### 3.98 $$\int \frac{(a+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^7} \, dx$$

Optimal. Leaf size=404 $-\frac{\left (a+c x^2\right )^{3/2} (a h-c g x) \left (6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )+6 c^2 d g^2\right )}{24 (g+h x)^4 \left (a h^2+c g^2\right )^3}-\frac{a c \sqrt{a+c x^2} (a h-c g x) \left (6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )+6 c^2 d g^2\right )}{16 (g+h x)^2 \left (a h^2+c g^2\right )^4}-\frac{a^2 c^2 \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )+6 c^2 d g^2\right )}{16 \left (a h^2+c g^2\right )^{9/2}}+\frac{\left (a+c x^2\right )^{5/2} \left (6 a h^2 (2 f g-e h)+c g \left (h (e g-7 d h)+5 f g^2\right )\right )}{30 h (g+h x)^5 \left (a h^2+c g^2\right )^2}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{6 h (g+h x)^6 \left (a h^2+c g^2\right )}$

[Out]

-(a*c*(6*c^2*d*g^2 + 6*a^2*f*h^2 - a*c*(f*g^2 - h*(7*e*g - d*h)))*(a*h - c*g*x)*Sqrt[a + c*x^2])/(16*(c*g^2 +
a*h^2)^4*(g + h*x)^2) - ((6*c^2*d*g^2 + 6*a^2*f*h^2 - a*c*(f*g^2 - h*(7*e*g - d*h)))*(a*h - c*g*x)*(a + c*x^2)
^(3/2))/(24*(c*g^2 + a*h^2)^3*(g + h*x)^4) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(5/2))/(6*h*(c*g^2 + a*h^2)*
(g + h*x)^6) + ((6*a*h^2*(2*f*g - e*h) + c*g*(5*f*g^2 + h*(e*g - 7*d*h)))*(a + c*x^2)^(5/2))/(30*h*(c*g^2 + a*
h^2)^2*(g + h*x)^5) - (a^2*c^2*(6*c^2*d*g^2 + 6*a^2*f*h^2 - a*c*(f*g^2 - h*(7*e*g - d*h)))*ArcTanh[(a*h - c*g*
x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(16*(c*g^2 + a*h^2)^(9/2))

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Rubi [A]  time = 0.552242, antiderivative size = 403, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.172, Rules used = {1651, 807, 721, 725, 206} $-\frac{\left (a+c x^2\right )^{3/2} (a h-c g x) \left (6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )+6 c^2 d g^2\right )}{24 (g+h x)^4 \left (a h^2+c g^2\right )^3}-\frac{a c \sqrt{a+c x^2} (a h-c g x) \left (6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )+6 c^2 d g^2\right )}{16 (g+h x)^2 \left (a h^2+c g^2\right )^4}-\frac{a^2 c^2 \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )+6 c^2 d g^2\right )}{16 \left (a h^2+c g^2\right )^{9/2}}+\frac{\left (a+c x^2\right )^{5/2} \left (6 a h^2 (2 f g-e h)+c g h (e g-7 d h)+5 c f g^3\right )}{30 h (g+h x)^5 \left (a h^2+c g^2\right )^2}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{6 h (g+h x)^6 \left (a h^2+c g^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^7,x]

[Out]

