frac{b \left (3 b^4-17 a b^2 c+19 a^2 c^2\right )+c \left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx}{a \left (b^2-4 a c\right )}\\ &=-\frac{3 b^2-8 a c}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac{b \left (3 b^2-11 a c\right )}{a^3 \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^2 \left (a+b x+c x^2\right )}+\frac{\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac{\int \frac{b \left (3 b^4-17 a b^2 c+19 a^2 c^2\right )+c \left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{a+b x+c x^2} \, dx}{a^4 \left (b^2-4 a c\right )}\\ &=-\frac{3 b^2-8 a c}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac{b \left (3 b^2-11 a c\right )}{a^3 \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^2 \left (a+b x+c x^2\right )}+\frac{\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac{\left (3 b^2-2 a c\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 a^4}-\frac{\left (b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 a^4 \left (b^2-4 a c\right )}\\ &=-\frac{3 b^2-8 a c}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac{b \left (3 b^2-11 a c\right )}{a^3 \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^2 \left (a+b x+c x^2\right )}+\frac{\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}+\frac{\left (b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^4 \left (b^2-4 a c\right )}\\ &=-\frac{3 b^2-8 a c}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac{b \left (3 b^2-11 a c\right )}{a^3 \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^2 \left (a+b x+c x^2\right )}+\frac{b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{3/2}}+\frac{\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}\\ \end{align*}

Mathematica [A]  time = 0.427668, size = 175, normalized size = 0.87 \[ \frac{\frac{2 a \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^3 c x+b^4\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{2 b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}-\frac{a^2}{x^2}+2 \log (x) \left (3 b^2-2 a c\right )+\left (2 a c-3 b^2\right ) \log (a+x (b+c x))+\frac{4 a b}{x}}{2 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^2/x^2) + (4*a*b)/x + (2*a*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x - 3*a*b*c^2*x))/((b^2 - 4*a*c)*(a + x*(b
+ c*x))) + (2*b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2
) + 2*(3*b^2 - 2*a*c)*Log[x] + (-3*b^2 + 2*a*c)*Log[a + x*(b + c*x)])/(2*a^4)

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Maple [B]  time = 0.016, size = 418, normalized size = 2.1 \begin{align*} -{\frac{1}{2\,{a}^{2}{x}^{2}}}-2\,{\frac{\ln \left ( x \right ) c}{{a}^{3}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{4}}}+2\,{\frac{b}{{a}^{3}x}}+3\,{\frac{{c}^{2}bx}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{3}cx}{{a}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{{c}^{2}}{a \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+4\,{\frac{{b}^{2}c}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{4}}{{a}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+4\,{\frac{{c}^{2}\ln \left ( c{x}^{2}+bx+a \right ) }{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }}-7\,{\frac{c\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}}{{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{3\,\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}}{2\,{a}^{4} \left ( 4\,ac-{b}^{2} \right ) }}+30\,{\frac{{c}^{2}b}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-20\,{\frac{{b}^{3}c}{{a}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+3\,{\frac{{b}^{5}}{{a}^{4} \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(x/(c*x^4+b*x^3+a*x^2)^2,x)

[Out]

-1/2/a^2/x^2-2/a^3*ln(x)*c+3/a^4*ln(x)*b^2+2/a^3*b/x+3/a^2/(c*x^2+b*x+a)*b*c^2/(4*a*c-b^2)*x-1/a^3/(c*x^2+b*x+
a)*b^3*c/(4*a*c-b^2)*x-2/a/(c*x^2+b*x+a)/(4*a*c-b^2)*c^2+4/a^2/(c*x^2+b*x+a)/(4*a*c-b^2)*b^2*c-1/a^3/(c*x^2+b*
x+a)/(4*a*c-b^2)*b^4+4/a^2/(4*a*c-b^2)*c^2*ln(c*x^2+b*x+a)-7/a^3/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b^2+3/2/a^4/(4*
a*c-b^2)*ln(c*x^2+b*x+a)*b^4+30/a^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^2-20/a^3/(4*a*c-
b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*c+3/a^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2
))*b^5

