3.87 \(\int \frac{\sqrt{a+c x^2} (d+e x+f x^2)}{(g+h x)^6} \, dx\)

Optimal. Leaf size=433 \[ -\frac{c \sqrt{a+c x^2} (a h-c g x) \left (a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )+4 c^2 d g^3\right )}{8 (g+h x)^2 \left (a h^2+c g^2\right )^4}-\frac{\left (a+c x^2\right )^{3/2} \left (20 a^2 f h^4-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )-c^2 g^2 \left (h (2 e g-27 d h)+3 f g^2\right )\right )}{60 h (g+h x)^3 \left (a h^2+c g^2\right )^3}-\frac{a c^2 \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )+4 c^2 d g^3\right )}{8 \left (a h^2+c g^2\right )^{9/2}}+\frac{\left (a+c x^2\right )^{3/2} \left (5 a h^2 (2 f g-e h)+c g \left (h (2 e g-7 d h)+3 f g^2\right )\right )}{20 h (g+h x)^4 \left (a h^2+c g^2\right )^2}-\frac{\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )} \]

[Out]

-(c*(4*c^2*d*g^3 + a^2*h^2*(6*f*g - e*h) - a*c*g*(f*g^2 - 3*h*(2*e*g - d*h)))*(a*h - c*g*x)*Sqrt[a + c*x^2])/(
8*(c*g^2 + a*h^2)^4*(g + h*x)^2) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(3/2))/(5*h*(c*g^2 + a*h^2)*(g + h*x)^
5) + ((5*a*h^2*(2*f*g - e*h) + c*g*(3*f*g^2 + h*(2*e*g - 7*d*h)))*(a + c*x^2)^(3/2))/(20*h*(c*g^2 + a*h^2)^2*(
g + h*x)^4) - ((20*a^2*f*h^4 - c^2*g^2*(3*f*g^2 + h*(2*e*g - 27*d*h)) - a*c*h^2*(18*f*g^2 - h*(33*e*g - 8*d*h)
))*(a + c*x^2)^(3/2))/(60*h*(c*g^2 + a*h^2)^3*(g + h*x)^3) - (a*c^2*(4*c^2*d*g^3 + a^2*h^2*(6*f*g - e*h) - a*c
*g*(f*g^2 - 3*h*(2*e*g - d*h)))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(8*(c*g^2 + a*h^
2)^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.742644, antiderivative size = 432, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1651, 835, 807, 721, 725, 206} \[ -\frac{c \sqrt{a+c x^2} (a h-c g x) \left (a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )+4 c^2 d g^3\right )}{8 (g+h x)^2 \left (a h^2+c g^2\right )^4}-\frac{\left (a+c x^2\right )^{3/2} \left (20 a^2 f h^4-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )-c^2 \left (g^2 h (2 e g-27 d h)+3 f g^4\right )\right )}{60 h (g+h x)^3 \left (a h^2+c g^2\right )^3}-\frac{a c^2 \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )+4 c^2 d g^3\right )}{8 \left (a h^2+c g^2\right )^{9/2}}+\frac{\left (a+c x^2\right )^{3/2} \left (5 a h^2 (2 f g-e h)+c g h (2 e g-7 d h)+3 c f g^3\right )}{20 h (g+h x)^4 \left (a h^2+c g^2\right )^2}-\frac{\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*(4*c^2*d*g^3 + a^2*h^2*(6*f*g - e*h) - a*c*g*(f*g^2 - 3*h*(2*e*g - d*h)))*(a*h - c*g*x)*Sqrt[a + c*x^2])/(
8*(c*g^2 + a*h^2)^4*(g + h*x)^2) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(3/2))/(5*h*(c*g^2 + a*h^2)*(g + h*x)^
5) + ((3*c*f*g^3 + c*g*h*(2*e*g - 7*d*h) + 5*a*h^2*(2*f*g - e*h))*(a + c*x^2)^(3/2))/(20*h*(c*g^2 + a*h^2)^2*(
g + h*x)^4) - ((20*a^2*f*h^4 - c^2*(3*f*g^4 + g^2*h*(2*e*g - 27*d*h)) - a*c*h^2*(18*f*g^2 - h*(33*e*g - 8*d*h)
))*(a + c*x^2)^(3/2))/(60*h*(c*g^2 + a*h^2)^3*(g + h*x)^3) - (a*c^2*(4*c^2*d*g^3 + a^2*h^2*(6*f*g - e*h) - a*c
*g*(f*g^2 - 3*h*(2*e*g - d*h)))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(8*(c*g^2 + a*h^
2)^(9/2))