-(a*c*(6*c^2*d*g^2 + 6*a^2*f*h^2 - a*c*(f*g^2 - h*(7*e*g - d*h)))*(a*h - c*g*x)*Sqrt[a + c*x^2])/(16*(c*g^2 +
a*h^2)^4*(g + h*x)^2) - ((6*c^2*d*g^2 + 6*a^2*f*h^2 - a*c*(f*g^2 - h*(7*e*g - d*h)))*(a*h - c*g*x)*(a + c*x^2)
^(3/2))/(24*(c*g^2 + a*h^2)^3*(g + h*x)^4) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(5/2))/(6*h*(c*g^2 + a*h^2)*
(g + h*x)^6) + ((5*c*f*g^3 + c*g*h*(e*g - 7*d*h) + 6*a*h^2*(2*f*g - e*h))*(a + c*x^2)^(5/2))/(30*h*(c*g^2 + a*
h^2)^2*(g + h*x)^5) - (a^2*c^2*(6*c^2*d*g^2 + 6*a^2*f*h^2 - a*c*(f*g^2 - h*(7*e*g - d*h)))*ArcTanh[(a*h - c*g*
x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(16*(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 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{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{6 h \left (c g^2+a h^2\right ) (g+h x)^6}-\frac{\int \frac{\left (-6 (c d g-a f g+a e h)-\left (6 a f h+c \left (e g+\frac{5 f g^2}{h}-d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^6} \, dx}{6 \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 )^{5/2}}{6 h \left (c g^2+a h^2\right ) (g+h x)^6}+\frac{\left (5 c f g^3+c g h (e g-7 d h)+6 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{30 h \left (c g^2+a h^2\right )^2 (g+h x)^5}+\frac{\left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )\right ) \int \frac{\left (a+c x^2\right )^{3/2}}{(g+h x)^5} \, dx}{6 \left (c g^2+a h^2\right )^2}\\ &=-\frac{\left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )\right ) (a h-c g x) \left (a+c x^2\right )^{3/2}}{24 \left (c g^2+a h^2\right )^3 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{6 h \left (c g^2+a h^2\right ) (g+h x)^6}+\frac{\left (5 c f g^3+c g h (e g-7 d h)+6 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{30 h \left (c g^2+a h^2\right )^2 (g+h x)^5}+\frac{\left (a c \left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )\right )\right ) \int \frac{\sqrt{a+c x^2}}{(g+h x)^3} \, dx}{8 \left (c g^2+a h^2\right )^3}\\ &=-\frac{a c \left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )\right ) (a h-c g x) \sqrt{a+c x^2}}{16 \left (c g^2+a h^2\right )^4 (g+h x)^2}-\frac{\left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )\right ) (a h-c g x) \left (a+c x^2\right )^{3/2}}{24 \left (c g^2+a h^2\right )^3 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{6 h \left (c g^2+a h^2\right ) (g+h x)^6}+\frac{\left (5 c f g^3+c g h (e g-7 d h)+6 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{30 h \left (c g^2+a h^2\right )^2 (g+h x)^5}+\frac{\left (a^2 c^2 \left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+c x^2}} \, dx}{16 \left (c g^2+a h^2\right )^4}\\ &=-\frac{a c \left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )\right ) (a h-c g x) \sqrt{a+c x^2}}{16 \left (c g^2+a h^2\right )^4 (g+h x)^2}-\frac{\left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )\right ) (a h-c g x) \left (a+c x^2\right )^{3/2}}{24 \left (c g^2+a h^2\right )^3 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{6 h \left (c g^2+a h^2\right ) (g+h x)^6}+\frac{\left (5 c f g^3+c g h (e g-7 d h)+6 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{30 h \left (c g^2+a h^2\right )^2 (g+h x)^5}-\frac{\left (a^2 c^2 \left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 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 )}{16 \left (c g^2+a h^2\right )^4}\\ &=-\frac{a c \left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )\right ) (a h-c g x) \sqrt{a+c x^2}}{16 \left (c g^2+a h^2\right )^4 (g+h x)^2}-\frac{\left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 e g-d h)\right )\right ) (a h-c g x) \left (a+c x^2\right )^{3/2}}{24 \left (c g^2+a h^2\right )^3 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{6 h \left (c g^2+a h^2\right ) (g+h x)^6}+\frac{\left (5 c f g^3+c g h (e g-7 d h)+6 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{30 h \left (c g^2+a h^2\right )^2 (g+h x)^5}-\frac{a^2 c^2 \left (6 c^2 d g^2+6 a^2 f h^2-a c \left (f g^2-h (7 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 )}{16 \left (c g^2+a h^2\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 2.47528, size = 696, normalized size = 1.72 $\frac{1}{240} \left (-\frac{\sqrt{a+c x^2} \left (2 (g+h x)^2 \left (a h^2+c g^2\right )^3 \left (30 a^2 f h^4+a c h^2 \left (h (35 d h-101 e g)+227 f g^2\right )+2 c^2 \left (g^2 h (19 d h-52 e g)+100 f g^4\right )\right )-2 c (g+h x)^3 \left (a h^2+c g^2\right )^2 \left (6 a^2 h^4 (31 f g-8 e h)+3 a c g h^2 \left (h (3 d h-37 e g)+131 f g^2\right )+2 c^2 \left (g^3 h (d h-28 e g)+100 f g^5\right )\right )+c (g+h x)^4 \left (a h^2+c g^2\right ) \left (3 a^2 c h^4 \left (h (5 d h-19 e g)+193 f g^2\right )+150 a^3 f h^6+6 a c^2 g^2 h^2 \left (99 f g^2-h (4 d h+5 e g)\right )+4 c^3 \left (50 f g^6-g^4 h (d h+2 e g)\right )\right )-c^2 (g+h x)^5 \left (3 a^2 c g h^4 \left (h (29 e g-27 d h)+89 f g^2\right )+6 a^3 h^6 (41 f g-8 e h)+2 a c^2 g^3 h^2 \left (h (14 d h+19 e g)+83 f g^2\right )+4 c^3 \left (g^5 h (d h+2 e g)+10 f g^7\right )\right )-8 (g+h x) \left (a h^2+c g^2\right )^4 \left (-6 a h^2 (e h-2 f g)+c g h (13 d h-19 e g)+25 c f g^3\right )+40 \left (a h^2+c g^2\right )^5 \left (h (d h-e g)+f g^2\right )\right )}{h^5 (g+h x)^6 \left (a h^2+c g^2\right )^4}-\frac{15 a^2 c^2 \log \left (\sqrt{a+c x^2} \sqrt{a h^2+c g^2}+a h-c g x\right ) \left (6 a^2 f h^2-a c \left (h (d h-7 e g)+f g^2\right )+6 c^2 d g^2\right )}{\left (a h^2+c g^2\right )^{9/2}}+\frac{15 a^2 c^2 \log (g+h x) \left (6 a^2 f h^2-a c \left (h (d h-7 e g)+f g^2\right )+6 c^2 d g^2\right )}{\left (a h^2+c g^2\right )^{9/2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^7,x]

