3.429 \(\int \frac{1}{(c+\frac{a}{x^2}+\frac{b}{x})^2 x^6} \, dx\)

Optimal. Leaf size=148 \[ -\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac{2 \left (b^2-3 a c\right )}{a^2 x \left (b^2-4 a c\right )}+\frac{b \log \left (a+b x+c x^2\right )}{a^3}-\frac{2 b \log (x)}{a^3}+\frac{-2 a c+b^2+b c x}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.182122, antiderivative size = 148, 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 = {1354, 740, 800, 634, 618, 206, 628} \[ -\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac{2 \left (b^2-3 a c\right )}{a^2 x \left (b^2-4 a c\right )}+\frac{b \log \left (a+b x+c x^2\right )}{a^3}-\frac{2 b \log (x)}{a^3}+\frac{-2 a c+b^2+b c x}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + a/x^2 + b/x)^2*x^6),x]

[Out]

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

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

Mathematica [A]  time = 0.293125, size = 131, normalized size = 0.89 \[ -\frac{\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{a \left (-3 a b c-2 a c^2 x+b^2 c x+b^3\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-b \log (a+x (b+c x))+\frac{a}{x}+2 b \log (x)}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + a/x^2 + b/x)^2*x^6),x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.014, size = 328, normalized size = 2.2 \begin{align*} -{\frac{1}{x{a}^{2}}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}-2\,{\frac{{c}^{2}x}{a \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{cx{b}^{2}}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-3\,{\frac{bc}{a \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{{b}^{3}}{{a}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+4\,{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) b}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}}{{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }}-12\,{\frac{{c}^{2}}{a \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{{b}^{2}c}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{{b}^{4}}{{a}^{3} \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+a/x^2+b/x)^2/x^6,x)

[Out]

-1/a^2/x-2*b*ln(x)/a^3-2/a/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x+1/a^2/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*b^2-3/a/(c*x^2+
b*x+a)*b/(4*a*c-b^2)*c+1/a^2/(c*x^2+b*x+a)*b^3/(4*a*c-b^2)+4/a^2/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b-1/a^3/(4*a*c-
b^2)*ln(c*x^2+b*x+a)*b^3-12/a/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2+12/a^2/(4*a*c-b^2)^(3/
2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c-2/a^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4

________________________________________________________________________________________

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)^2/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.8916, size = 2049, normalized size = 13.84 \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)^2/x^6,x, algorithm="fricas")

[Out]

[-(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + 2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*x^2 + ((b^4*c - 6*a*b^2*c^2 +
6*a^2*c^3)*x^3 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x^2 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*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)) + (2*a*b^5 - 15*a^
2*b^3*c + 28*a^3*b*c^2)*x - ((b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x^2
+ (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*log(c*x^2 + b*x + a) + 2*((b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3
+ (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x^2 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*log(x))/((a^3*b^4*c - 8*a^4
*b^2*c^2 + 16*a^5*c^3)*x^3 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)
*x), -(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + 2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*x^2 + 2*((b^4*c - 6*a*b^2
*c^2 + 6*a^2*c^3)*x^3 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x^2 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*x)*sqrt(-b^2 +
4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2)*x - ((
b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x^2 + (a*b^5 - 8*a^2*b^3*c + 16*a
^3*b*c^2)*x)*log(c*x^2 + b*x + a) + 2*((b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 8*a*b^4*c + 16*a^2*b^
2*c^2)*x^2 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*log(x))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^3 + (
a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*x)]

________________________________________________________________________________________

Sympy [B]  time = 11.4878, size = 2672, normalized size = 18.05 \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)**2/x**6,x)

[Out]

