3.99 \(\int \frac{(a+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^8} \, dx\)

Optimal. Leaf size=532 \[ -\frac{a c^2 \sqrt{a+c x^2} (a h-c g x) \left (a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )+6 c^2 d g^3\right )}{16 (g+h x)^2 \left (a h^2+c g^2\right )^5}-\frac{c \left (a+c x^2\right )^{3/2} (a h-c g x) \left (a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )+6 c^2 d g^3\right )}{24 (g+h x)^4 \left (a h^2+c g^2\right )^4}-\frac{\left (a+c x^2\right )^{5/2} \left (42 a^2 f h^4-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )-c^2 g^2 \left (h (2 e g-51 d h)+5 f g^2\right )\right )}{210 h (g+h x)^5 \left (a h^2+c g^2\right )^3}-\frac{a^2 c^3 \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )+6 c^2 d g^3\right )}{16 \left (a h^2+c g^2\right )^{11/2}}+\frac{\left (a+c x^2\right )^{5/2} \left (7 a h^2 (2 f g-e h)+c g \left (h (2 e g-9 d h)+5 f g^2\right )\right )}{42 h (g+h x)^6 \left (a h^2+c g^2\right )^2}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{7 h (g+h x)^7 \left (a h^2+c g^2\right )} \]

[Out]

-(a*c^2*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 - h*(8*e*g - 3*d*h)))*(a*h - c*g*x)*Sqrt[a + c*x^2
])/(16*(c*g^2 + a*h^2)^5*(g + h*x)^2) - (c*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 - h*(8*e*g - 3*
d*h)))*(a*h - c*g*x)*(a + c*x^2)^(3/2))/(24*(c*g^2 + a*h^2)^4*(g + h*x)^4) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x
^2)^(5/2))/(7*h*(c*g^2 + a*h^2)*(g + h*x)^7) + ((7*a*h^2*(2*f*g - e*h) + c*g*(5*f*g^2 + h*(2*e*g - 9*d*h)))*(a
+ c*x^2)^(5/2))/(42*h*(c*g^2 + a*h^2)^2*(g + h*x)^6) - ((42*a^2*f*h^4 - c^2*g^2*(5*f*g^2 + h*(2*e*g - 51*d*h)
) - a*c*h^2*(26*f*g^2 - h*(61*e*g - 12*d*h)))*(a + c*x^2)^(5/2))/(210*h*(c*g^2 + a*h^2)^3*(g + h*x)^5) - (a^2*
c^3*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 - h*(8*e*g - 3*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)^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.888746, antiderivative size = 531, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{a c^2 \sqrt{a+c x^2} (a h-c g x) \left (a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )+6 c^2 d g^3\right )}{16 (g+h x)^2 \left (a h^2+c g^2\right )^5}-\frac{c \left (a+c x^2\right )^{3/2} (a h-c g x) \left (a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )+6 c^2 d g^3\right )}{24 (g+h x)^4 \left (a h^2+c g^2\right )^4}-\frac{\left (a+c x^2\right )^{5/2} \left (42 a^2 f h^4-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )-c^2 \left (g^2 h (2 e g-51 d h)+5 f g^4\right )\right )}{210 h (g+h x)^5 \left (a h^2+c g^2\right )^3}-\frac{a^2 c^3 \tanh ^{-1}\left (\frac{a h-c g x}{\sqrt{a+c x^2} \sqrt{a h^2+c g^2}}\right ) \left (a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )+6 c^2 d g^3\right )}{16 \left (a h^2+c g^2\right )^{11/2}}+\frac{\left (a+c x^2\right )^{5/2} \left (7 a h^2 (2 f g-e h)+c g h (2 e g-9 d h)+5 c f g^3\right )}{42 h (g+h x)^6 \left (a h^2+c g^2\right )^2}-\frac{\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{7 h (g+h x)^7 \left (a h^2+c g^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8,x]

[Out]

