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

Optimal. Leaf size=255 \[ \frac{20 a^2 c^2+3 b c x^2 \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{4 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{3 \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{3 b \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac{3 b \log (x)}{a^4}+\frac{-2 a c+b^2+b c x^2}{4 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

(-3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(2*a^3*(b^2 - 4*a*c)^2*x^2) + (b^2 - 2*a*c + b*c*x^2)/(4*a*(b^2 - 4*a*c)*x^2*
(a + b*x^2 + c*x^4)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*x^2)/(4*a^2*(b^2 - 4*a*c)^2*x^
2*(a + b*x^2 + c*x^4)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 -
4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[x])/a^4 + (3*b*Log[a + b*x^2 + c*x^4])/(4*a^4)

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Rubi [A]  time = 0.391344, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {1114, 740, 822, 800, 634, 618, 206, 628} \[ \frac{20 a^2 c^2+3 b c x^2 \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{4 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{3 \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{3 b \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac{3 b \log (x)}{a^4}+\frac{-2 a c+b^2+b c x^2}{4 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^2 + c*x^4)^3),x]

[Out]

(-3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(2*a^3*(b^2 - 4*a*c)^2*x^2) + (b^2 - 2*a*c + b*c*x^2)/(4*a*(b^2 - 4*a*c)*x^2*
(a + b*x^2 + c*x^4)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*x^2)/(4*a^2*(b^2 - 4*a*c)^2*x^
2*(a + b*x^2 + c*x^4)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 -
4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[x])/a^4 + (3*b*Log[a + b*x^2 + c*x^4])/(4*a^4)

Rule 1114

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

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 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*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*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

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^3 \left (a+b x^2+c x^4\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac{b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-3 b^2+10 a c-4 b c x}{x^2 \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 a \left (b^2-4 a c\right )}\\ &=\frac{b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}+\frac{3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{6 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )+6 b c \left (b^2-6 a c\right ) x}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac{b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}+\frac{3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}+\frac{\operatorname{Subst}\left (\int \left (\frac{6 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a x^2}-\frac{6 b \left (-b^2+4 a c\right )^2}{a^2 x}+\frac{6 \left (b^6-9 a b^4 c+23 a^2 b^2 c^2-10 a^3 c^3+b c \left (b^2-4 a c\right )^2 x\right )}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac{b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}+\frac{3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}-\frac{3 b \log (x)}{a^4}+\frac{3 \operatorname{Subst}\left (\int \frac{b^6-9 a b^4 c+23 a^2 b^2 c^2-10 a^3 c^3+b c \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^4 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac{b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}+\frac{3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}-\frac{3 b \log (x)}{a^4}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4}+\frac{\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac{b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}+\frac{3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}-\frac{3 b \log (x)}{a^4}+\frac{3 b \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac{\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 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^2\right )}{2 a^4 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac{b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}+\frac{3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}-\frac{3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}-\frac{3 b \log (x)}{a^4}+\frac{3 b \log \left (a+b x^2+c x^4\right )}{4 a^4}\\ \end{align*}

Mathematica [A]  time = 0.614424, size = 402, normalized size = 1.58 \[ \frac{\frac{a^2 \left (-3 a b c-2 a c^2 x^2+b^2 c x^2+b^3\right )}{\left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}-\frac{a \left (46 a^2 b c^2+28 a^2 c^3 x^2-26 a b^2 c^2 x^2-29 a b^3 c+4 b^4 c x^2+4 b^5\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 \left (30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}-20 a^3 c^3+b^5 \sqrt{b^2-4 a c}-10 a b^4 c-8 a b^3 c \sqrt{b^2-4 a c}+b^6\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 \left (-30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}+20 a^3 c^3+b^5 \sqrt{b^2-4 a c}+10 a b^4 c-8 a b^3 c \sqrt{b^2-4 a c}-b^6\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{2 a}{x^2}-12 b \log (x)}{4 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a)/x^2 + (a^2*(b^3 - 3*a*b*c + b^2*c*x^2 - 2*a*c^2*x^2))/((-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) - (a*(4*b
^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x^2 - 26*a*b^2*c^2*x^2 + 28*a^2*c^3*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2
+ c*x^4)) - 12*b*Log[x] + (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^
3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5
/2) + (3*(-b^6 + 10*a*b^4*c - 30*a^2*b^2*c^2 + 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c
] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/(4*a^4)

