3.40 \(\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{(a+b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=468 \[ \frac{x^2 \left (-\left (-2 a c i+b^2 i-b c g+2 c^2 e\right )\right )-b (a i+c e)+2 a c g}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt{b^2-4 a c}}+a b h-2 a c f+b c d\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt{b^2-4 a c}}+a b h-2 a c f+b c d\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) (2 a i-b g+2 c e)}{\left (b^2-4 a c\right )^{3/2}} \]

[Out]

(x*(b^2*d - a*b*f - 2*a*(c*d - a*h) + (b*c*d - 2*a*c*f + a*b*h)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))
+ (2*a*c*g - b*(c*e + a*i) - (2*c^2*e - b*c*g + b^2*i - 2*a*c*i)*x^2)/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))
+ ((b*c*d - 2*a*c*f + a*b*h + (4*a*b*c*f + b^2*(c*d - a*h) - 4*a*c*(3*c*d + a*h))/Sqrt[b^2 - 4*a*c])*ArcTan[(S
qrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*Sqrt[c]*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]
) + ((b*c*d - 2*a*c*f + a*b*h - (4*a*b*c*f + b^2*(c*d - a*h) - 4*a*c*(3*c*d + a*h))/Sqrt[b^2 - 4*a*c])*ArcTan[
(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*Sqrt[c]*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c
]]) + ((2*c*e - b*g + 2*a*i)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 1.11765, antiderivative size = 468, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.225, Rules used = {1673, 1678, 1166, 205, 1663, 1660, 12, 618, 206} \[ \frac{x^2 \left (-\left (-2 a c i+b^2 i-b c g+2 c^2 e\right )\right )-b (a i+c e)+2 a c g}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt{b^2-4 a c}}+a b h-2 a c f+b c d\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{b^2 (c d-a h)+4 a b c f-4 a c (a h+3 c d)}{\sqrt{b^2-4 a c}}+a b h-2 a c f+b c d\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) (2 a i-b g+2 c e)}{\left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x^2 + c*x^4)^2,x]

[Out]

(x*(b^2*d - a*b*f - 2*a*(c*d - a*h) + (b*c*d - 2*a*c*f + a*b*h)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))
+ (2*a*c*g - b*(c*e + a*i) - (2*c^2*e - b*c*g + b^2*i - 2*a*c*i)*x^2)/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4))
