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

Optimal. Leaf size=252 \[ -\frac{2 \left (5 a^2 c^2-9 a b^2 c+2 b^4\right )}{a^4 x \left (b^2-4 a c\right )}-\frac{2 \left (30 a^2 b^2 c^2-10 a^3 c^3-15 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{3/2}}+\frac{b \left (2 b^2-7 a c\right )}{a^3 x^2 \left (b^2-4 a c\right )}-\frac{2 \left (2 b^2-5 a c\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}+\frac{b \left (2 b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{a^5}-\frac{2 b \log (x) \left (2 b^2-3 a c\right )}{a^5}+\frac{-2 a c+b^2+b c x}{a x^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[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

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Rubi [A]  time = 0.323337, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {1594, 740, 800, 634, 618, 206, 628} \[ -\frac{2 \left (5 a^2 c^2-9 a b^2 c+2 b^4\right )}{a^4 x \left (b^2-4 a c\right )}-\frac{2 \left (30 a^2 b^2 c^2-10 a^3 c^3-15 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{3/2}}+\frac{b \left (2 b^2-7 a c\right )}{a^3 x^2 \left (b^2-4 a c\right )}-\frac{2 \left (2 b^2-5 a c\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}+\frac{b \left (2 b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{a^5}-\frac{2 b \log (x) \left (2 b^2-3 a c\right )}{a^5}+\frac{-2 a c+b^2+b c x}{a x^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[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 1594

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]

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}{\left (a x^2+b x^3+c x^4\right )^2} \, dx &=\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 ) \operatorname{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]  time = 0.341258, size = 218, normalized size = 0.87 \[ \frac{-\frac{3 a \left (5 a^2 b c^2+2 a^2 c^3 x-4 a b^2 c^2 x-5 a b^3 c+b^4 c x+b^5\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{6 \left (30 a^2 b^2 c^2-10 a^3 c^3-15 a b^4 c+2 b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{3 a^2 b}{x^2}-\frac{a^3}{x^3}+\frac{3 a \left (2 a c-3 b^2\right )}{x}+6 \log (x) \left (3 a b c-2 b^3\right )+3 \left (2 b^3-3 a b c\right ) \log (a+x (b+c x))}{3 a^5} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.016, size = 515, normalized size = 2. \begin{align*} -{\frac{1}{3\,{a}^{2}{x}^{3}}}+2\,{\frac{c}{{a}^{3}x}}-3\,{\frac{{b}^{2}}{{a}^{4}x}}+{\frac{b}{{x}^{2}{a}^{3}}}+6\,{\frac{b\ln \left ( x \right ) c}{{a}^{4}}}-4\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{5}}}+2\,{\frac{{c}^{3}x}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-4\,{\frac{{c}^{2}x{b}^{2}}{{a}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{cx{b}^{4}}{{a}^{4} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+5\,{\frac{{c}^{2}b}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-5\,{\frac{{b}^{3}c}{{a}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{{b}^{5}}{{a}^{4} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-12\,{\frac{{c}^{2}\ln \left ( c{x}^{2}+bx+a \right ) b}{{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }}+11\,{\frac{c\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}}{{a}^{4} \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{5}}{{a}^{5} \left ( 4\,ac-{b}^{2} \right ) }}+20\,{\frac{{c}^{3}}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-60\,{\frac{{c}^{2}{b}^{2}}{{a}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+30\,{\frac{{b}^{4}c}{{a}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{6}}{{a}^{5} \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/(c*x^4+b*x^3+a*x^2)^2,x)

[Out]

