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

Optimal. Leaf size=196 \[ \frac{b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 \left (3 b^2-8 a c\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac{b x \left (3 b^2-11 a c\right )}{c^3 \left (b^2-4 a c\right )}+\frac{x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b x^3}{c \left (b^2-4 a c\right )} \]

[Out]

-((b*(3*b^2 - 11*a*c)*x)/(c^3*(b^2 - 4*a*c))) + ((3*b^2 - 8*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)) - (b*x^3)/(c*(b^2
- 4*a*c)) + (x^4*(2*a + b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTanh
[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*Log[a + b*x + c*x^2])/(2*c^4)

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Rubi [A]  time = 0.204786, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {1354, 738, 800, 634, 618, 206, 628} \[ \frac{b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 \left (3 b^2-8 a c\right )}{2 c^2 \left (b^2-4 a c\right )}+\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac{b x \left (3 b^2-11 a c\right )}{c^3 \left (b^2-4 a c\right )}+\frac{x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b x^3}{c \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(c + a/x^2 + b/x)^2,x]

[Out]

-((b*(3*b^2 - 11*a*c)*x)/(c^3*(b^2 - 4*a*c))) + ((3*b^2 - 8*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)) - (b*x^3)/(c*(b^2
- 4*a*c)) + (x^4*(2*a + b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTanh
[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*Log[a + b*x + c*x^2])/(2*c^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 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, 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] && GtQ[m, 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{x}{\left (c+\frac{a}{x^2}+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^5}{\left (a+b x+c x^2\right )^2} \, dx\\ &=\frac{x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \frac{x^3 (8 a+3 b x)}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac{x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \left (\frac{b \left (3 b^2-11 a c\right )}{c^3}-\frac{\left (3 b^2-8 a c\right ) x}{c^2}+\frac{3 b x^2}{c}-\frac{a b \left (3 b^2-11 a c\right )+\left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx}{-b^2+4 a c}\\ &=-\frac{b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac{\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac{b x^3}{c \left (b^2-4 a c\right )}+\frac{x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \frac{a b \left (3 b^2-11 a c\right )+\left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=-\frac{b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac{\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac{b x^3}{c \left (b^2-4 a c\right )}+\frac{x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (3 b^2-2 a c\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}-\frac{\left (b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )}\\ &=-\frac{b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac{\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac{b x^3}{c \left (b^2-4 a c\right )}+\frac{x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{\left (b \left (3 b^4-20 a b^2 c+30 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 )}{c^4 \left (b^2-4 a c\right )}\\ &=-\frac{b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac{\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac{b x^3}{c \left (b^2-4 a c\right )}+\frac{x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end{align*}

Mathematica [A]  time = 0.22912, size = 163, normalized size = 0.83 \[ \frac{\frac{2 \left (a^2 b c (5 c x-4 b)+2 a^3 c^2+a b^3 (b-5 c x)+b^5 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{2 b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\left (3 b^2-2 a c\right ) \log (a+x (b+c x))-4 b c x+c^2 x^2}{2 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(c + a/x^2 + b/x)^2,x]

[Out]

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

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Maple [B]  time = 0.01, size = 434, normalized size = 2.2 \begin{align*}{\frac{{x}^{2}}{2\,{c}^{2}}}-2\,{\frac{bx}{{c}^{3}}}-5\,{\frac{bx{a}^{2}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+5\,{\frac{{b}^{3}xa}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{5}x}{{c}^{4} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{{a}^{3}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+4\,{\frac{{a}^{2}{b}^{2}}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{a{b}^{4}}{{c}^{4} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-4\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}}{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}}+7\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}}{{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{3\,\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}}{2\,{c}^{4} \left ( 4\,ac-{b}^{2} \right ) }}+30\,{\frac{{a}^{2}b}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-20\,{\frac{a{b}^{3}}{{c}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+3\,{\frac{{b}^{5}}{{c}^{4} \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(x/(c+a/x^2+b/x)^2,x)

[Out]