+ ((b*c*d - 2*a*c*f + a*b*h + (4*a*b*c*f + b^2*(c*d - a*h) - 4*a*c*(3*c*d + a*h))/Sqrt[b^2 - 4*a*c])*ArcTan[(S
qrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*Sqrt[c]*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]
) + ((b*c*d - 2*a*c*f + a*b*h - (4*a*b*c*f + b^2*(c*d - a*h) - 4*a*c*(3*c*d + a*h))/Sqrt[b^2 - 4*a*c])*ArcTan[
(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*Sqrt[c]*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c
]]) + ((2*c*e - b*g + 2*a*i)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2
+ c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*
a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*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[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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])

Rubi steps

\begin{align*} \int \frac{d+e x+f x^2+g x^3+h x^4+40 x^5}{\left (a+b x^2+c x^4\right )^2} \, dx &=\int \frac{x \left (e+g x^2+40 x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx+\int \frac{d+f x^2+h x^4}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{e+g x+40 x^2}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )-\frac{\int \frac{-b^2 d-a b f+2 a (3 c d+a h)+(-b c d+2 a c f-a b h) x^2}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac{40 a b+b c e-2 a c g+\left (40 b^2-2 c (40 a-c e)-b c g\right ) x^2}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{80 a+2 c e-b g}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}+\frac{\left (b c d-2 a c f+a b h-\frac{4 a b c f+b^2 (c d-a h)-4 a c (3 c d+a h)}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a \left (b^2-4 a c\right )}+\frac{\left (b c d-2 a c f+a b h+\frac{4 a b c f+b^2 (c d-a h)-4 a c (3 c d+a h)}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a \left (b^2-4 a c\right )}\\ &=-\frac{40 a b+b c e-2 a c g+\left (40 b^2-2 c (40 a-c e)-b c g\right ) x^2}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (b c d-2 a c f+a b h+\frac{4 a b c f+b^2 (c d-a h)-4 a c (3 c d+a h)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (b c d-2 a c f+a b h-\frac{4 a b c f+b^2 (c d-a h)-4 a c (3 c d+a h)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{(80 a+2 c e-b g) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac{40 a b+b c e-2 a c g+\left (40 b^2-2 c (40 a-c e)-b c g\right ) x^2}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (b c d-2 a c f+a b h+\frac{4 a b c f+b^2 (c d-a h)-4 a c (3 c d+a h)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (b c d-2 a c f+a b h-\frac{4 a b c f+b^2 (c d-a h)-4 a c (3 c d+a h)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{(80 a+2 c e-b g) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{b^2-4 a c}\\ &=-\frac{40 a b+b c e-2 a c g+\left (40 b^2-2 c (40 a-c e)-b c g\right ) x^2}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x \left (b^2 d-a b f-2 a (c d-a h)+(b c d-2 a c f+a b h) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (b c d-2 a c f+a b h+\frac{4 a b c f+b^2 (c d-a h)-4 a c (3 c d+a h)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (b c d-2 a c f+a b h-\frac{4 a b c f+b^2 (c d-a h)-4 a c (3 c d+a h)}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{(80 a+2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 2.55582, size = 524, normalized size = 1.12 \[ \frac{1}{4} \left (\frac{2 \left (a^2 (b i-2 c (g+x (h+i x)))+a \left (b^2 i x^2+b c (e+x (f-x (g+h x)))+2 c^2 x (d+x (e+f x))\right )-b c d x \left (b+c x^2\right )\right )}{a c \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (b \left (c d \sqrt{b^2-4 a c}+a h \sqrt{b^2-4 a c}+4 a c f\right )-2 a c \left (f \sqrt{b^2-4 a c}+2 a h+6 c d\right )+b^2 (c d-a h)\right )}{a \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (b \left (c d \sqrt{b^2-4 a c}+a h \sqrt{b^2-4 a c}-4 a c f\right )+2 a c \left (-f \sqrt{b^2-4 a c}+2 a h+6 c d\right )+b^2 (a h-c d)\right )}{a \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right ) (-2 a i+b g-2 c e)}{\left (b^2-4 a c\right )^{3/2}}+\frac{2 \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) (2 a i-b g+2 c e)}{\left (b^2-4 a c\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x^2 + c*x^4)^2,x]

[Out]

((2*(-(b*c*d*x*(b + c*x^2)) + a^2*(b*i - 2*c*(g + x*(h + i*x))) + a*(b^2*i*x^2 + 2*c^2*x*(d + x*(e + f*x)) + b
*c*(e + x*(f - x*(g + h*x))))))/(a*c*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(b^2*(c*d - a*h) - 2*a*c*(