[Out]

(-((Sqrt[a + c*x^2]*(40*(c*g^2 + a*h^2)^5*(f*g^2 + h*(-(e*g) + d*h)) - 8*(c*g^2 + a*h^2)^4*(25*c*f*g^3 + c*g*h
*(-19*e*g + 13*d*h) - 6*a*h^2*(-2*f*g + e*h))*(g + h*x) + 2*(c*g^2 + a*h^2)^3*(30*a^2*f*h^4 + 2*c^2*(100*f*g^4
+ g^2*h*(-52*e*g + 19*d*h)) + a*c*h^2*(227*f*g^2 + h*(-101*e*g + 35*d*h)))*(g + h*x)^2 - 2*c*(c*g^2 + a*h^2)^
2*(6*a^2*h^4*(31*f*g - 8*e*h) + 2*c^2*(100*f*g^5 + g^3*h*(-28*e*g + d*h)) + 3*a*c*g*h^2*(131*f*g^2 + h*(-37*e*
g + 3*d*h)))*(g + h*x)^3 + c*(c*g^2 + a*h^2)*(150*a^3*f*h^6 + 4*c^3*(50*f*g^6 - g^4*h*(2*e*g + d*h)) + 6*a*c^2
*g^2*h^2*(99*f*g^2 - h*(5*e*g + 4*d*h)) + 3*a^2*c*h^4*(193*f*g^2 + h*(-19*e*g + 5*d*h)))*(g + h*x)^4 - c^2*(6*
a^3*h^6*(41*f*g - 8*e*h) + 3*a^2*c*g*h^4*(89*f*g^2 + h*(29*e*g - 27*d*h)) + 4*c^3*(10*f*g^7 + g^5*h*(2*e*g + d
*h)) + 2*a*c^2*g^3*h^2*(83*f*g^2 + h*(19*e*g + 14*d*h)))*(g + h*x)^5))/(h^5*(c*g^2 + a*h^2)^4*(g + h*x)^6)) +
(15*a^2*c^2*(6*c^2*d*g^2 + 6*a^2*f*h^2 - a*c*(f*g^2 + h*(-7*e*g + d*h)))*Log[g + h*x])/(c*g^2 + a*h^2)^(9/2) -
(15*a^2*c^2*(6*c^2*d*g^2 + 6*a^2*f*h^2 - a*c*(f*g^2 + h*(-7*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2
]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(9/2))/240

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Maple [B]  time = 0.27, size = 17026, normalized size = 42.1 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^7,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((c*x^2+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^7,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((c*x^2+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^7,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((c*x**2+a)**(3/2)*(f*x**2+e*x+d)/(h*x+g)**7,x)

[Out]

Timed out

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Giac [B]  time = 2.11242, size = 8265, normalized size = 20.46 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^7,x, algorithm="giac")