-1/3/a^2/x^3+2/a^3/x*c-3/a^4/x*b^2+1/a^3*b/x^2+6*b/a^4*ln(x)*c-4*b^3/a^5*ln(x)+2/a^2/(c*x^2+b*x+a)*c^3/(4*a*c-
b^2)*x-4/a^3/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*b^2+1/a^4/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*b^4+5/a^2/(c*x^2+b*x+a)*b
/(4*a*c-b^2)*c^2-5/a^3/(c*x^2+b*x+a)*b^3/(4*a*c-b^2)*c+1/a^4/(c*x^2+b*x+a)*b^5/(4*a*c-b^2)-12/a^3/(4*a*c-b^2)*
c^2*ln(c*x^2+b*x+a)*b+11/a^4/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b^3-2/a^5/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^5+20/a^2/(4
*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^3-60/a^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^
(1/2))*b^2*c^2+30/a^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*c-4/a^5/(4*a*c-b^2)^(3/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^6

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.10804, size = 2989, normalized size = 11.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [B]  time = 31.0241, size = 4774, normalized size = 18.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)
/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-4928*a**16*b*c**6*(-b*(3*a*c - 2*b*
*2)/a**5 - sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c
**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 10032*a**15*b**3*c**5*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(
4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2
*c**2 + 12*a*b**4*c - b**6)))**2 - 7980*a**14*b**5*c**4*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(4*a*c - b**2)**3)*(
10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6)))**2 + 3249*a**13*b**7*c**3*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a
**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 800
*a**13*c**8*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4
*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 723*a**12*b**9*c**2*(-b*(3*a*c
- 2*b**2)/a**5 - sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*
a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 6704*a**12*b**2*c**7*(-b*(3*a*c - 2*b**2)/a**5 - sq
rt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2
*b**2*c**2 + 12*a*b**4*c - b**6))) + 84*a**11*b**11*c*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(4*a*c - b**2)**3)*(10
*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c -
b**6)))**2 - 15182*a**11*b**4*c**6*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a*
*2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 4*a**10
*b**13*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c -
2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 12844*a**10*b**6*c**5*(-b*(3*a*c
- 2*b**2)/a**5 - sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*
a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 5546*a**9*b**8*c**4*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-
(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**
2*c**2 + 12*a*b**4*c - b**6))) - 4800*a**9*b*c**9 + 1306*a**8*b**10*c**3*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(4*
a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c
**2 + 12*a*b**4*c - b**6))) + 140384*a**8*b**3*c**8 - 160*a**7*b**12*c**2*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(4
*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*
c**2 + 12*a*b**4*c - b**6))) - 479788*a**7*b**5*c**7 + 8*a**6*b**14*c*(-b*(3*a*c - 2*b**2)/a**5 - sqrt(-(4*a*c
- b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6))) + 709872*a**6*b**7*c**6 - 575864*a**5*b**9*c**5 + 279640*a**4*b**11*c**4 - 83528*a**3
*b**13*c**3 + 15056*a**2*b**15*c**2 - 1504*a*b**17*c + 64*b**19)/(1000*a**9*c**10 + 42840*a**8*b**2*c**9 - 232
020*a**7*b**4*c**8 + 431760*a**6*b**6*c**7 - 406368*a**5*b**8*c**6 + 219600*a**4*b**10*c**5 - 71160*a**3*b**12
*c**4 + 13680*a**2*b**14*c**3 - 1440*a*b**16*c**2 + 64*b**18*c)) + (-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c -
b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 +
12*a*b**4*c - b**6)))*log(x + (-4928*a**16*b*c**6*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**