6*c*d + Sqrt[b^2 - 4*a*c]*f + 2*a*h) + b*(c*Sqrt[b^2 - 4*a*c]*d + 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*h))*ArcTan[(Sq
rt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (
Sqrt[2]*(b^2*(-(c*d) + a*h) + 2*a*c*(6*c*d - Sqrt[b^2 - 4*a*c]*f + 2*a*h) + b*(c*Sqrt[b^2 - 4*a*c]*d - 4*a*c*f
+ a*Sqrt[b^2 - 4*a*c]*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*Sqrt[c]*(b^2 - 4*a*c)^(
3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (2*(-2*c*e + b*g - 2*a*i)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*
a*c)^(3/2) + (2*(2*c*e - b*g + 2*a*i)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/4

________________________________________________________________________________________

Maple [B]  time = 0.034, size = 1917, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 22.393, size = 8384, normalized size = 17.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/16*(16*a^2*b^5*c - 128*a^3*b^3*c^2 + 64*a^2*b^4*c^2 + 256*a^4*b*c^3 - 256*a^3*b^2*c^3 + 64*a^2*b^3*c^3 + 8*
a^2*b^4*c - 64*a^3*b^2*c^2 + 64*a^2*b^3*c^2 + 128*a^4*c^3 - 256*a^3*b*c^3 + 96*a^2*b^2*c^3 + 16*a^2*b^2*c^2 -
64*a^3*c^3 + 48*a^2*b*c^3 + 8*a^2*c^3 + (2*b^6*c - 40*a*b^4*c^2 + 8*b^5*c^2 + 224*a^2*b^2*c^3 - 128*a*b^3*c^3
+ 8*b^4*c^3 - 384*a^3*c^4 + 384*a^2*b*c^4 - 96*a*b^2*c^4 - b^5*c + 8*a*b^3*c^2 - 16*a^2*b*c^3 - 48*a*b^2*c^3 +
4*b^3*c^3 + 192*a^2*c^4 - 96*a*b*c^4 - 2*b^3*c^2 + 8*a*b*c^3 - 2*b^2*c^3 - 24*a*c^4 - b*c^3)*sqrt(2*b*c + c)*
d + 2*(4*a*b^5*c - 32*a^2*b^3*c^2 + 16*a*b^4*c^2 + 64*a^3*b*c^3 - 64*a^2*b^2*c^3 + 16*a*b^3*c^3 + a*b^4*c - 8*
a^2*b^2*c^2 + 12*a*b^3*c^2 + 16*a^3*c^3 - 48*a^2*b*c^3 + 20*a*b^2*c^3 + 2*a*b^2*c^2 - 8*a^2*c^3 + 8*a*b*c^3 +
a*c^3)*sqrt(2*b*c + c)*f - (2*a*b^6 - 8*a^2*b^4*c + 8*a*b^5*c - 32*a^3*b^2*c^2 + 8*a*b^4*c^2 + 128*a^4*c^3 - 1
28*a^3*b*c^3 + 32*a^2*b^2*c^3 + a*b^5 - 8*a^2*b^3*c + 8*a*b^4*c + 16*a^3*b*c^2 - 16*a^2*b^2*c^2 + 12*a*b^3*c^2
- 64*a^3*c^3 + 32*a^2*b*c^3 + 2*a*b^3*c - 8*a^2*b*c^2 + 6*a*b^2*c^2 + 8*a^2*c^3 + a*b*c^2)*sqrt(2*b*c + c)*h
+ 4*(2*a*b^6*c*i - 16*a^2*b^4*c^2*i + 8*a*b^5*c^2*i + 32*a^3*b^2*c^3*i - 32*a^2*b^3*c^3*i + 8*a*b^4*c^3*i + a*
b^5*c*i - 8*a^2*b^3*c^2*i + 8*a*b^4*c^2*i + 16*a^3*b*c^3*i - 32*a^2*b^2*c^3*i + 12*a*b^3*c^3*i + 2*a*b^3*c^2*i