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

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{\sqrt{a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}-\frac{\int \frac{\left (-5 (c d g-a f g+a e h)-\left (5 a f h+c \left (2 e g+\frac{3 f g^2}{h}-2 d h\right )\right ) x\right ) \sqrt{a+c x^2}}{(g+h x)^5} \, dx}{5 \left (c g^2+a h^2\right )}\\ &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac{\left (3 c f g^3+c g h (2 e g-7 d h)+5 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{3/2}}{20 h \left (c g^2+a h^2\right )^2 (g+h x)^4}+\frac{\int \frac{\left (4 \left (5 c^2 d g^2+5 a^2 f h^2-a c \left (2 f g^2-h (7 e g-2 d h)\right )\right )+\frac{c \left (3 c f g^3+c g h (2 e g-7 d h)+5 a h^2 (2 f g-e h)\right ) x}{h}\right ) \sqrt{a+c x^2}}{(g+h x)^4} \, dx}{20 \left (c g^2+a h^2\right )^2}\\ &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac{\left (3 c f g^3+c g h (2 e g-7 d h)+5 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{3/2}}{20 h \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac{\left (20 a^2 f h^4-c^2 \left (3 f g^4+g^2 h (2 e g-27 d h)\right )-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )\right ) \left (a+c x^2\right )^{3/2}}{60 h \left (c g^2+a h^2\right )^3 (g+h x)^3}+\frac{\left (c \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )\right )\right ) \int \frac{\sqrt{a+c x^2}}{(g+h x)^3} \, dx}{4 \left (c g^2+a h^2\right )^3}\\ &=-\frac{c \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )\right ) (a h-c g x) \sqrt{a+c x^2}}{8 \left (c g^2+a h^2\right )^4 (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac{\left (3 c f g^3+c g h (2 e g-7 d h)+5 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{3/2}}{20 h \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac{\left (20 a^2 f h^4-c^2 \left (3 f g^4+g^2 h (2 e g-27 d h)\right )-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )\right ) \left (a+c x^2\right )^{3/2}}{60 h \left (c g^2+a h^2\right )^3 (g+h x)^3}+\frac{\left (a c^2 \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{8 \left (c g^2+a h^2\right )^4}\\ &=-\frac{c \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )\right ) (a h-c g x) \sqrt{a+c x^2}}{8 \left (c g^2+a h^2\right )^4 (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac{\left (3 c f g^3+c g h (2 e g-7 d h)+5 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{3/2}}{20 h \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac{\left (20 a^2 f h^4-c^2 \left (3 f g^4+g^2 h (2 e g-27 d h)\right )-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )\right ) \left (a+c x^2\right )^{3/2}}{60 h \left (c g^2+a h^2\right )^3 (g+h x)^3}-\frac{\left (a c^2 \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c g^2+a h^2-x^2} \, dx,x,\frac{a h-c g x}{\sqrt{a+c x^2}}\right )}{8 \left (c g^2+a h^2\right )^4}\\ &=-\frac{c \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )\right ) (a h-c g x) \sqrt{a+c x^2}}{8 \left (c g^2+a h^2\right )^4 (g+h x)^2}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac{\left (3 c f g^3+c g h (2 e g-7 d h)+5 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{3/2}}{20 h \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac{\left (20 a^2 f h^4-c^2 \left (3 f g^4+g^2 h (2 e g-27 d h)\right )-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )\right ) \left (a+c x^2\right )^{3/2}}{60 h \left (c g^2+a h^2\right )^3 (g+h x)^3}-\frac{a c^2 \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{c g^2+a h^2} \sqrt{a+c x^2}}\right )}{8 \left (c g^2+a h^2\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.52842, size = 583, normalized size = 1.35 \[ -\frac{\sqrt{a+c x^2} \left (2 (g+h x)^2 \left (a h^2+c g^2\right )^2 \left (20 a^2 f h^4+a c h^2 \left (h (4 d h-9 e g)+54 f g^2\right )+c^2 \left (27 f g^4-g^2 h (3 d h+2 e g)\right )\right )-c (g+h x)^3 \left (a h^2+c g^2\right ) \left (5 a^2 h^4 (10 f g-3 e h)+a c g h^2 \left (h (24 e g-29 d h)+21 f g^2\right )+c^2 \left (2 g^3 h (3 d h+2 e g)+6 f g^5\right )\right )-c (g+h x)^4 \left (a^2 c h^4 \left (h (16 d h-81 e g)+86 f g^2\right )-40 a^3 f h^6+a c^2 g^2 h^2 \left (h (28 e g-83 d h)+27 f g^2\right )+c^3 \left (2 g^4 h (3 d h+2 e g)+6 f g^6\right )\right )-6 (g+h x) \left (a h^2+c g^2\right )^3 \left (-5 a h^2 (e h-2 f g)+c g h (d h-6 e g)+11 c f g^3\right )+24 \left (a h^2+c g^2\right )^4 \left (h (d h-e g)+f g^2\right )\right )}{120 h^3 (g+h x)^5 \left (a h^2+c g^2\right )^4}-\frac{a c^2 \log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (a^2 h^2 (6 f g-e h)-a c g \left (3 h (d h-2 e g)+f g^2\right )+4 c^2 d g^3\right )}{8 \left (a h^2+c g^2\right )^{9/2}}+\frac{a c^2 \log (g+h x) \left (a^2 h^2 (6 f g-e h)-a c g \left (3 h (d h-2 e g)+f g^2\right )+4 c^2 d g^3\right )}{8 \left (a h^2+c g^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + c*x^2]*(24*(c*g^2 + a*h^2)^4*(f*g^2 + h*(-(e*g) + d*h)) - 6*(c*g^2 + a*h^2)^3*(11*c*f*g^3 + c*g*h*(
-6*e*g + d*h) - 5*a*h^2*(-2*f*g + e*h))*(g + h*x) + 2*(c*g^2 + a*h^2)^2*(20*a^2*f*h^4 + c^2*(27*f*g^4 - g^2*h*
(2*e*g + 3*d*h)) + a*c*h^2*(54*f*g^2 + h*(-9*e*g + 4*d*h)))*(g + h*x)^2 - c*(c*g^2 + a*h^2)*(5*a^2*h^4*(10*f*g
- 3*e*h) + a*c*g*h^2*(21*f*g^2 + h*(24*e*g - 29*d*h)) + c^2*(6*f*g^5 + 2*g^3*h*(2*e*g + 3*d*h)))*(g + h*x)^3
- c*(-40*a^3*f*h^6 + a*c^2*g^2*h^2*(27*f*g^2 + h*(28*e*g - 83*d*h)) + c^3*(6*f*g^6 + 2*g^4*h*(2*e*g + 3*d*h))
+ a^2*c*h^4*(86*f*g^2 + h*(-81*e*g + 16*d*h)))*(g + h*x)^4))/(120*h^3*(c*g^2 + a*h^2)^4*(g + h*x)^5) + (a*c^2*
(4*c^2*d*g^3 + a^2*h^2*(6*f*g - e*h) - a*c*g*(f*g^2 + 3*h*(-2*e*g + d*h)))*Log[g + h*x])/(8*(c*g^2 + a*h^2)^(9
/2)) - (a*c^2*(4*c^2*d*g^3 + a^2*h^2*(6*f*g - e*h) - a*c*g*(f*g^2 + 3*h*(-2*e*g + d*h)))*Log[a*h - c*g*x + Sqr
t[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(8*(c*g^2 + a*h^2)^(9/2))

