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

Optimal. Leaf size=137 \[ -\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c}}+\frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac{b^2-a c}{a^3 x}-\frac{b \log (x) \left (b^2-2 a c\right )}{a^4}+\frac{b}{2 a^2 x^2}-\frac{1}{3 a x^3} \]

[Out]

-1/(3*a*x^3) + b/(2*a^2*x^2) - (b^2 - a*c)/(a^3*x) - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sqrt[b
^2 - 4*a*c]])/(a^4*Sqrt[b^2 - 4*a*c]) - (b*(b^2 - 2*a*c)*Log[x])/a^4 + (b*(b^2 - 2*a*c)*Log[a + b*x + c*x^2])/
(2*a^4)

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Rubi [A]  time = 0.1928, antiderivative size = 137, 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, 709, 800, 634, 618, 206, 628} \[ -\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c}}+\frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac{b^2-a c}{a^3 x}-\frac{b \log (x) \left (b^2-2 a c\right )}{a^4}+\frac{b}{2 a^2 x^2}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + a/x^2 + b/x)*x^6),x]

[Out]

-1/(3*a*x^3) + b/(2*a^2*x^2) - (b^2 - a*c)/(a^3*x) - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sqrt[b
^2 - 4*a*c]])/(a^4*Sqrt[b^2 - 4*a*c]) - (b*(b^2 - 2*a*c)*Log[x])/a^4 + (b*(b^2 - 2*a*c)*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 709

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d - b*e - c
*e*x, x])/(a + b*x + c*x^2), 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[m, -1]

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

Mathematica [A]  time = 0.100566, size = 131, normalized size = 0.96 \[ \frac{\frac{6 \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{3 a^2 b}{x^2}-\frac{2 a^3}{x^3}+\frac{6 a \left (a c-b^2\right )}{x}-6 \log (x) \left (b^3-2 a b c\right )+3 \left (b^3-2 a b c\right ) \log (a+x (b+c x))}{6 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 214, normalized size = 1.6 \begin{align*} -{\frac{1}{3\,a{x}^{3}}}+{\frac{c}{x{a}^{2}}}-{\frac{{b}^{2}}{{a}^{3}x}}+2\,{\frac{b\ln \left ( x \right ) c}{{a}^{3}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}}+{\frac{b}{2\,{a}^{2}{x}^{2}}}-{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) b}{{a}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}}{2\,{a}^{4}}}+2\,{\frac{{c}^{2}}{{a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{c{b}^{2}}{{a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}}{{a}^{4}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c+a/x^2+b/x)/x^6,x)

[Out]

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

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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)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.26154, size = 979, normalized size = 7.15 \begin{align*} \left [\frac{3 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \sqrt{b^{2} - 4 \, a c} x^{3} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 2 \, a^{3} b^{2} + 8 \, a^{4} c + 3 \,{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} x^{3} \log \left (c x^{2} + b x + a\right ) - 6 \,{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} x^{3} \log \left (x\right ) - 6 \,{\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x}{6 \,{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{3}}, -\frac{6 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c} x^{3} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, a^{3} b^{2} - 8 \, a^{4} c - 3 \,{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} x^{3} \log \left (c x^{2} + b x + a\right ) + 6 \,{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} x^{3} \log \left (x\right ) + 6 \,{\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} x^{2} - 3 \,{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x}{6 \,{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*sqrt(b^2 - 4*a*c)*x^3*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^2 + 8*a^4*c + 3*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*x^3*log(c*x
^2 + b*x + a) - 6*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*x^3*log(x) - 6*(a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*x^2 + 3*(a^
2*b^3 - 4*a^3*b*c)*x)/((a^4*b^2 - 4*a^5*c)*x^3), -1/6*(6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*sqrt(-b^2 + 4*a*c)*x^3*
arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*a^3*b^2 - 8*a^4*c - 3*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2
)*x^3*log(c*x^2 + b*x + a) + 6*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*x^3*log(x) + 6*(a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2
)*x^2 - 3*(a^2*b^3 - 4*a^3*b*c)*x)/((a^4*b^2 - 4*a^5*c)*x^3)]

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Sympy [B]  time = 8.59681, size = 2105, normalized size = 15.36 \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)/x**6,x)

[Out]