- 8*a^2*b*c^3*i + 6*a*b^2*c^3*i + a*b*c^3*i)*g - 8*(2*a*b^5*c^2*i - 16*a^2*b^3*c^3*i + 8*a*b^4*c^3*i + 32*a^3
*b*c^4*i - 32*a^2*b^2*c^4*i + 8*a*b^3*c^4*i + a*b^4*c^2*i - 8*a^2*b^2*c^3*i + 8*a*b^3*c^3*i + 16*a^3*c^4*i - 3
2*a^2*b*c^4*i + 12*a*b^2*c^4*i + 2*a*b^2*c^3*i - 8*a^2*c^4*i + 6*a*b*c^4*i + a*c^4*i)*e)*log(x + 1/2*sqrt(-(2*
a*b^3*i - 8*a^2*b*c*i + sqrt(-4*(a*b^3 - 4*a^2*b*c)^2 + 16*(a^2*b^2 - 4*a^3*c)*(a*b^2*c - 4*a^2*c^2)))/(a*b^2*
c*i - 4*a^2*c^2*i)))/(a*b^8*c*i - 16*a^2*b^6*c^2*i + 4*a*b^7*c^2*i + 96*a^3*b^4*c^3*i - 48*a^2*b^5*c^3*i + 4*a
*b^6*c^3*i - 256*a^4*b^2*c^4*i + 192*a^3*b^3*c^4*i - 32*a^2*b^4*c^4*i + 256*a^5*c^5*i - 256*a^4*b*c^5*i + 64*a
^3*b^2*c^5*i + 2*a*b^6*c^2*i - 24*a^2*b^4*c^3*i + 4*a*b^5*c^3*i + 96*a^3*b^2*c^4*i - 32*a^2*b^3*c^4*i - 128*a^
4*c^5*i + 64*a^3*b*c^5*i + a*b^4*c^3*i - 8*a^2*b^2*c^4*i + 16*a^3*c^5*i - (a*b^7*c - 12*a^2*b^5*c^2 + 4*a*b^6*
c^2 + 48*a^3*b^3*c^3 - 32*a^2*b^4*c^3 + 4*a*b^5*c^3 - 64*a^4*b*c^4 + 64*a^3*b^2*c^4 - 16*a^2*b^3*c^4 + 2*a*b^5
*c^2 - 16*a^2*b^3*c^3 + 4*a*b^4*c^3 + 32*a^3*b*c^4 - 16*a^2*b^2*c^4 + a*b^3*c^3 - 4*a^2*b*c^4)*sqrt(-b^2 + 4*a
*c)) - 1/16*(16*a^2*b^5*c - 128*a^3*b^3*c^2 + 64*a^2*b^4*c^2 + 256*a^4*b*c^3 - 256*a^3*b^2*c^3 + 64*a^2*b^3*c^
3 + 8*a^2*b^4*c - 64*a^3*b^2*c^2 + 64*a^2*b^3*c^2 + 128*a^4*c^3 - 256*a^3*b*c^3 + 96*a^2*b^2*c^3 + 16*a^2*b^2*
c^2 - 64*a^3*c^3 + 48*a^2*b*c^3 + 8*a^2*c^3 - (2*b^6*c - 40*a*b^4*c^2 + 8*b^5*c^2 + 224*a^2*b^2*c^3 - 128*a*b^
3*c^3 + 8*b^4*c^3 - 384*a^3*c^4 + 384*a^2*b*c^4 - 96*a*b^2*c^4 - b^5*c + 8*a*b^3*c^2 - 16*a^2*b*c^3 - 48*a*b^2
*c^3 + 4*b^3*c^3 + 192*a^2*c^4 - 96*a*b*c^4 - 2*b^3*c^2 + 8*a*b*c^3 - 2*b^2*c^3 - 24*a*c^4 - b*c^3)*sqrt(2*b*c
+ c)*d - 2*(4*a*b^5*c - 32*a^2*b^3*c^2 + 16*a*b^4*c^2 + 64*a^3*b*c^3 - 64*a^2*b^2*c^3 + 16*a*b^3*c^3 + a*b^4*
c - 8*a^2*b^2*c^2 + 12*a*b^3*c^2 + 16*a^3*c^3 - 48*a^2*b*c^3 + 20*a*b^2*c^3 + 2*a*b^2*c^2 - 8*a^2*c^3 + 8*a*b*
c^3 + a*c^3)*sqrt(2*b*c + c)*f + (2*a*b^6 - 8*a^2*b^4*c + 8*a*b^5*c - 32*a^3*b^2*c^2 + 8*a*b^4*c^2 + 128*a^4*c
^3 - 128*a^3*b*c^3 + 32*a^2*b^2*c^3 + a*b^5 - 8*a^2*b^3*c + 8*a*b^4*c + 16*a^3*b*c^2 - 16*a^2*b^2*c^2 + 12*a*b
^3*c^2 - 64*a^3*c^3 + 32*a^2*b*c^3 + 2*a*b^3*c - 8*a^2*b*c^2 + 6*a*b^2*c^2 + 8*a^2*c^3 + a*b*c^2)*sqrt(2*b*c +