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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(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 = 3.11668, size = 2606, normalized size = 12.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 - 2*(3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*x^3 - (6*a*b^6 - 49*
a^2*b^4*c + 108*a^3*b^2*c^2 - 32*a^4*c^3)*x^2 + ((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*x^4 + (3*b^6 - 20*a*b
^4*c + 30*a^2*b^2*c^2)*x^3 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*x^2)*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)) - 3*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b
*c^2)*x + ((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*x^4 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 -
32*a^3*b*c^3)*x^3 + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*x^2)*log(c*x^2 + b*x + a) - 2*((3*
b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*x^4 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)
*x^3 + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*x^2)*log(x))/((a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^
6*c^3)*x^4 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^3 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x^2), -1/2*(a^3
*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 - 2*(3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*x^3 - (6*a*b^6 - 49*a^2*b^4*c
+ 108*a^3*b^2*c^2 - 32*a^4*c^3)*x^2 - 2*((3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*x^4 + (3*b^6 - 20*a*b^4*c + 3
0*a^2*b^2*c^2)*x^3 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c
)*(2*c*x + b)/(b^2 - 4*a*c)) - 3*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x + ((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*
b^2*c^3 - 32*a^3*c^4)*x^4 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*x^3 + (3*a*b^6 - 26*a^2*b^4*c
+ 64*a^3*b^2*c^2 - 32*a^4*c^3)*x^2)*log(c*x^2 + b*x + a) - 2*((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a
^3*c^4)*x^4 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*x^3 + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*
c^2 - 32*a^4*c^3)*x^2)*log(x))/((a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)*x^4 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6
*b*c^2)*x^3 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x^2)]