(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 +
12*a*b**4*c - b**6)))*log(x + (-1728*a**11*b*c**5*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*
c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 2256*a**10*b**3*c**4*(b/a**3 -
sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4
*c - b**6)))**2 - 1172*a**9*b**5*c**3*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a*
*3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 288*a**9*c**6*(b/a**3 - sqrt(-(4*a*c - b**2)
**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 303*a
**8*b**7*c**2*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a*
*2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 432*a**8*b**2*c**5*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 -
6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 39*a**7*b**9*c*(b/a**3 -
sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**
4*c - b**6)))**2 + 558*a**7*b**4*c**4*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a*
*3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 2*a**6*b**11*(b/a**3 - sqrt(-(4*a*c - b**2)**3)
*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 212*a*
*6*b**6*c**3*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**
2*b**2*c**2 + 12*a*b**4*c - b**6))) - 576*a**6*b*c**6 + 34*a**5*b**8*c**2*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(
6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 6048*a**5*b
**3*c**5 - 2*a**4*b**10*c*(b/a**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*
c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 7908*a**4*b**5*c**4 + 4264*a**3*b**7*c**3 - 1144*a**2*b**9*
c**2 + 152*a*b**11*c - 8*b**13)/(216*a**6*c**7 + 2808*a**5*b**2*c**6 - 5292*a**4*b**4*c**5 + 3384*a**3*b**6*c*
*4 - 1008*a**2*b**8*c**3 + 144*a*b**10*c**2 - 8*b**12*c)) + (b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 -
6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-1728*a**11*b*c**5
*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6)))**2 + 2256*a**10*b**3*c**4*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c
+ b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 1172*a**9*b**5*c**3*(b/a**3 + sq
rt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6)))**2 - 288*a**9*c**6*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a
**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 303*a**8*b**7*c**2*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(
6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 432*a**8
*b**2*c**5*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*
b**2*c**2 + 12*a*b**4*c - b**6))) - 39*a**7*b**9*c*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*
c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 + 558*a**7*b**4*c**4*(b/a**3 + sq
rt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6))) + 2*a**6*b**11*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*
c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))**2 - 212*a**6*b**6*c**3*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6
*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 576*a**6*b*c
**6 + 34*a**5*b**8*c**2*(b/a**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c*
*3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 6048*a**5*b**3*c**5 - 2*a**4*b**10*c*(b/a**3 + sqrt(-(4*a*c -
b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(a**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) -
7908*a**4*b**5*c**4 + 4264*a**3*b**7*c**3 - 1144*a**2*b**9*c**2 + 152*a*b**11*c - 8*b**13)/(216*a**6*c**7 + 2
808*a**5*b**2*c**6 - 5292*a**4*b**4*c**5 + 3384*a**3*b**6*c**4 - 1008*a**2*b**8*c**3 + 144*a*b**10*c**2 - 8*b*
*12*c)) - (4*a**2*c - a*b**2 + x**2*(6*a*c**2 - 2*b**2*c) + x*(7*a*b*c - 2*b**3))/(x**3*(4*a**3*c**2 - a**2*b*
*2*c) + x**2*(4*a**3*b*c - a**2*b**3) + x*(4*a**4*c - a**3*b**2)) - 2*b*log(x)/a**3

________________________________________________________________________________________

Giac [A]  time = 1.13286, size = 231, normalized size = 1.56 \begin{align*} \frac{2 \,{\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{2} c x^{2} - 6 \, a c^{2} x^{2} + 2 \, b^{3} x - 7 \, a b c x + a b^{2} - 4 \, a^{2} c}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )}{\left (c x^{3} + b x^{2} + a x\right )}} + \frac{b \log \left (c x^{2} + b x + a\right )}{a^{3}} - \frac{2 \, b \log \left ({\left | x \right |}\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x^2+b/x)^2/x^6,x, algorithm="giac")

[Out]

2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^3*b^2 - 4*a^4*c)*sqrt(-b^2 + 4*a*c)
) - (2*b^2*c*x^2 - 6*a*c^2*x^2 + 2*b^3*x - 7*a*b*c*x + a*b^2 - 4*a^2*c)/((a^2*b^2 - 4*a^3*c)*(c*x^3 + b*x^2 +
a*x)) + b*log(c*x^2 + b*x + a)/a^3 - 2*b*log(abs(x))/a^3