3.439 \(\int \frac{1}{(c+\frac{a}{x^2}+\frac{b}{x})^3 x^8} \, dx\)

Optimal. Leaf size=239 \[ \frac{20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{2 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{5/2}}-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 x \left (b^2-4 a c\right )^2}+\frac{3 b \log \left (a+b x+c x^2\right )}{2 a^4}-\frac{3 b \log (x)}{a^4}+\frac{-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.277369, antiderivative size = 239, 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 = {1354, 740, 822, 800, 634, 618, 206, 628} \[ \frac{20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{2 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{5/2}}-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 x \left (b^2-4 a c\right )^2}+\frac{3 b \log \left (a+b x+c x^2\right )}{2 a^4}-\frac{3 b \log (x)}{a^4}+\frac{-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + a/x^2 + b/x)^3*x^8),x]

[Out]

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

Rule 1354

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + 2*n*p)*(c + b/x^n +
a/x^(2*n))^p, x] /; FreeQ[{a, b, c, m, n}, x] && EqQ[n2, 2*n] && ILtQ[p, 0] && NegQ[n]

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}{\left (c+\frac{a}{x^2}+\frac{b}{x}\right )^3 x^8} \, dx &=\int \frac{1}{x^2 \left (a+b x+c x^2\right )^3} \, dx\\ &=\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^2}-\frac{\int \frac{-3 b^2+10 a c-4 b c x}{x^2 \left (a+b x+c x^2\right )^2} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\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 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}+\frac{\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}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\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 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}+\frac{\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}{2 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 )}{a^3 \left (b^2-4 a c\right )^2 x}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\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 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac{3 b \log (x)}{a^4}+\frac{3 \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}{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 )}{a^3 \left (b^2-4 a c\right )^2 x}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\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 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac{3 b \log (x)}{a^4}+\frac{(3 b) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 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 ) \int \frac{1}{a+b x+c x^2} \, dx}{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 )}{a^3 \left (b^2-4 a c\right )^2 x}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\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 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac{3 b \log (x)}{a^4}+\frac{3 b \log \left (a+b x+c x^2\right )}{2 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\right )}{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 )}{a^3 \left (b^2-4 a c\right )^2 x}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\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 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\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}{\sqrt{b^2-4 a c}}\right )}{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+c x^2\right )}{2 a^4}\\ \end{align*}

Mathematica [A]  time = 0.471998, size = 221, normalized size = 0.92 \[ \frac{\frac{a^2 \left (-3 a b c-2 a c^2 x+b^2 c x+b^3\right )}{\left (4 a c-b^2\right ) (a+x (b+c x))^2}-\frac{a \left (46 a^2 b c^2+28 a^2 c^3 x-26 a b^2 c^2 x-29 a b^3 c+4 b^4 c x+4 b^5\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{6 \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+3 b \log (a+x (b+c x))-\frac{2 a}{x}-6 b \log (x)}{2 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + a/x^2 + b/x)^3*x^8),x]

[Out]

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

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Maple [B]  time = 0.02, size = 954, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c+a/x^2+b/x)^3/x^8,x)

[Out]

-1/a^3/x-3*b*ln(x)/a^4-14/a/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+13/a^2/(c*x^2+b*x+a)^2*c^3/(16*
a^2*c^2-8*a*b^2*c+b^4)*x^3*b^2-2/a^3/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b^4-37/a/(c*x^2+b*x+a)
^2*b*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+55/2/a^2/(c*x^2+b*x+a)^2*b^3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-4/a^3/
(c*x^2+b*x+a)^2*b^5*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-18/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*c^3-7/a/(
c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^2*c^2+12/a^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^4*c-
2/a^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^6-29/(c*x^2+b*x+a)^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2+18/
a/(c*x^2+b*x+a)^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c-5/2/a^2/(c*x^2+b*x+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)+24/a
^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*ln(c*x^2+b*x+a)*b-12/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln(c*x^2+b*x+a)*b^3+3/
2/a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*b^5-60/a/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan(
(2*c*x+b)/(4*a*c-b^2)^(1/2))*c^3+90/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+b)/(4*a*c-b
^2)^(1/2))*b^2*c^2-30/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+b)/(4*a*c-b^2)^(1/2))*b^4
*c+3/a^4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((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+a/x^2+b/x)^3/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 6.04604, size = 4849, normalized size = 20.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x^2+b/x)^3/x^8,x, algorithm="fricas")

[Out]