[Out]

1/8*(6*a^2*c^4*d*g^2 - a^3*c^3*f*g^2 - a^3*c^3*d*h^2 + 6*a^4*c^2*f*h^2 + 7*a^3*c^3*g*h*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)) + 1/120*(240*(sqrt(c)*x - sqrt(c*x^2 + a))^11*c^6*f*g^8*h^5 + 960
*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^5*f*g^6*h^7 + 1440*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^4*f*g^4*h^9 -
90*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^4*d*g^2*h^11 + 975*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^3*f*g^2*h^
11 + 15*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^3*d*h^13 + 150*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^4*c^2*f*h^13
- 105*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^3*g*h^12*e + 1200*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(13/2)*f*g^9
*h^4 + 4800*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a*c^(11/2)*f*g^7*h^6 + 7200*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*
c^(9/2)*f*g^5*h^8 - 990*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(9/2)*d*g^3*h^10 + 4965*(sqrt(c)*x - sqrt(c*x^2
+ a))^10*a^3*c^(7/2)*f*g^3*h^10 + 165*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(7/2)*d*g*h^12 + 210*(sqrt(c)*x
- sqrt(c*x^2 + a))^10*a^4*c^(5/2)*f*g*h^12 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(13/2)*g^8*h^5*e + 960*(sq
rt(c)*x - sqrt(c*x^2 + a))^10*a*c^(11/2)*g^6*h^7*e + 1440*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(9/2)*g^4*h^9
*e - 195*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(7/2)*g^2*h^11*e + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^4*c^
(5/2)*h^13*e + 3200*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^7*f*g^10*h^3 + 320*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^7*d
*g^8*h^5 + 12080*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*f*g^8*h^5 + 1280*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*
d*g^6*h^7 + 16320*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^5*f*g^6*h^7 - 2520*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2
*c^5*d*g^4*h^9 + 9220*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^4*f*g^4*h^9 + 2530*(sqrt(c)*x - sqrt(c*x^2 + a))^9
*a^3*c^4*d*g^2*h^11 - 4205*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^3*f*g^2*h^11 + 235*(sqrt(c)*x - sqrt(c*x^2 +
a))^9*a^4*c^3*d*h^13 - 210*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^5*c^2*f*h^13 + 640*(sqrt(c)*x - sqrt(c*x^2 + a))^
9*c^7*g^9*h^4*e + 2560*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*g^7*h^6*e + 3840*(sqrt(c)*x - sqrt(c*x^2 + a))^9*
a^2*c^5*g^5*h^8*e - 2620*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^4*g^3*h^10*e + 1235*(sqrt(c)*x - sqrt(c*x^2 + a
))^9*a^4*c^3*g*h^12*e + 4800*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(15/2)*f*g^11*h^2 + 480*(sqrt(c)*x - sqrt(c*x^2
+ a))^8*c^(15/2)*d*g^9*h^4 + 15120*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(13/2)*f*g^9*h^4 + 1920*(sqrt(c)*x - s
qrt(c*x^2 + a))^8*a*c^(13/2)*d*g^7*h^6 + 12480*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*f*g^7*h^6 - 7380*(
sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*d*g^5*h^8 - 3570*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(9/2)*f*g^5
*h^8 + 8220*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(9/2)*d*g^3*h^10 - 22545*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4
*c^(7/2)*f*g^3*h^10 - 285*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(7/2)*d*g*h^12 + 510*(sqrt(c)*x - sqrt(c*x^2 +
a))^8*a^5*c^(5/2)*f*g*h^12 + 960*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(15/2)*g^10*h^3*e + 3600*(sqrt(c)*x - sqrt
(c*x^2 + a))^8*a*c^(13/2)*g^8*h^5*e + 4800*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*g^6*h^7*e - 9570*(sqrt
(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(9/2)*g^4*h^9*e + 5355*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(7/2)*g^2*h^11*e
- 240*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^5*c^(5/2)*h^13*e + 3840*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*f*g^12*h
+ 384*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*d*g^10*h^3 + 6336*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^7*f*g^10*h^3 +
1728*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^7*d*g^8*h^5 - 11808*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*f*g^8*h^