c)*h + 4*(2*a*b^6*c*i - 16*a^2*b^4*c^2*i + 8*a*b^5*c^2*i + 32*a^3*b^2*c^3*i - 32*a^2*b^3*c^3*i + 8*a*b^4*c^3*
i + a*b^5*c*i - 8*a^2*b^3*c^2*i + 8*a*b^4*c^2*i + 16*a^3*b*c^3*i - 32*a^2*b^2*c^3*i + 12*a*b^3*c^3*i + 2*a*b^3
*c^2*i - 8*a^2*b*c^3*i + 6*a*b^2*c^3*i + a*b*c^3*i)*g - 8*(2*a*b^5*c^2*i - 16*a^2*b^3*c^3*i + 8*a*b^4*c^3*i +
32*a^3*b*c^4*i - 32*a^2*b^2*c^4*i + 8*a*b^3*c^4*i + a*b^4*c^2*i - 8*a^2*b^2*c^3*i + 8*a*b^3*c^3*i + 16*a^3*c^4
*i - 32*a^2*b*c^4*i + 12*a*b^2*c^4*i + 2*a*b^2*c^3*i - 8*a^2*c^4*i + 6*a*b*c^4*i + a*c^4*i)*e)*log(x - 1/2*sqr
t(-(2*a*b^3*i - 8*a^2*b*c*i + sqrt(-4*(a*b^3 - 4*a^2*b*c)^2 + 16*(a^2*b^2 - 4*a^3*c)*(a*b^2*c - 4*a^2*c^2)))/(
a*b^2*c*i - 4*a^2*c^2*i)))/(a*b^8*c*i - 16*a^2*b^6*c^2*i + 4*a*b^7*c^2*i + 96*a^3*b^4*c^3*i - 48*a^2*b^5*c^3*i
+ 4*a*b^6*c^3*i - 256*a^4*b^2*c^4*i + 192*a^3*b^3*c^4*i - 32*a^2*b^4*c^4*i + 256*a^5*c^5*i - 256*a^4*b*c^5*i
+ 64*a^3*b^2*c^5*i + 2*a*b^6*c^2*i - 24*a^2*b^4*c^3*i + 4*a*b^5*c^3*i + 96*a^3*b^2*c^4*i - 32*a^2*b^3*c^4*i -
128*a^4*c^5*i + 64*a^3*b*c^5*i + a*b^4*c^3*i - 8*a^2*b^2*c^4*i + 16*a^3*c^5*i - (a*b^7*c - 12*a^2*b^5*c^2 + 4*
a*b^6*c^2 + 48*a^3*b^3*c^3 - 32*a^2*b^4*c^3 + 4*a*b^5*c^3 - 64*a^4*b*c^4 + 64*a^3*b^2*c^4 - 16*a^2*b^3*c^4 + 2
*a*b^5*c^2 - 16*a^2*b^3*c^3 + 4*a*b^4*c^3 + 32*a^3*b*c^4 - 16*a^2*b^2*c^4 + a*b^3*c^3 - 4*a^2*b*c^4)*sqrt(-b^2
+ 4*a*c)) - 1/16*(16*a^2*b^5*c - 128*a^3*b^3*c^2 + 64*a^2*b^4*c^2 + 256*a^4*b*c^3 - 256*a^3*b^2*c^3 + 64*a^2*
b^3*c^3 - 8*a^2*b^4*c + 64*a^3*b^2*c^2 - 64*a^2*b^3*c^2 - 128*a^4*c^3 + 256*a^3*b*c^3 - 96*a^2*b^2*c^3 + 16*a^
2*b^2*c^2 - 64*a^3*c^3 + 48*a^2*b*c^3 - 8*a^2*c^3 - (2*b^6*c - 40*a*b^4*c^2 + 8*b^5*c^2 + 224*a^2*b^2*c^3 - 12
8*a*b^3*c^3 + 8*b^4*c^3 - 384*a^3*c^4 + 384*a^2*b*c^4 - 96*a*b^2*c^4 + b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3 + 48
*a*b^2*c^3 - 4*b^3*c^3 - 192*a^2*c^4 + 96*a*b*c^4 - 2*b^3*c^2 + 8*a*b*c^3 - 2*b^2*c^3 - 24*a*c^4 + b*c^3)*sqrt
(2*b*c - c)*d - 2*(4*a*b^5*c - 32*a^2*b^3*c^2 + 16*a*b^4*c^2 + 64*a^3*b*c^3 - 64*a^2*b^2*c^3 + 16*a*b^3*c^3 -
a*b^4*c + 8*a^2*b^2*c^2 - 12*a*b^3*c^2 - 16*a^3*c^3 + 48*a^2*b*c^3 - 20*a*b^2*c^3 + 2*a*b^2*c^2 - 8*a^2*c^3 +