[-1/2*(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^4 + 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^3 + 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^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^5 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^4 + (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^3 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*x^2 + (a^2*
b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3)*x)*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)) + (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 - 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^5 + 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^4 + (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^3 + 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^2 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5
*b*c^3)*x)*log(c*x^2 + b*x + a) + 6*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 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^4 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 1
28*a^4*b*c^4)*x^3 + 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^2 + (a^2*b^7 - 12*a^3*b^5*c +
48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*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^5 +
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^4 + (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^3 + 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^2 + (a^6*b
^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*x), -1/2*(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^4 + 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^3 + 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^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^5 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3
- 20*a^3*b*c^4)*x^4 + (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^3 + 2*(a*b^7 - 10*a^
2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*x^2 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3)*x)*sqrt(
-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (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 - 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^5 + 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^4 + (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^3 + 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^2 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*
b^3*c^2 - 64*a^5*b*c^3)*x)*log(c*x^2 + b*x + a) + 6*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*
x^5 + 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^4 + (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^3 + 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^2 + (a^2*b^7
- 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*log(x))/((a^4*b^6*c^2 - 12*a^5*b^4*c^3 + 48*a^6*b^2*c^4 - 6
4*a^7*c^5)*x^5 + 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^4 + (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^3 + 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^2 + (a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*x)]

________________________________________________________________________________________

Sympy [B]  time = 52.6089, size = 5722, normalized size = 23.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x**2+b/x)**3/x**8,x)

[Out]

(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(10
24*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))*log(x +
(-108544*a**16*b*c**8*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4
*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8
*c - b**10)))**2 + 224768*a**15*b**3*c**7*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b
**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b
**6*c**2 + 20*a*b**8*c - b**10)))**2 - 202752*a**14*b**5*c**6*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a
**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b*
*4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 + 104128*a**13*b**7*c**5*(3*b/(2*a**4) - 3*sqrt(-(4*a
*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**
2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 19200*a**13*c**9*(3*b/(2*a**4)
- 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 -
1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 33320*a**12*b**9*c**
4*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(
1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 4
4736*a**12*b**2*c**8*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*
c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*
c - b**10))) + 6806*a**11*b**11*c**3*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c
**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c
**2 + 20*a*b**8*c - b**10)))**2 + 101232*a**11*b**4*c**7*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c
**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c*
*3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 867*a**10*b**13*c**2*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)
**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 6
40*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 77268*a**10*b**6*c**6*(3*b/(2*a**4) - 3*s
qrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280
*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 63*a**9*b**15*c*(3*b/(2*a
**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c
**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 + 31368*a**9*b
**8*c**5*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2
*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))
- 57600*a**9*b*c**9 - 2*a**8*b**17*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c*
*2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c*
*2 + 20*a*b**8*c - b**10)))**2 - 7545*a**8*b**10*c**4*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3
- 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3
- 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 842688*a**8*b**3*c**8 + 1086*a**7*b**12*c**3*(3*b/(2*a**4) - 3
*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 12
80*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 1719216*a**7*b**5*c**7
- 87*a**6*b**14*c**2*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*
c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*
c - b**10))) + 1592964*a**6*b**7*c**6 + 3*a**5*b**16*c*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**
3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3
- 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 843048*a**5*b**9*c**5 + 277245*a**4*b**11*c**4 - 57996*a**3*b
**13*c**3 + 7542*a**2*b**15*c**2 - 558*a*b**17*c + 18*b**19)/(18000*a**9*c**10 + 333720*a**8*b**2*c**9 - 99198
0*a**7*b**4*c**8 + 1099710*a**6*b**6*c**7 - 651186*a**5*b**8*c**6 + 231795*a**4*b**10*c**5 - 51480*a**3*b**12*
c**4 + 7020*a**2*b**14*c**3 - 540*a*b**16*c**2 + 18*b**18*c)) + (3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20
*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*
b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))*log(x + (-108544*a**16*b*c**8*(3*b/(2*a**4) + 3*sqrt(-
(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4
*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 + 224768*a**15*b**3*c**7*(3*b
/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a
**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 202752*
a**14*b**5*c**6*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b
**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b
**10)))**2 + 104128*a**13*b**7*c**5*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c*
*2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c*
*2 + 20*a*b**8*c - b**10)))**2 - 19200*a**13*c**9*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 3
0*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 16
0*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 33320*a**12*b**9*c**4*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(
20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**
3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 44736*a**12*b**2*c**8*(3*b/(2*a**4) + 3*sqrt(-(
4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*
b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 6806*a**11*b**11*c**3*(3*b/(2*a
**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c
**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 + 101232*a**11
*b**4*c**7*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/
(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)
)) - 867*a**10*b**13*c**2*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*
b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*
b**8*c - b**10)))**2 - 77268*a**10*b**6*c**6*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**
2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**
2*b**6*c**2 + 20*a*b**8*c - b**10))) + 63*a**9*b**15*c*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**
3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3
- 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 + 31368*a**9*b**8*c**5*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2
)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 +
640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 57600*a**9*b*c**9 - 2*a**8*b**17*(3*b/(2*a*
*4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c*
*5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 7545*a**8*b**
10*c**4*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*
a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))
+ 842688*a**8*b**3*c**8 + 1086*a**7*b**12*c**3*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a
**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a
**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 1719216*a**7*b**5*c**7 - 87*a**6*b**14*c**2*(3*b/(2*a**4) + 3*sqrt(-(
4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*
b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 1592964*a**6*b**7*c**6 + 3*a**5
*b**16*c*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2
*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))
- 843048*a**5*b**9*c**5 + 277245*a**4*b**11*c**4 - 57996*a**3*b**13*c**3 + 7542*a**2*b**15*c**2 - 558*a*b**17
*c + 18*b**19)/(18000*a**9*c**10 + 333720*a**8*b**2*c**9 - 991980*a**7*b**4*c**8 + 1099710*a**6*b**6*c**7 - 65
1186*a**5*b**8*c**6 + 231795*a**4*b**10*c**5 - 51480*a**3*b**12*c**4 + 7020*a**2*b**14*c**3 - 540*a*b**16*c**2
+ 18*b**18*c)) - (32*a**4*c**2 - 16*a**3*b**2*c + 2*a**2*b**4 + x**4*(60*a**2*c**4 - 42*a*b**2*c**3 + 6*b**4*
c**2) + x**3*(138*a**2*b*c**3 - 87*a*b**3*c**2 + 12*b**5*c) + x**2*(100*a**3*c**3 + 14*a**2*b**2*c**2 - 36*a*b
**4*c + 6*b**6) + x*(122*a**3*b*c**2 - 68*a**2*b**3*c + 9*a*b**5))/(x**5*(32*a**5*c**4 - 16*a**4*b**2*c**3 + 2
*a**3*b**4*c**2) + x**4*(64*a**5*b*c**3 - 32*a**4*b**3*c**2 + 4*a**3*b**5*c) + x**3*(64*a**6*c**3 - 12*a**4*b*
*4*c + 2*a**3*b**6) + x**2*(64*a**6*b*c**2 - 32*a**5*b**3*c + 4*a**4*b**5) + x*(32*a**7*c**2 - 16*a**6*b**2*c
+ 2*a**5*b**4)) - 3*b*log(x)/a**4