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Maple [B]  time = 0.25, size = 8546, normalized size = 19.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^6,x)

[Out]

result too large to display

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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((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^6,x, algorithm="fricas")

[Out]

Timed out

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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((f*x**2+e*x+d)*(c*x**2+a)**(1/2)/(h*x+g)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.62264, size = 5686, normalized size = 13.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^6,x, algorithm="giac")

[Out]

-1/4*(4*a*c^4*d*g^3 - a^2*c^3*f*g^3 - 3*a^2*c^3*d*g*h^2 + 6*a^3*c^2*f*g*h^2 + 6*a^2*c^3*g^2*h*e - a^3*c^2*h^3*
e)*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c^4*g^8 + 4*a*c^3*g^6*h^2 + 6*
a^2*c^2*g^4*h^4 + 4*a^3*c*g^2*h^6 + a^4*h^8)*sqrt(-c*g^2 - a*h^2)) - 1/60*(60*(sqrt(c)*x - sqrt(c*x^2 + a))^9*
a*c^4*d*g^3*h^8 - 15*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^3*f*g^3*h^8 - 45*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^
2*c^3*d*g*h^10 + 90*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^2*f*g*h^10 + 90*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*
c^3*g^2*h^9*e - 15*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^2*h^11*e - 120*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(11/
2)*f*g^8*h^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(9/2)*f*g^6*h^5 + 540*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a
*c^(9/2)*d*g^4*h^7 - 855*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*f*g^4*h^7 - 405*(sqrt(c)*x - sqrt(c*x^2 +
a))^8*a^2*c^(7/2)*d*g^2*h^9 + 330*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*f*g^2*h^9 - 120*(sqrt(c)*x - sq
rt(c*x^2 + a))^8*a^4*c^(3/2)*f*h^11 + 810*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*g^3*h^8*e - 135*(sqrt(c)
*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*g*h^10*e - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*f*g^9*h^2 - 960*(sqrt(c
)*x - sqrt(c*x^2 + a))^7*a*c^5*f*g^7*h^4 + 1880*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^5*d*g^5*h^6 - 1910*(sqrt(c
)*x - sqrt(c*x^2 + a))^7*a^2*c^4*f*g^5*h^6 - 1690*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4*d*g^3*h^8 + 1930*(sq
rt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*f*g^3*h^8 + 210*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*d*g*h^10 - 660*(s
qrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^2*f*g*h^10 - 160*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*g^8*h^3*e - 640*(sqrt
(c)*x - sqrt(c*x^2 + a))^7*a*c^5*g^6*h^5*e + 1860*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4*g^4*h^7*e - 1530*(sq
rt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*g^2*h^9*e - 90*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^2*h^11*e - 240*(sqrt
(c)*x - sqrt(c*x^2 + a))^6*c^(13/2)*f*g^10*h - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(13/2)*d*g^8*h^3 - 720*(s
qrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(11/2)*f*g^8*h^3 + 2120*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(11/2)*d*g^6*h^5
- 1250*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(9/2)*f*g^6*h^5 - 5710*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(9/
2)*d*g^4*h^7 + 5590*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(7/2)*f*g^4*h^7 + 510*(sqrt(c)*x - sqrt(c*x^2 + a))^
6*a^3*c^(7/2)*d*g^2*h^9 - 2220*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(5/2)*f*g^2*h^9 - 240*(sqrt(c)*x - sqrt(c
*x^2 + a))^6*a^4*c^(5/2)*d*h^11 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(3/2)*f*h^11 - 160*(sqrt(c)*x - sq
rt(c*x^2 + a))^6*c^(13/2)*g^9*h^2*e - 640*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(11/2)*g^7*h^4*e + 3660*(sqrt(c)
*x - sqrt(c*x^2 + a))^6*a^2*c^(9/2)*g^5*h^6*e - 4350*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(7/2)*g^3*h^8*e + 3
30*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(5/2)*g*h^10*e - 96*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^7*f*g^11 - 96*(
sqrt(c)*x - sqrt(c*x^2 + a))^5*c^7*d*g^9*h^2 + 48*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^6*f*g^9*h^2 + 1808*(sqrt
(c)*x - sqrt(c*x^2 + a))^5*a*c^6*d*g^7*h^4 + 604*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^5*f*g^7*h^4 - 7076*(sqr
t(c)*x - sqrt(c*x^2 + a))^5*a^2*c^5*d*g^5*h^6 + 6710*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^4*f*g^5*h^6 + 3770*
(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^4*d*g^3*h^8 - 5780*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^3*f*g^3*h^8 - 4
