3.125 \(\int \frac{x (d+e x^2+f x^4+g x^6)}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=149 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )}{2 c^3 \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-c (a g+b f)+b^2 g+c^2 e\right )}{4 c^3}+\frac{x^2 (c f-b g)}{2 c^2}+\frac{g x^4}{4 c} \]

[Out]

((c*f - b*g)*x^2)/(2*c^2) + (g*x^4)/(4*c) - ((2*c^3*d - c^2*(b*e + 2*a*f) - b^3*g + b*c*(b*f + 3*a*g))*ArcTanh
[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*Sqrt[b^2 - 4*a*c]) + ((c^2*e + b^2*g - c*(b*f + a*g))*Log[a + b*x^2
+ c*x^4])/(4*c^3)

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Rubi [A]  time = 0.294841, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1663, 1657, 634, 618, 206, 628} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )}{2 c^3 \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-c (a g+b f)+b^2 g+c^2 e\right )}{4 c^3}+\frac{x^2 (c f-b g)}{2 c^2}+\frac{g x^4}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x^2 + f*x^4 + g*x^6))/(a + b*x^2 + c*x^4),x]

[Out]

((c*f - b*g)*x^2)/(2*c^2) + (g*x^4)/(4*c) - ((2*c^3*d - c^2*(b*e + 2*a*f) - b^3*g + b*c*(b*f + 3*a*g))*ArcTanh
[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*Sqrt[b^2 - 4*a*c]) + ((c^2*e + b^2*g - c*(b*f + a*g))*Log[a + b*x^2
+ c*x^4])/(4*c^3)

Rule 1663

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

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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

Mathematica [A]  time = 0.127054, size = 142, normalized size = 0.95 \[ \frac{\frac{2 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (-c^2 (2 a f+b e)+b c (3 a g+b f)+b^3 (-g)+2 c^3 d\right )}{\sqrt{4 a c-b^2}}+\log \left (a+b x^2+c x^4\right ) \left (-c (a g+b f)+b^2 g+c^2 e\right )+2 c x^2 (c f-b g)+c^2 g x^4}{4 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(d + e*x^2 + f*x^4 + g*x^6))/(a + b*x^2 + c*x^4),x]

[Out]

(2*c*(c*f - b*g)*x^2 + c^2*g*x^4 + (2*(2*c^3*d - c^2*(b*e + 2*a*f) - b^3*g + b*c*(b*f + 3*a*g))*ArcTan[(b + 2*
c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + (c^2*e + b^2*g - c*(b*f + a*g))*Log[a + b*x^2 + c*x^4])/(4*c^
3)

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Maple [B]  time = 0.005, size = 357, normalized size = 2.4 \begin{align*}{\frac{g{x}^{4}}{4\,c}}-{\frac{b{x}^{2}g}{2\,{c}^{2}}}+{\frac{f{x}^{2}}{2\,c}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) ag}{4\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}g}{4\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bf}{4\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) e}{4\,c}}+{\frac{3\,abg}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{af}{c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{d\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}g}{2\,{c}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}f}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{be}{2\,c}\arctan \left ({(2\,c{x}^{2}+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(x*(g*x^6+f*x^4+e*x^2+d)/(c*x^4+b*x^2+a),x)

[Out]

1/4*g*x^4/c-1/2/c^2*x^2*b*g+1/2*f*x^2/c-1/4/c^2*ln(c*x^4+b*x^2+a)*a*g+1/4/c^3*ln(c*x^4+b*x^2+a)*b^2*g-1/4/c^2*
ln(c*x^4+b*x^2+a)*b*f+1/4/c*ln(c*x^4+b*x^2+a)*e+3/2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2)
)*a*b*g-1/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*a*f+1/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)
/(4*a*c-b^2)^(1/2))*d-1/2/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^3*g+1/2/c^2/(4*a*c-b^2
)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^2*f-1/2/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/
2))*b*e