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Giac [A]  time = 1.14143, size = 417, normalized size = 1.74 \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 + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{3 \, b \log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} - \frac{3 \, b \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{2 \, a^{3} b^{4} - 16 \, a^{4} b^{2} c + 32 \, a^{5} c^{2} + 6 \,{\left (a b^{4} c^{2} - 7 \, a^{2} b^{2} c^{3} + 10 \, a^{3} c^{4}\right )} x^{4} + 3 \,{\left (4 \, a b^{5} c - 29 \, a^{2} b^{3} c^{2} + 46 \, a^{3} b c^{3}\right )} x^{3} + 2 \,{\left (3 \, a b^{6} - 18 \, a^{2} b^{4} c + 7 \, a^{3} b^{2} c^{2} + 50 \, a^{4} c^{3}\right )} x^{2} +{\left (9 \, a^{2} b^{5} - 68 \, a^{3} b^{3} c + 122 \, a^{4} b c^{2}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )}^{2} a^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x^2+b/x)^3/x^8,x, algorithm="giac")

[Out]

3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctan((2*c*x + 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)) + 3/2*b*log(c*x^2 + b*x + a)/a^4 - 3*b*log(abs(x))/a^4 - 1/2*(2*a^3*b^4
- 16*a^4*b^2*c + 32*a^5*c^2 + 6*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*x^4 + 3*(4*a*b^5*c - 29*a^2*b^3*c^2 +
46*a^3*b*c^3)*x^3 + 2*(3*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2 + 50*a^4*c^3)*x^2 + (9*a^2*b^5 - 68*a^3*b^3*c +
122*a^4*b*c^2)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*a^4*x)