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Maple [B]  time = 0.192, size = 1002, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a^3/x^2-3*b*ln(x)/a^4-7/a/(c*x^4+b*x^2+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+13/2/a^2/(c*x^4+b*x^2+a)^2
*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*b^2-1/a^3/(c*x^4+b*x^2+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*b^4-37/2/a/
(c*x^4+b*x^2+a)^2*b*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+55/4/a^2/(c*x^4+b*x^2+a)^2*b^3*c^2/(16*a^2*c^2-8*a*b^2*
c+b^4)*x^4-2/a^3/(c*x^4+b*x^2+a)^2*b^5*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-9/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^
2*c+b^4)*x^2*c^3-7/2/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^2*c^2+6/a^2/(c*x^4+b*x^2+a)^2/(16*a^
2*c^2-8*a*b^2*c+b^4)*x^2*b^4*c-1/a^3/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^6-29/2/(c*x^4+b*x^2+a)
^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2+9/a/(c*x^4+b*x^2+a)^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c-5/4/a^2/(c*x^4+b*x^
2+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)+12/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*ln(c*x^4+b*x^2+a)*b-6/a^3/(16*a^2*
c^2-8*a*b^2*c+b^4)*c*ln(c*x^4+b*x^2+a)*b^3+3/4/a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*b^5-30/a/(16*a
^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*c^3+45/a^2/(16*a^2*c^2-8*a*b^2*c
+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^2*c^2-15/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c
-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^4*c+3/2/a^4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*a
rctan((2*c*x^2+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/x^3/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 6.09586, size = 4906, normalized size = 19.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*a^3*b^6 - 24*a^4*b^4*c + 96*a^5*b^2*c^2 - 128*a^6*c^3 + 6*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^
4 - 40*a^4*c^5)*x^8 + 3*(4*a*b^7*c - 45*a^2*b^5*c^2 + 162*a^3*b^3*c^3 - 184*a^4*b*c^4)*x^6 + 2*(3*a*b^8 - 30*a
^2*b^6*c + 79*a^3*b^4*c^2 + 22*a^4*b^2*c^3 - 200*a^5*c^4)*x^4 + (9*a^2*b^7 - 104*a^3*b^5*c + 394*a^4*b^3*c^2 -
488*a^5*b*c^3)*x^2 + 3*((b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^10 + 2*(b^7*c - 10*a*b^5*c^2
+ 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^8 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^6 +
2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*x^4 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*
a^5*c^3)*x^2)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c
*x^4 + b*x^2 + a)) - 3*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^10 + 2*(b^8*c - 12*a*b^6*c^
2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^8 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 128*a^4*b*c^4
)*x^6 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^4 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*
c^2 - 64*a^5*b*c^3)*x^2)*log(c*x^4 + b*x^2 + a) + 12*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)
*x^10 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^8 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 +
32*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^6 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^4 + (a^2*b^
7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x^2)*log(x))/((a^4*b^6*c^2 - 12*a^5*b^4*c^3 + 48*a^6*b^2*c^4
- 64*a^7*c^5)*x^10 + 2*(a^4*b^7*c - 12*a^5*b^5*c^2 + 48*a^6*b^3*c^3 - 64*a^7*b*c^4)*x^8 + (a^4*b^8 - 10*a^5*b