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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*(g*x^6+f*x^4+e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.35085, size = 1021, normalized size = 6.85 \begin{align*} \left [\frac{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} g x^{4} + 2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} f -{\left (b^{3} c - 4 \, a b c^{2}\right )} g\right )} x^{2} +{\left (2 \, c^{3} d - b c^{2} e +{\left (b^{2} c - 2 \, a c^{2}\right )} f -{\left (b^{3} - 3 \, a b c\right )} g\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c -{\left (2 \, c x^{2} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e -{\left (b^{3} c - 4 \, a b c^{2}\right )} f +{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} g\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} g x^{4} + 2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} f -{\left (b^{3} c - 4 \, a b c^{2}\right )} g\right )} x^{2} - 2 \,{\left (2 \, c^{3} d - b c^{2} e +{\left (b^{2} c - 2 \, a c^{2}\right )} f -{\left (b^{3} - 3 \, a b c\right )} g\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e -{\left (b^{3} c - 4 \, a b c^{2}\right )} f +{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} g\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(g*x^6+f*x^4+e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

[1/4*((b^2*c^2 - 4*a*c^3)*g*x^4 + 2*((b^2*c^2 - 4*a*c^3)*f - (b^3*c - 4*a*b*c^2)*g)*x^2 + (2*c^3*d - b*c^2*e +
(b^2*c - 2*a*c^2)*f - (b^3 - 3*a*b*c)*g)*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)) + ((b^2*c^2 - 4*a*c^3)*e - (b^3*c - 4*a*b*c^2)*f + (b^4 - 5*a*b
^2*c + 4*a^2*c^2)*g)*log(c*x^4 + b*x^2 + a))/(b^2*c^3 - 4*a*c^4), 1/4*((b^2*c^2 - 4*a*c^3)*g*x^4 + 2*((b^2*c^2
- 4*a*c^3)*f - (b^3*c - 4*a*b*c^2)*g)*x^2 - 2*(2*c^3*d - b*c^2*e + (b^2*c - 2*a*c^2)*f - (b^3 - 3*a*b*c)*g)*s
qrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + ((b^2*c^2 - 4*a*c^3)*e - (b^3*c -
4*a*b*c^2)*f + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*g)*log(c*x^4 + b*x^2 + a))/(b^2*c^3 - 4*a*c^4)]

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Sympy [B]  time = 49.2751, size = 789, normalized size = 5.3 \begin{align*} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c g - 2 a c^{2} f - b^{3} g + b^{2} c f - b c^{2} e + 2 c^{3} d\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c g - b^{2} g + b c f - c^{2} e}{4 c^{3}}\right ) \log{\left (x^{2} + \frac{2 a^{2} c g - a b^{2} g + a b c f + 8 a c^{3} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c g - 2 a c^{2} f - b^{3} g + b^{2} c f - b c^{2} e + 2 c^{3} d\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c g - b^{2} g + b c f - c^{2} e}{4 c^{3}}\right ) - 2 a c^{2} e - 2 b^{2} c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c g - 2 a c^{2} f - b^{3} g + b^{2} c f - b c^{2} e + 2 c^{3} d\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c g - b^{2} g + b c f - c^{2} e}{4 c^{3}}\right ) + b c^{2} d}{3 a b c g - 2 a c^{2} f - b^{3} g + b^{2} c f - b c^{2} e + 2 c^{3} d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c g - 2 a c^{2} f - b^{3} g + b^{2} c f - b c^{2} e + 2 c^{3} d\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c g - b^{2} g + b c f - c^{2} e}{4 c^{3}}\right ) \log{\left (x^{2} + \frac{2 a^{2} c g - a b^{2} g + a b c f + 8 a c^{3} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c g - 2 a c^{2} f - b^{3} g + b^{2} c f - b c^{2} e + 2 c^{3} d\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c g - b^{2} g + b c f - c^{2} e}{4 c^{3}}\right ) - 2 a c^{2} e - 2 b^{2} c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c g - 2 a c^{2} f - b^{3} g + b^{2} c f - b c^{2} e + 2 c^{3} d\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c g - b^{2} g + b c f - c^{2} e}{4 c^{3}}\right ) + b c^{2} d}{3 a b c g - 2 a c^{2} f - b^{3} g + b^{2} c f - b c^{2} e + 2 c^{3} d} \right )} + \frac{g x^{4}}{4 c} - \frac{x^{2} \left (b g - c f\right )}{2 c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(g*x**6+f*x**4+e*x**2+d)/(c*x**4+b*x**2+a),x)