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Sympy [B]  time = 19.1743, size = 4083, normalized size = 20.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 +
12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))*log(x + (3072*a**14*c**6*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a
**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3
*b**2)/(2*a**4))**2 - 9408*a**13*b**2*c**5*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/
(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 + 9040*a**12*
b**4*c**4*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b
**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 - 4116*a**11*b**6*c**3*(-b*sqrt(-(4*a*c - b**2
)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) +
(2*a*c - 3*b**2)/(2*a**4))**2 + 3072*a**11*c**7*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*
b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 987*a**1
0*b**8*c**2*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2
*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 - 7536*a**10*b**2*c**6*(-b*sqrt(-(4*a*c - b*
*2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))
+ (2*a*c - 3*b**2)/(2*a**4)) - 121*a**9*b**10*c*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*
b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 + 8152*
a**9*b**4*c**5*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a
**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 6*a**8*b**12*(-b*sqrt(-(4*a*c - b**2)**3)*
(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*
c - 3*b**2)/(2*a**4))**2 - 4343*a**8*b**6*c**4*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b*
*4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) - 6144*a**8*
c**8 + 1198*a**7*b**8*c**3*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3
*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 50208*a**7*b**2*c**7 - 165*a**
6*b**10*c**2*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**
2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) - 137792*a**6*b**4*c**6 + 9*a**5*b**12*c*(-b*s
qrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a
*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 176474*a**5*b**6*c**5 - 119275*a**4*b**8*c**4 + 45448*a**3*b**
10*c**3 - 9846*a**2*b**12*c**2 + 1134*a*b**14*c - 54*b**16)/(17280*a**7*b*c**8 - 69570*a**6*b**3*c**7 + 112428
*a**5*b**5*c**6 - 88605*a**4*b**7*c**5 + 37600*a**3*b**9*c**4 - 8820*a**2*b**11*c**3 + 1080*a*b**13*c**2 - 54*
b**15*c)) + (b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*
b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))*log(x + (3072*a**14*c**6*(b*sqrt(-(4*a*c - b**2)
**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) +
(2*a*c - 3*b**2)/(2*a**4))**2 - 9408*a**13*b**2*c**5*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c +
3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 + 90
40*a**12*b**4*c**4*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 4
8*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 - 4116*a**11*b**6*c**3*(b*sqrt(-(4*a*c
- b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b
**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 + 3072*a**11*c**7*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*
c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 98
7*a**10*b**8*c**2*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48
*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 - 7536*a**10*b**2*c**6*(b*sqrt(-(4*a*c
- b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b*
*6)) + (2*a*c - 3*b**2)/(2*a**4)) - 121*a**9*b**10*c*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c +
3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4))**2 + 81
52*a**9*b**4*c**5*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48
*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 6*a**8*b**12*(b*sqrt(-(4*a*c - b**2)**3)
*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a
*c - 3*b**2)/(2*a**4))**2 - 4343*a**8*b**6*c**4*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b*
*4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) - 6144*a**8*
c**8 + 1198*a**7*b**8*c**3*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*
c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 50208*a**7*b**2*c**7 - 165*a**6
*b**10*c**2*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*
b**2*c**2 + 12*a*b**4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) - 137792*a**6*b**4*c**6 + 9*a**5*b**12*c*(b*sqrt
(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*a**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b*
*4*c - b**6)) + (2*a*c - 3*b**2)/(2*a**4)) + 176474*a**5*b**6*c**5 - 119275*a**4*b**8*c**4 + 45448*a**3*b**10*
c**3 - 9846*a**2*b**12*c**2 + 1134*a*b**14*c - 54*b**16)/(17280*a**7*b*c**8 - 69570*a**6*b**3*c**7 + 112428*a*
*5*b**5*c**6 - 88605*a**4*b**7*c**5 + 37600*a**3*b**9*c**4 - 8820*a**2*b**11*c**3 + 1080*a*b**13*c**2 - 54*b**
15*c)) + (-4*a**3*c + a**2*b**2 + x**3*(22*a*b*c**2 - 6*b**3*c) + x**2*(-8*a**2*c**2 + 25*a*b**2*c - 6*b**4) +
x*(12*a**2*b*c - 3*a*b**3))/(x**4*(8*a**4*c**2 - 2*a**3*b**2*c) + x**3*(8*a**4*b*c - 2*a**3*b**3) + x**2*(8*a
**5*c - 2*a**4*b**2)) - (2*a*c - 3*b**2)*log(x + (-6144*a**8*c**8 + 50208*a**7*b**2*c**7 - 3072*a**7*c**7*(2*a
*c - 3*b**2) - 137792*a**6*b**4*c**6 + 7536*a**6*b**2*c**6*(2*a*c - 3*b**2) + 3072*a**6*c**6*(2*a*c - 3*b**2)*
*2 + 176474*a**5*b**6*c**5 - 8152*a**5*b**4*c**5*(2*a*c - 3*b**2) - 9408*a**5*b**2*c**5*(2*a*c - 3*b**2)**2 -
119275*a**4*b**8*c**4 + 4343*a**4*b**6*c**4*(2*a*c - 3*b**2) + 9040*a**4*b**4*c**4*(2*a*c - 3*b**2)**2 + 45448
*a**3*b**10*c**3 - 1198*a**3*b**8*c**3*(2*a*c - 3*b**2) - 4116*a**3*b**6*c**3*(2*a*c - 3*b**2)**2 - 9846*a**2*
b**12*c**2 + 165*a**2*b**10*c**2*(2*a*c - 3*b**2) + 987*a**2*b**8*c**2*(2*a*c - 3*b**2)**2 + 1134*a*b**14*c -
9*a*b**12*c*(2*a*c - 3*b**2) - 121*a*b**10*c*(2*a*c - 3*b**2)**2 - 54*b**16 + 6*b**12*(2*a*c - 3*b**2)**2)/(17
280*a**7*b*c**8 - 69570*a**6*b**3*c**7 + 112428*a**5*b**5*c**6 - 88605*a**4*b**7*c**5 + 37600*a**3*b**9*c**4 -
8820*a**2*b**11*c**3 + 1080*a*b**13*c**2 - 54*b**15*c))/a**4

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Giac [A]  time = 1.12117, size = 309, normalized size = 1.53 \begin{align*} -\frac{{\left (3 \, b^{5} - 20 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (3 \, b^{2} - 2 \, a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} + \frac{{\left (3 \, b^{2} - 2 \, a c\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{a^{3} b^{2} - 4 \, a^{4} c - 2 \,{\left (3 \, a b^{3} c - 11 \, a^{2} b c^{2}\right )} x^{3} -{\left (6 \, a b^{4} - 25 \, a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} x^{2} - 3 \,{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} a^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*b^5 - 20*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^2 - 4*a^5*c)*sqrt(-b^2 + 4
*a*c)) - 1/2*(3*b^2 - 2*a*c)*log(c*x^2 + b*x + a)/a^4 + (3*b^2 - 2*a*c)*log(abs(x))/a^4 - 1/2*(a^3*b^2 - 4*a^4
*c - 2*(3*a*b^3*c - 11*a^2*b*c^2)*x^3 - (6*a*b^4 - 25*a^2*b^2*c + 8*a^3*c^2)*x^2 - 3*(a^2*b^3 - 4*a^3*b*c)*x)/
((c*x^2 + b*x + a)*(b^2 - 4*a*c)*a^4*x^2)