^6*c + 24*a^6*b^4*c^2 + 32*a^7*b^2*c^3 - 128*a^8*c^4)*x^6 + 2*(a^5*b^7 - 12*a^6*b^5*c + 48*a^7*b^3*c^2 - 64*a^
8*b*c^3)*x^4 + (a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*x^2), -1/4*(2*a^3*b^6 - 24*a^4*b^4*c + 9
6*a^5*b^2*c^2 - 128*a^6*c^3 + 6*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a^4*c^5)*x^8 + 3*(4*a*b^7*c
- 45*a^2*b^5*c^2 + 162*a^3*b^3*c^3 - 184*a^4*b*c^4)*x^6 + 2*(3*a*b^8 - 30*a^2*b^6*c + 79*a^3*b^4*c^2 + 22*a^4*
b^2*c^3 - 200*a^5*c^4)*x^4 + (9*a^2*b^7 - 104*a^3*b^5*c + 394*a^4*b^3*c^2 - 488*a^5*b*c^3)*x^2 + 6*((b^6*c^2 -
10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^10 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x
^8 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^6 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b
^3*c^2 - 20*a^4*b*c^3)*x^4 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3)*x^2)*sqrt(-b^2 + 4*a*c)*ar
ctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - 3*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b
*c^5)*x^10 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^8 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c
^2 + 32*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^6 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^4 + (a
^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x^2)*log(c*x^4 + b*x^2 + a) + 12*((b^7*c^2 - 12*a*b^5*c
^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^10 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^8 + (b
^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^6 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*
c^2 - 64*a^4*b^2*c^3)*x^4 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x^2)*log(x))/((a^4*b^6*c^
2 - 12*a^5*b^4*c^3 + 48*a^6*b^2*c^4 - 64*a^7*c^5)*x^10 + 2*(a^4*b^7*c - 12*a^5*b^5*c^2 + 48*a^6*b^3*c^3 - 64*a
^7*b*c^4)*x^8 + (a^4*b^8 - 10*a^5*b^6*c + 24*a^6*b^4*c^2 + 32*a^7*b^2*c^3 - 128*a^8*c^4)*x^6 + 2*(a^5*b^7 - 12
*a^6*b^5*c + 48*a^7*b^3*c^2 - 64*a^8*b*c^3)*x^4 + (a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 31.9524, size = 516, normalized size = 2.02 \begin{align*} \frac{3 \,{\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{9 \, b^{5} c^{2} x^{8} - 72 \, a b^{3} c^{3} x^{8} + 144 \, a^{2} b c^{4} x^{8} + 18 \, b^{6} c x^{6} - 136 \, a b^{4} c^{2} x^{6} + 236 \, a^{2} b^{2} c^{3} x^{6} + 56 \, a^{3} c^{4} x^{6} + 9 \, b^{7} x^{4} - 38 \, a b^{5} c x^{4} - 110 \, a^{2} b^{3} c^{2} x^{4} + 436 \, a^{3} b c^{3} x^{4} + 26 \, a b^{6} x^{2} - 192 \, a^{2} b^{4} c x^{2} + 316 \, a^{3} b^{2} c^{2} x^{2} + 72 \, a^{4} c^{3} x^{2} + 19 \, a^{2} b^{5} - 144 \, a^{3} b^{3} c + 260 \, a^{4} b c^{2}}{8 \,{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} + \frac{3 \, b \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} - \frac{3 \, b \log \left (x^{2}\right )}{2 \, a^{4}} + \frac{3 \, b x^{2} - a}{2 \, a^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^4 - 8*a^
5*b^2*c + 16*a^6*c^2)*sqrt(-b^2 + 4*a*c)) - 1/8*(9*b^5*c^2*x^8 - 72*a*b^3*c^3*x^8 + 144*a^2*b*c^4*x^8 + 18*b^6
*c*x^6 - 136*a*b^4*c^2*x^6 + 236*a^2*b^2*c^3*x^6 + 56*a^3*c^4*x^6 + 9*b^7*x^4 - 38*a*b^5*c*x^4 - 110*a^2*b^3*c
^2*x^4 + 436*a^3*b*c^3*x^4 + 26*a*b^6*x^2 - 192*a^2*b^4*c*x^2 + 316*a^3*b^2*c^2*x^2 + 72*a^4*c^3*x^2 + 19*a^2*
b^5 - 144*a^3*b^3*c + 260*a^4*b*c^2)/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*(c*x^4 + b*x^2 + a)^2) + 3/4*b*log(
c*x^4 + b*x^2 + a)/a^4 - 3/2*b*log(x^2)/a^4 + 1/2*(3*b*x^2 - a)/(a^4*x^2)