1/2/c^2*x^2-2/c^3*b*x-5/c^2/(c*x^2+b*x+a)*b/(4*a*c-b^2)*x*a^2+5/c^3/(c*x^2+b*x+a)*b^3/(4*a*c-b^2)*x*a-1/c^4/(c
*x^2+b*x+a)*b^5/(4*a*c-b^2)*x-2/c^2/(c*x^2+b*x+a)*a^3/(4*a*c-b^2)+4/c^3/(c*x^2+b*x+a)*a^2/(4*a*c-b^2)*b^2-1/c^
4/(c*x^2+b*x+a)*a/(4*a*c-b^2)*b^4-4/c^2/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a^2+7/c^3/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*b^
2-3/2/c^4/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^4+30/c^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b-2
0/c^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3+3/c^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*
a*c-b^2)^(1/2))*b^5

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.79405, size = 2201, normalized size = 11.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*a*b^6 - 16*a^2*b^4*c + 36*a^3*b^2*c^2 - 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 - 3*(b^5
*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 - (4*b^6*c - 33*a*b^4*c^2 + 72*a^2*b^2*c^3 - 16*a^3*c^4)*x^2 - (3*a*b^5
- 20*a^2*b^3*c + 30*a^3*b*c^2 + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*x^2 + (3*b^6 - 20*a*b^4*c + 30*a^2*b^
2*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*(b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*x + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 -
32*a^4*c^3 + (3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*x^2 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^
2 - 32*a^3*b*c^3)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^4 - 8*a^2*b^2*c^5 + 16*a^3*c^6 + (b^4*c^5 - 8*a*b^2*c^6 +
16*a^2*c^7)*x^2 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x), 1/2*(2*a*b^6 - 16*a^2*b^4*c + 36*a^3*b^2*c^2 - 16
*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 - 3*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 - (4*b^6*c
- 33*a*b^4*c^2 + 72*a^2*b^2*c^3 - 16*a^3*c^4)*x^2 + 2*(3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2 + (3*b^5*c - 20*a
*b^3*c^2 + 30*a^2*b*c^3)*x^2 + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*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*(b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*x + (3*a*b^6 - 26*a^
2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3 + (3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*x^2 + (3*b^7 -
26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^4 - 8*a^2*b^2*c^5 + 16*a^3*c^6 +
(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^2 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x)]

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Sympy [B]  time = 2.09912, size = 1012, normalized size = 5.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b*x/c**3 + (-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 48*a*
*2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4))*log(x + (16*a**3*c**2 - 17*a**2*b**2*c + 16*a
**2*c**5*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b*
*2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)) + 3*a*b**4 - 8*a*b**2*c**4*(-b*sqrt(-(4*a*c - b**2
)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) -
(2*a*c - 3*b**2)/(2*c**4)) + b**4*c**3*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*
c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)))/(30*a**2*b*c**2 -
20*a*b**3*c + 3*b**5)) + (b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c*
*3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4))*log(x + (16*a**3*c**2 - 17*a**2*b**
2*c + 16*a**2*c**5*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 4
8*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)) + 3*a*b**4 - 8*a*b**2*c**4*(b*sqrt(-(4*a*
c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c -
b**6)) - (2*a*c - 3*b**2)/(2*c**4)) + b**4*c**3*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b*
*4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)))/(30*a**2*b*
c**2 - 20*a*b**3*c + 3*b**5)) - (2*a**3*c**2 - 4*a**2*b**2*c + a*b**4 + x*(5*a**2*b*c**2 - 5*a*b**3*c + b**5))
/(4*a**2*c**5 - a*b**2*c**4 + x**2*(4*a*c**6 - b**2*c**5) + x*(4*a*b*c**5 - b**3*c**4)) + x**2/(2*c**2)

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Giac [A]  time = 1.12732, size = 254, normalized size = 1.3 \begin{align*} -\frac{{\left (3 \, b^{5} - 20 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (3 \, b^{2} - 2 \, a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{c^{2} x^{2} - 4 \, b c x}{2 \, c^{4}} + \frac{a b^{4} - 4 \, a^{2} b^{2} c + 2 \, a^{3} c^{2} +{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} x}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*b^5 - 20*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^4 - 4*a*c^5)*sqrt(-b^2 + 4
*a*c)) + 1/2*(3*b^2 - 2*a*c)*log(c*x^2 + b*x + a)/c^4 + 1/2*(c^2*x^2 - 4*b*c*x)/c^4 + (a*b^4 - 4*a^2*b^2*c + 2
*a^3*c^2 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^4)