80*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^3*d*g*h^10 + 1200*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^2*f*g*h^10 -
64*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^7*g^10*h*e - 128*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^6*g^8*h^3*e + 3416*(
sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^5*g^6*h^5*e - 7320*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^4*g^4*h^7*e + 24
30*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^3*g^2*h^9*e + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(13/2)*f*g^10*h
+ 240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(13/2)*d*g^8*h^3 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(11/2)
*f*g^8*h^3 - 5240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(11/2)*d*g^6*h^5 + 2450*(sqrt(c)*x - sqrt(c*x^2 + a))^
4*a^3*c^(9/2)*f*g^6*h^5 + 5590*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(9/2)*d*g^4*h^7 - 7660*(sqrt(c)*x - sqrt(
c*x^2 + a))^4*a^4*c^(7/2)*f*g^4*h^7 - 2240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(7/2)*d*g^2*h^9 + 3440*(sqrt(
c)*x - sqrt(c*x^2 + a))^4*a^5*c^(5/2)*f*g^2*h^9 - 80*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(5/2)*d*h^11 - 160*
(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(3/2)*f*h^11 + 160*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(13/2)*g^9*h^2*e
+ 1120*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(11/2)*g^7*h^4*e - 6140*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(9/
2)*g^5*h^6*e + 5650*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(7/2)*g^3*h^8*e - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^
4*a^5*c^(5/2)*g*h^10*e - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^6*f*g^9*h^2 - 480*(sqrt(c)*x - sqrt(c*x^2 +
a))^3*a^2*c^6*d*g^7*h^4 - 960*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^5*f*g^7*h^4 + 5000*(sqrt(c)*x - sqrt(c*x^
2 + a))^3*a^3*c^5*d*g^5*h^6 - 3890*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^4*f*g^5*h^6 - 2910*(sqrt(c)*x - sqrt(
c*x^2 + a))^3*a^4*c^4*d*g^3*h^8 + 4710*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^3*f*g^3*h^8 + 430*(sqrt(c)*x - sq
rt(c*x^2 + a))^3*a^5*c^3*d*g*h^10 - 940*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^2*f*g*h^10 - 160*(sqrt(c)*x - sq
rt(c*x^2 + a))^3*a^2*c^6*g^8*h^3*e - 1440*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^5*g^6*h^5*e + 5740*(sqrt(c)*x
- sqrt(c*x^2 + a))^3*a^4*c^4*g^4*h^7*e - 1710*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^3*g^2*h^9*e + 90*(sqrt(c)*
x - sqrt(c*x^2 + a))^3*a^6*c^2*h^11*e + 120*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(11/2)*f*g^8*h^3 + 240*(sqrt
(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(11/2)*d*g^6*h^5 + 570*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(9/2)*f*g^6*h^5
- 2810*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(9/2)*d*g^4*h^7 + 2450*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(7/2
)*f*g^4*h^7 + 650*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(7/2)*d*g^2*h^9 - 1700*(sqrt(c)*x - sqrt(c*x^2 + a))^2
*a^6*c^(5/2)*f*g^2*h^9 - 80*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(5/2)*d*h^11 + 80*(sqrt(c)*x - sqrt(c*x^2 +
a))^2*a^7*c^(3/2)*f*h^11 + 160*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(11/2)*g^7*h^4*e + 1100*(sqrt(c)*x - sqrt
(c*x^2 + a))^2*a^4*c^(9/2)*g^5*h^6*e - 2570*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(7/2)*g^3*h^8*e + 270*(sqrt(
c)*x - sqrt(c*x^2 + a))^2*a^6*c^(5/2)*g*h^10*e - 60*(sqrt(c)*x - sqrt(c*x^2 + a))*a^4*c^5*f*g^7*h^4 - 60*(sqrt
(c)*x - sqrt(c*x^2 + a))*a^4*c^5*d*g^5*h^6 - 270*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^4*f*g^5*h^6 + 770*(sqrt(c
)*x - sqrt(c*x^2 + a))*a^5*c^4*d*g^3*h^8 - 845*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^3*f*g^3*h^8 - 115*(sqrt(c)*
x - sqrt(c*x^2 + a))*a^6*c^3*d*g*h^10 + 310*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^2*f*g*h^10 - 40*(sqrt(c)*x - s
qrt(c*x^2 + a))*a^4*c^5*g^6*h^5*e - 280*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^4*g^4*h^7*e + 720*(sqrt(c)*x - sqr
t(c*x^2 + a))*a^6*c^3*g^2*h^9*e + 15*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^2*h^11*e + 6*a^5*c^(9/2)*f*g^6*h^5 +
6*a^5*c^(9/2)*d*g^4*h^7 + 27*a^6*c^(7/2)*f*g^4*h^7 - 83*a^6*c^(7/2)*d*g^2*h^9 + 86*a^7*c^(5/2)*f*g^2*h^9 + 16*
a^7*c^(5/2)*d*h^11 - 40*a^8*c^(3/2)*f*h^11 + 4*a^5*c^(9/2)*g^5*h^6*e + 28*a^6*c^(7/2)*g^3*h^8*e - 81*a^7*c^(5/
2)*g*h^10*e)/((c^4*g^8*h^4 + 4*a*c^3*g^6*h^6 + 6*a^2*c^2*g^4*h^8 + 4*a^3*c*g^2*h^10 + a^4*h^12)*((sqrt(c)*x -
sqrt(c*x^2 + a))^2*h + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*g - a*h)^5)