[Out]

(-sqrt(-4*a*c + b**2)*(3*a*b*c*g - 2*a*c**2*f - b**3*g + b**2*c*f - b*c**2*e + 2*c**3*d)/(4*c**3*(4*a*c - b**2
)) - (a*c*g - b**2*g + b*c*f - c**2*e)/(4*c**3))*log(x**2 + (2*a**2*c*g - a*b**2*g + a*b*c*f + 8*a*c**3*(-sqrt
(-4*a*c + b**2)*(3*a*b*c*g - 2*a*c**2*f - b**3*g + b**2*c*f - b*c**2*e + 2*c**3*d)/(4*c**3*(4*a*c - b**2)) - (
a*c*g - b**2*g + b*c*f - c**2*e)/(4*c**3)) - 2*a*c**2*e - 2*b**2*c**2*(-sqrt(-4*a*c + b**2)*(3*a*b*c*g - 2*a*c
**2*f - b**3*g + b**2*c*f - b*c**2*e + 2*c**3*d)/(4*c**3*(4*a*c - b**2)) - (a*c*g - b**2*g + b*c*f - c**2*e)/(
4*c**3)) + b*c**2*d)/(3*a*b*c*g - 2*a*c**2*f - b**3*g + b**2*c*f - b*c**2*e + 2*c**3*d)) + (sqrt(-4*a*c + b**2
)*(3*a*b*c*g - 2*a*c**2*f - b**3*g + b**2*c*f - b*c**2*e + 2*c**3*d)/(4*c**3*(4*a*c - b**2)) - (a*c*g - b**2*g
+ b*c*f - c**2*e)/(4*c**3))*log(x**2 + (2*a**2*c*g - a*b**2*g + a*b*c*f + 8*a*c**3*(sqrt(-4*a*c + b**2)*(3*a*
b*c*g - 2*a*c**2*f - b**3*g + b**2*c*f - b*c**2*e + 2*c**3*d)/(4*c**3*(4*a*c - b**2)) - (a*c*g - b**2*g + b*c*
f - c**2*e)/(4*c**3)) - 2*a*c**2*e - 2*b**2*c**2*(sqrt(-4*a*c + b**2)*(3*a*b*c*g - 2*a*c**2*f - b**3*g + b**2*
c*f - b*c**2*e + 2*c**3*d)/(4*c**3*(4*a*c - b**2)) - (a*c*g - b**2*g + b*c*f - c**2*e)/(4*c**3)) + b*c**2*d)/(
3*a*b*c*g - 2*a*c**2*f - b**3*g + b**2*c*f - b*c**2*e + 2*c**3*d)) + g*x**4/(4*c) - x**2*(b*g - c*f)/(2*c**2)

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Giac [A]  time = 1.17534, size = 197, normalized size = 1.32 \begin{align*} \frac{c g x^{4} + 2 \, c f x^{2} - 2 \, b g x^{2}}{4 \, c^{2}} - \frac{{\left (b c f - b^{2} g + a c g - c^{2} e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} + \frac{{\left (2 \, c^{3} d + b^{2} c f - 2 \, a c^{2} f - b^{3} g + 3 \, a b c g - b c^{2} e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(g*x^6+f*x^4+e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

1/4*(c*g*x^4 + 2*c*f*x^2 - 2*b*g*x^2)/c^2 - 1/4*(b*c*f - b^2*g + a*c*g - c^2*e)*log(c*x^4 + b*x^2 + a)/c^3 + 1
/2*(2*c^3*d + b^2*c*f - 2*a*c^2*f - b^3*g + 3*a*b*c*g - b*c^2*e)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqr
t(-b^2 + 4*a*c)*c^3)