8*a*b*c^3 - a*c^3)*sqrt(2*b*c - c)*f + (2*a*b^6 - 8*a^2*b^4*c + 8*a*b^5*c - 32*a^3*b^2*c^2 + 8*a*b^4*c^2 + 128
*a^4*c^3 - 128*a^3*b*c^3 + 32*a^2*b^2*c^3 - a*b^5 + 8*a^2*b^3*c - 8*a*b^4*c - 16*a^3*b*c^2 + 16*a^2*b^2*c^2 -
12*a*b^3*c^2 + 64*a^3*c^3 - 32*a^2*b*c^3 + 2*a*b^3*c - 8*a^2*b*c^2 + 6*a*b^2*c^2 + 8*a^2*c^3 - a*b*c^2)*sqrt(2
*b*c - c)*h + 4*(2*a*b^6*c*i - 16*a^2*b^4*c^2*i + 8*a*b^5*c^2*i + 32*a^3*b^2*c^3*i - 32*a^2*b^3*c^3*i + 8*a*b^
4*c^3*i - a*b^5*c*i + 8*a^2*b^3*c^2*i - 8*a*b^4*c^2*i - 16*a^3*b*c^3*i + 32*a^2*b^2*c^3*i - 12*a*b^3*c^3*i + 2
*a*b^3*c^2*i - 8*a^2*b*c^3*i + 6*a*b^2*c^3*i - a*b*c^3*i)*g - 8*(2*a*b^5*c^2*i - 16*a^2*b^3*c^3*i + 8*a*b^4*c^
3*i + 32*a^3*b*c^4*i - 32*a^2*b^2*c^4*i + 8*a*b^3*c^4*i - a*b^4*c^2*i + 8*a^2*b^2*c^3*i - 8*a*b^3*c^3*i - 16*a
^3*c^4*i + 32*a^2*b*c^4*i - 12*a*b^2*c^4*i + 2*a*b^2*c^3*i - 8*a^2*c^4*i + 6*a*b*c^4*i - a*c^4*i)*e)*log(x + 1
/2*sqrt(-(2*a*b^3*i - 8*a^2*b*c*i - sqrt(-4*(a*b^3 - 4*a^2*b*c)^2 + 16*(a^2*b^2 - 4*a^3*c)*(a*b^2*c - 4*a^2*c^
2)))/(a*b^2*c*i - 4*a^2*c^2*i)))/(a*b^8*c*i - 16*a^2*b^6*c^2*i + 4*a*b^7*c^2*i + 96*a^3*b^4*c^3*i - 48*a^2*b^5
*c^3*i + 4*a*b^6*c^3*i - 256*a^4*b^2*c^4*i + 192*a^3*b^3*c^4*i - 32*a^2*b^4*c^4*i + 256*a^5*c^5*i - 256*a^4*b*
c^5*i + 64*a^3*b^2*c^5*i - 2*a*b^6*c^2*i + 24*a^2*b^4*c^3*i - 4*a*b^5*c^3*i - 96*a^3*b^2*c^4*i + 32*a^2*b^3*c^
4*i + 128*a^4*c^5*i - 64*a^3*b*c^5*i + a*b^4*c^3*i - 8*a^2*b^2*c^4*i + 16*a^3*c^5*i + (a*b^7*c - 12*a^2*b^5*c^
2 + 4*a*b^6*c^2 + 48*a^3*b^3*c^3 - 32*a^2*b^4*c^3 + 4*a*b^5*c^3 - 64*a^4*b*c^4 + 64*a^3*b^2*c^4 - 16*a^2*b^3*c
^4 - 2*a*b^5*c^2 + 16*a^2*b^3*c^3 - 4*a*b^4*c^3 - 32*a^3*b*c^4 + 16*a^2*b^2*c^4 + a*b^3*c^3 - 4*a^2*b*c^4)*sqr
t(-b^2 + 4*a*c)) - 1/16*(16*a^2*b^5*c - 128*a^3*b^3*c^2 + 64*a^2*b^4*c^2 + 256*a^4*b*c^3 - 256*a^3*b^2*c^3 + 6
4*a^2*b^3*c^3 - 8*a^2*b^4*c + 64*a^3*b^2*c^2 - 64*a^2*b^3*c^2 - 128*a^4*c^3 + 256*a^3*b*c^3 - 96*a^2*b^2*c^3 +
16*a^2*b^2*c^2 - 64*a^3*c^3 + 48*a^2*b*c^3 - 8*a^2*c^3 + (2*b^6*c - 40*a*b^4*c^2 + 8*b^5*c^2 + 224*a^2*b^2*c^
3 - 128*a*b^3*c^3 + 8*b^4*c^3 - 384*a^3*c^4 + 384*a^2*b*c^4 - 96*a*b^2*c^4 + b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^
3 + 48*a*b^2*c^3 - 4*b^3*c^3 - 192*a^2*c^4 + 96*a*b*c^4 - 2*b^3*c^2 + 8*a*b*c^3 - 2*b^2*c^3 - 24*a*c^4 + b*c^3