5 - 9456*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*d*g^6*h^7 - 31704*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^5*f*g
^6*h^7 + 20760*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^5*d*g^4*h^9 - 39960*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c
^4*f*g^4*h^9 - 2700*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^4*d*g^2*h^11 + 12150*(sqrt(c)*x - sqrt(c*x^2 + a))^7
*a^5*c^3*f*g^2*h^11 + 390*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^3*d*h^13 + 60*(sqrt(c)*x - sqrt(c*x^2 + a))^7*
a^6*c^2*f*h^13 + 768*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*g^11*h^2*e + 1728*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c
^7*g^9*h^4*e - 768*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*g^7*h^6*e - 19608*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a
^3*c^5*g^5*h^8*e + 14040*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^4*g^3*h^10*e - 2730*(sqrt(c)*x - sqrt(c*x^2 + a
))^7*a^5*c^3*g*h^12*e + 1280*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(17/2)*f*g^13 + 128*(sqrt(c)*x - sqrt(c*x^2 + a
))^6*c^(17/2)*d*g^11*h^2 - 4288*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(15/2)*f*g^11*h^2 - 64*(sqrt(c)*x - sqrt(c
*x^2 + a))^6*a*c^(15/2)*d*g^9*h^4 - 24096*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*f*g^9*h^4 - 8592*(sqrt(
c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*d*g^7*h^6 - 26728*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(11/2)*f*g^7*h^
6 + 24440*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(11/2)*d*g^5*h^8 - 12640*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c
^(9/2)*f*g^5*h^8 - 14860*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(9/2)*d*g^3*h^10 + 41610*(sqrt(c)*x - sqrt(c*x^
2 + a))^6*a^5*c^(7/2)*f*g^3*h^10 + 810*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(7/2)*d*g*h^12 - 2460*(sqrt(c)*x
- sqrt(c*x^2 + a))^6*a^6*c^(5/2)*f*g*h^12 + 256*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(17/2)*g^12*h*e - 704*(sqrt(
c)*x - sqrt(c*x^2 + a))^6*a*c^(15/2)*g^10*h^3*e - 4896*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*g^8*h^5*e
- 15656*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(11/2)*g^6*h^7*e + 26800*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(
9/2)*g^4*h^9*e - 9510*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(7/2)*g^2*h^11*e + 480*(sqrt(c)*x - sqrt(c*x^2 + a
))^6*a^6*c^(5/2)*h^13*e - 3840*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^8*f*g^12*h - 384*(sqrt(c)*x - sqrt(c*x^2 +
a))^5*a*c^8*d*g^10*h^3 - 6336*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^7*f*g^10*h^3 - 1728*(sqrt(c)*x - sqrt(c*x^
2 + a))^5*a^2*c^7*d*g^8*h^5 + 11808*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^6*f*g^8*h^5 + 19056*(sqrt(c)*x - sqr
t(c*x^2 + a))^5*a^3*c^6*d*g^6*h^7 + 29304*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^5*f*g^6*h^7 - 21480*(sqrt(c)*x
- sqrt(c*x^2 + a))^5*a^4*c^5*d*g^4*h^9 + 46080*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^4*f*g^4*h^9 + 7020*(sqrt
(c)*x - sqrt(c*x^2 + a))^5*a^5*c^4*d*g^2*h^11 - 17370*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^3*f*g^2*h^11 + 390
*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^3*d*h^13 + 60*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^7*c^2*f*h^13 - 768*(sqr
t(c)*x - sqrt(c*x^2 + a))^5*a*c^8*g^11*h^2*e - 1728*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^7*g^9*h^4*e - 192*(s
qrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^6*g^7*h^6*e + 26808*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^5*g^5*h^8*e - 19
440*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^4*g^3*h^10*e + 3030*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^3*g*h^12*e
+ 4800*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(15/2)*f*g^11*h^2 + 480*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(1
5/2)*d*g^9*h^4 + 15120*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(13/2)*f*g^9*h^4 + 3840*(sqrt(c)*x - sqrt(c*x^2 +
a))^4*a^3*c^(13/2)*d*g^7*h^6 + 12360*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(11/2)*f*g^7*h^6 - 18720*(sqrt(c)*
x - sqrt(c*x^2 + a))^4*a^4*c^(11/2)*d*g^5*h^8 + 1020*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(9/2)*f*g^5*h^8 + 1