3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**
6)))**2 + 10032*a**15*b**3*c**5*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b
**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 7980*a**
14*b**5*c**4*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**
4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 3249*a**13*b**7*c**3*(-b*(3
*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5
*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 800*a**13*c**8*(-b*(3*a*c - 2*b**2)/a**5 + sqr
t(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*
b**2*c**2 + 12*a*b**4*c - b**6))) - 723*a**12*b**9*c**2*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(
10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6)))**2 + 6704*a**12*b**2*c**7*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a
**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 84*a**
11*b**11*c*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*
c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 15182*a**11*b**4*c**6*(-b*(3*
a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*
(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 4*a**10*b**13*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4
*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*
c**2 + 12*a*b**4*c - b**6)))**2 + 12844*a**10*b**6*c**5*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(
10*a**3*c**3 - 30*a**2*b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6))) - 5546*a**9*b**8*c**4*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*
b**2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 4800*a**9*
b*c**9 + 1306*a**8*b**10*c**3*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**
2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 140384*a**8*b
**3*c**8 - 160*a**7*b**12*c**2*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b*
*2*c**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 479788*a**7*
b**5*c**7 + 8*a**6*b**14*c*(-b*(3*a*c - 2*b**2)/a**5 + sqrt(-(4*a*c - b**2)**3)*(10*a**3*c**3 - 30*a**2*b**2*c
**2 + 15*a*b**4*c - 2*b**6)/(a**5*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 709872*a**6*b**7
*c**6 - 575864*a**5*b**9*c**5 + 279640*a**4*b**11*c**4 - 83528*a**3*b**13*c**3 + 15056*a**2*b**15*c**2 - 1504*
a*b**17*c + 64*b**19)/(1000*a**9*c**10 + 42840*a**8*b**2*c**9 - 232020*a**7*b**4*c**8 + 431760*a**6*b**6*c**7
- 406368*a**5*b**8*c**6 + 219600*a**4*b**10*c**5 - 71160*a**3*b**12*c**4 + 13680*a**2*b**14*c**3 - 1440*a*b**1
6*c**2 + 64*b**18*c)) + (-4*a**4*c + a**3*b**2 + x**4*(30*a**2*c**3 - 54*a*b**2*c**2 + 12*b**4*c) + x**3*(51*a
**2*b*c**2 - 60*a*b**3*c + 12*b**5) + x**2*(20*a**3*c**2 - 29*a**2*b**2*c + 6*a*b**4) + x*(8*a**3*b*c - 2*a**2
*b**3))/(x**5*(12*a**5*c**2 - 3*a**4*b**2*c) + x**4*(12*a**5*b*c - 3*a**4*b**3) + x**3*(12*a**6*c - 3*a**5*b**
2)) + 2*b*(3*a*c - 2*b**2)*log(x + (-4800*a**9*b*c**9 + 140384*a**8*b**3*c**8 + 1600*a**8*b*c**8*(3*a*c - 2*b*
*2) - 479788*a**7*b**5*c**7 + 13408*a**7*b**3*c**7*(3*a*c - 2*b**2) + 709872*a**6*b**7*c**6 - 30364*a**6*b**5*
c**6*(3*a*c - 2*b**2) - 19712*a**6*b**3*c**6*(3*a*c - 2*b**2)**2 - 575864*a**5*b**9*c**5 + 25688*a**5*b**7*c**
5*(3*a*c - 2*b**2) + 40128*a**5*b**5*c**5*(3*a*c - 2*b**2)**2 + 279640*a**4*b**11*c**4 - 11092*a**4*b**9*c**4*
(3*a*c - 2*b**2) - 31920*a**4*b**7*c**4*(3*a*c - 2*b**2)**2 - 83528*a**3*b**13*c**3 + 2612*a**3*b**11*c**3*(3*
a*c - 2*b**2) + 12996*a**3*b**9*c**3*(3*a*c - 2*b**2)**2 + 15056*a**2*b**15*c**2 - 320*a**2*b**13*c**2*(3*a*c
- 2*b**2) - 2892*a**2*b**11*c**2*(3*a*c - 2*b**2)**2 - 1504*a*b**17*c + 16*a*b**15*c*(3*a*c - 2*b**2) + 336*a*
b**13*c*(3*a*c - 2*b**2)**2 + 64*b**19 - 16*b**15*(3*a*c - 2*b**2)**2)/(1000*a**9*c**10 + 42840*a**8*b**2*c**9
- 232020*a**7*b**4*c**8 + 431760*a**6*b**6*c**7 - 406368*a**5*b**8*c**6 + 219600*a**4*b**10*c**5 - 71160*a**3
*b**12*c**4 + 13680*a**2*b**14*c**3 - 1440*a*b**16*c**2 + 64*b**18*c))/a**5

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Giac [A]  time = 1.10674, size = 381, normalized size = 1.51 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)