)*sqrt(2*b*c - c)*d + 2*(4*a*b^5*c - 32*a^2*b^3*c^2 + 16*a*b^4*c^2 + 64*a^3*b*c^3 - 64*a^2*b^2*c^3 + 16*a*b^3*
c^3 - a*b^4*c + 8*a^2*b^2*c^2 - 12*a*b^3*c^2 - 16*a^3*c^3 + 48*a^2*b*c^3 - 20*a*b^2*c^3 + 2*a*b^2*c^2 - 8*a^2*
c^3 + 8*a*b*c^3 - a*c^3)*sqrt(2*b*c - c)*f - (2*a*b^6 - 8*a^2*b^4*c + 8*a*b^5*c - 32*a^3*b^2*c^2 + 8*a*b^4*c^2
+ 128*a^4*c^3 - 128*a^3*b*c^3 + 32*a^2*b^2*c^3 - a*b^5 + 8*a^2*b^3*c - 8*a*b^4*c - 16*a^3*b*c^2 + 16*a^2*b^2*
c^2 - 12*a*b^3*c^2 + 64*a^3*c^3 - 32*a^2*b*c^3 + 2*a*b^3*c - 8*a^2*b*c^2 + 6*a*b^2*c^2 + 8*a^2*c^3 - a*b*c^2)*
sqrt(2*b*c - c)*h + 4*(2*a*b^6*c*i - 16*a^2*b^4*c^2*i + 8*a*b^5*c^2*i + 32*a^3*b^2*c^3*i - 32*a^2*b^3*c^3*i +
8*a*b^4*c^3*i - a*b^5*c*i + 8*a^2*b^3*c^2*i - 8*a*b^4*c^2*i - 16*a^3*b*c^3*i + 32*a^2*b^2*c^3*i - 12*a*b^3*c^3
*i + 2*a*b^3*c^2*i - 8*a^2*b*c^3*i + 6*a*b^2*c^3*i - a*b*c^3*i)*g - 8*(2*a*b^5*c^2*i - 16*a^2*b^3*c^3*i + 8*a*
b^4*c^3*i + 32*a^3*b*c^4*i - 32*a^2*b^2*c^4*i + 8*a*b^3*c^4*i - a*b^4*c^2*i + 8*a^2*b^2*c^3*i - 8*a*b^3*c^3*i
- 16*a^3*c^4*i + 32*a^2*b*c^4*i - 12*a*b^2*c^4*i + 2*a*b^2*c^3*i - 8*a^2*c^4*i + 6*a*b*c^4*i - a*c^4*i)*e)*log
(x - 1/2*sqrt(-(2*a*b^3*i - 8*a^2*b*c*i - sqrt(-4*(a*b^3 - 4*a^2*b*c)^2 + 16*(a^2*b^2 - 4*a^3*c)*(a*b^2*c - 4*
a^2*c^2)))/(a*b^2*c*i - 4*a^2*c^2*i)))/(a*b^8*c*i - 16*a^2*b^6*c^2*i + 4*a*b^7*c^2*i + 96*a^3*b^4*c^3*i - 48*a
^2*b^5*c^3*i + 4*a*b^6*c^3*i - 256*a^4*b^2*c^4*i + 192*a^3*b^3*c^4*i - 32*a^2*b^4*c^4*i + 256*a^5*c^5*i - 256*
a^4*b*c^5*i + 64*a^3*b^2*c^5*i - 2*a*b^6*c^2*i + 24*a^2*b^4*c^3*i - 4*a*b^5*c^3*i - 96*a^3*b^2*c^4*i + 32*a^2*
b^3*c^4*i + 128*a^4*c^5*i - 64*a^3*b*c^5*i + a*b^4*c^3*i - 8*a^2*b^2*c^4*i + 16*a^3*c^5*i + (a*b^7*c - 12*a^2*
b^5*c^2 + 4*a*b^6*c^2 + 48*a^3*b^3*c^3 - 32*a^2*b^4*c^3 + 4*a*b^5*c^3 - 64*a^4*b*c^4 + 64*a^3*b^2*c^4 - 16*a^2
*b^3*c^4 - 2*a*b^5*c^2 + 16*a^2*b^3*c^3 - 4*a*b^4*c^3 - 32*a^3*b*c^4 + 16*a^2*b^2*c^4 + a*b^3*c^3 - 4*a^2*b*c^
4)*sqrt(-b^2 + 4*a*c)) + 1/2*(b*c^2*d*i*x^3 - 2*a*c^2*f*i*x^3 + a*b*c*h*i*x^3 + a*b*c*g*i*x^2 - 2*a*c^2*i*x^2*
e + b^2*c*d*i*x - 2*a*c^2*d*i*x - a*b*c*f*i*x + 2*a^2*c*h*i*x + 2*a^2*c*g*i + a*b^2*x^2 - 2*a^2*c*x^2 - a*b*c*
i*e + a^2*b)/((a*b^2*c*i - 4*a^2*c^2*i)*(c*x^4 + b*x^2 + a))