(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2)))*l
og(x + (-52*a**11*b*c**3*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(
2*a**4*(4*a*c - b**2)))**2 + 57*a**10*b**3*c**2*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2
- 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2)))**2 - 19*a**9*b**5*c*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c +
b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2)))**2 + 4*a**9*c**5*(-b*(2*a*c - b**2)/(2*a**4)
- sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2))) + 2*a**8*b**7*(-b*(2*a*c - b
**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2)))**2 + 23*a**8*b*
*2*c**4*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b
**2))) - 26*a**7*b**4*c**3*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)
/(2*a**4*(4*a*c - b**2))) + 9*a**6*b**6*c**2*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 -
4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2))) - 8*a**6*b*c**6 - a**5*b**8*c*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-
4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2))) + 166*a**5*b**3*c**5 - 361*a**4*b**5*
c**4 + 312*a**3*b**7*c**3 - 130*a**2*b**9*c**2 + 26*a*b**11*c - 2*b**13)/(2*a**6*c**7 + 60*a**5*b**2*c**6 - 20
7*a**4*b**4*c**5 + 224*a**3*b**6*c**4 - 108*a**2*b**8*c**3 + 24*a*b**10*c**2 - 2*b**12*c)) + (-b*(2*a*c - b**2
)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2)))*log(x + (-52*a**11
*b*c**3*(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b
**2)))**2 + 57*a**10*b**3*c**2*(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b
**4)/(2*a**4*(4*a*c - b**2)))**2 - 19*a**9*b**5*c*(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c*
*2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2)))**2 + 4*a**9*c**5*(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c +
b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2))) + 2*a**8*b**7*(-b*(2*a*c - b**2)/(2*a**4) + s
qrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2)))**2 + 23*a**8*b**2*c**4*(-b*(2*a*
c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2))) - 26*a**7*
b**4*c**3*(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c -
b**2))) + 9*a**6*b**6*c**2*(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4
)/(2*a**4*(4*a*c - b**2))) - 8*a**6*b*c**6 - a**5*b**8*c*(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*
a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2))) + 166*a**5*b**3*c**5 - 361*a**4*b**5*c**4 + 312*a**3*b
**7*c**3 - 130*a**2*b**9*c**2 + 26*a*b**11*c - 2*b**13)/(2*a**6*c**7 + 60*a**5*b**2*c**6 - 207*a**4*b**4*c**5
+ 224*a**3*b**6*c**4 - 108*a**2*b**8*c**3 + 24*a*b**10*c**2 - 2*b**12*c)) + (-2*a**2 + 3*a*b*x + x**2*(6*a*c -
6*b**2))/(6*a**3*x**3) + b*(2*a*c - b**2)*log(x + (-8*a**6*b*c**6 + 166*a**5*b**3*c**5 + 4*a**5*b*c**5*(2*a*c
- b**2) - 361*a**4*b**5*c**4 + 23*a**4*b**3*c**4*(2*a*c - b**2) + 312*a**3*b**7*c**3 - 26*a**3*b**5*c**3*(2*a
*c - b**2) - 52*a**3*b**3*c**3*(2*a*c - b**2)**2 - 130*a**2*b**9*c**2 + 9*a**2*b**7*c**2*(2*a*c - b**2) + 57*a
**2*b**5*c**2*(2*a*c - b**2)**2 + 26*a*b**11*c - a*b**9*c*(2*a*c - b**2) - 19*a*b**7*c*(2*a*c - b**2)**2 - 2*b
**13 + 2*b**9*(2*a*c - b**2)**2)/(2*a**6*c**7 + 60*a**5*b**2*c**6 - 207*a**4*b**4*c**5 + 224*a**3*b**6*c**4 -
108*a**2*b**8*c**3 + 24*a*b**10*c**2 - 2*b**12*c))/a**4

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Giac [A]  time = 1.15574, size = 184, normalized size = 1.34 \begin{align*} \frac{{\left (b^{3} - 2 \, a b c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} - \frac{{\left (b^{3} - 2 \, a b c\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a^{4}} + \frac{3 \, a^{2} b x - 2 \, a^{3} - 6 \,{\left (a b^{2} - a^{2} c\right )} x^{2}}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^3 - 2*a*b*c)*log(c*x^2 + b*x + a)/a^4 - (b^3 - 2*a*b*c)*log(abs(x))/a^4 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)
*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^4) + 1/6*(3*a^2*b*x - 2*a^3 - 6*(a*b^2 - a^2*c)*
x^2)/(a^4*x^3)