1640*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(9/2)*d*g^3*h^10 - 32490*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(7/2
)*f*g^3*h^10 - 930*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(7/2)*d*g*h^12 + 3180*(sqrt(c)*x - sqrt(c*x^2 + a))^4
*a^7*c^(5/2)*f*g*h^12 + 960*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(15/2)*g^10*h^3*e + 3600*(sqrt(c)*x - sqrt(c
*x^2 + a))^4*a^3*c^(13/2)*g^8*h^5*e + 7080*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(11/2)*g^6*h^7*e - 22260*(sqr
t(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(9/2)*g^4*h^9*e + 7470*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(7/2)*g^2*h^11*
e - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^7*c^(5/2)*h^13*e - 3200*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^7*f*g^
10*h^3 - 320*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^7*d*g^8*h^5 - 12080*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^6
*f*g^8*h^5 - 2960*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^6*d*g^6*h^7 - 16440*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^
5*c^5*f*g^6*h^7 + 12120*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^5*d*g^4*h^9 - 14120*(sqrt(c)*x - sqrt(c*x^2 + a)
)^3*a^6*c^4*f*g^4*h^9 - 2330*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^4*d*g^2*h^11 + 10555*(sqrt(c)*x - sqrt(c*x^
2 + a))^3*a^7*c^3*f*g^2*h^11 + 235*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^3*d*h^13 - 210*(sqrt(c)*x - sqrt(c*x^
2 + a))^3*a^8*c^2*f*h^13 - 640*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^7*g^9*h^4*e - 3040*(sqrt(c)*x - sqrt(c*x^
2 + a))^3*a^4*c^6*g^7*h^6*e - 7800*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^5*g^5*h^8*e + 10280*(sqrt(c)*x - sqrt
(c*x^2 + a))^3*a^6*c^4*g^3*h^10*e - 1645*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^3*g*h^12*e + 1200*(sqrt(c)*x -
sqrt(c*x^2 + a))^2*a^4*c^(13/2)*f*g^9*h^4 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(13/2)*d*g^7*h^6 + 4920*
(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(11/2)*f*g^7*h^6 + 1656*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(11/2)*d*g
^5*h^8 + 7824*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(9/2)*f*g^5*h^8 - 4038*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6
*c^(9/2)*d*g^3*h^10 + 8193*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(7/2)*f*g^3*h^10 + 321*(sqrt(c)*x - sqrt(c*x^
2 + a))^2*a^7*c^(7/2)*d*g*h^12 - 1686*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^8*c^(5/2)*f*g*h^12 + 240*(sqrt(c)*x -
sqrt(c*x^2 + a))^2*a^4*c^(13/2)*g^8*h^5*e + 1272*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(11/2)*g^6*h^7*e + 3552
*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(9/2)*g^4*h^9*e - 3207*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(7/2)*g^2*
h^11*e + 48*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^8*c^(5/2)*h^13*e - 240*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^6*f*g
^8*h^5 - 48*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^6*d*g^6*h^7 - 1032*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^5*f*g^6
*h^7 - 336*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^5*d*g^4*h^9 - 1764*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^4*f*g^4*
h^9 + 882*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^4*d*g^2*h^11 - 1977*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^3*f*g^2*
h^11 + 15*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^3*d*h^13 + 150*(sqrt(c)*x - sqrt(c*x^2 + a))*a^9*c^2*f*h^13 - 96
*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^6*g^7*h^6*e - 456*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^5*g^5*h^8*e - 1044*
(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^4*g^3*h^10*e + 471*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^3*g*h^12*e + 40*a^6
*c^(11/2)*f*g^7*h^6 + 4*a^6*c^(11/2)*d*g^5*h^8 + 166*a^7*c^(9/2)*f*g^5*h^8 + 28*a^7*c^(9/2)*d*g^3*h^10 + 267*a
^8*c^(7/2)*f*g^3*h^10 - 81*a^8*c^(7/2)*d*g*h^12 + 246*a^9*c^(5/2)*f*g*h^12 + 8*a^6*c^(11/2)*g^6*h^7*e + 38*a^7
*c^(9/2)*g^4*h^9*e + 87*a^8*c^(7/2)*g^2*h^11*e - 48*a^9*c^(5/2)*h^13*e)/((c^4*g^8*h^6 + 4*a*c^3*g^6*h^8 + 6*a^
2*c^2*g^4*h^10 + 4*a^3*c*g^2*h^12 + a^4*h^14)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*h + 2*(sqrt(c)*x - sqrt(c*x^2 +
a))*sqrt(c)*g - a*h)^6)