-(a*c^2*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 - h*(8*e*g - 3*d*h)))*(a*h - c*g*x)*Sqrt[a + c*x^2
])/(16*(c*g^2 + a*h^2)^5*(g + h*x)^2) - (c*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 - h*(8*e*g - 3*
d*h)))*(a*h - c*g*x)*(a + c*x^2)^(3/2))/(24*(c*g^2 + a*h^2)^4*(g + h*x)^4) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x
^2)^(5/2))/(7*h*(c*g^2 + a*h^2)*(g + h*x)^7) + ((5*c*f*g^3 + c*g*h*(2*e*g - 9*d*h) + 7*a*h^2*(2*f*g - e*h))*(a
+ c*x^2)^(5/2))/(42*h*(c*g^2 + a*h^2)^2*(g + h*x)^6) - ((42*a^2*f*h^4 - c^2*(5*f*g^4 + g^2*h*(2*e*g - 51*d*h)
) - a*c*h^2*(26*f*g^2 - h*(61*e*g - 12*d*h)))*(a + c*x^2)^(5/2))/(210*h*(c*g^2 + a*h^2)^3*(g + h*x)^5) - (a^2*
c^3*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2 - h*(8*e*g - 3*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)^(11/2))

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{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^8} \, dx &=-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}-\frac{\int \frac{\left (-7 (c d g-a f g+a e h)-\left (7 a f h+c \left (2 e g+\frac{5 f g^2}{h}-2 d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^7} \, dx}{7 \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}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac{\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}+\frac{\int \frac{\left (6 \left (7 c^2 d g^2+7 a^2 f h^2-a c \left (2 f g^2-h (9 e g-2 d h)\right )\right )+\frac{c \left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) x}{h}\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^6} \, dx}{42 \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 )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac{\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac{\left (42 a^2 f h^4-c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}+\frac{\left (c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\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 )^3}\\ &=-\frac{c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 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 )^4 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac{\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac{\left (42 a^2 f h^4-c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}+\frac{\left (a c^2 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 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 )^4}\\ &=-\frac{a c^2 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \sqrt{a+c x^2}}{16 \left (c g^2+a h^2\right )^5 (g+h x)^2}-\frac{c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 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 )^4 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac{\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac{\left (42 a^2 f h^4-c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}+\frac{\left (a^2 c^3 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 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 )^5}\\ &=-\frac{a c^2 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \sqrt{a+c x^2}}{16 \left (c g^2+a h^2\right )^5 (g+h x)^2}-\frac{c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 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 )^4 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac{\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac{\left (42 a^2 f h^4-c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}-\frac{\left (a^2 c^3 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 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 )^5}\\ &=-\frac{a c^2 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 d h)\right )\right ) (a h-c g x) \sqrt{a+c x^2}}{16 \left (c g^2+a h^2\right )^5 (g+h x)^2}-\frac{c \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 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 )^4 (g+h x)^4}-\frac{\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{7 h \left (c g^2+a h^2\right ) (g+h x)^7}+\frac{\left (5 c f g^3+c g h (2 e g-9 d h)+7 a h^2 (2 f g-e h)\right ) \left (a+c x^2\right )^{5/2}}{42 h \left (c g^2+a h^2\right )^2 (g+h x)^6}-\frac{\left (42 a^2 f h^4-c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-a c h^2 \left (26 f g^2-h (61 e g-12 d h)\right )\right ) \left (a+c x^2\right )^{5/2}}{210 h \left (c g^2+a h^2\right )^3 (g+h x)^5}-\frac{a^2 c^3 \left (6 c^2 d g^3+a^2 h^2 (8 f g-e h)-a c g \left (f g^2-h (8 e g-3 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 )^{11/2}}\\ \end{align*}

Mathematica [A]  time = 2.63601, size = 863, normalized size = 1.62 \[ \frac{a^2 \left (6 c^2 d g^3-a c \left (f g^2+h (3 d h-8 e g)\right ) g+a^2 h^2 (8 f g-e h)\right ) \log (g+h x) c^3}{16 \left (c g^2+a h^2\right )^{11/2}}-\frac{a^2 \left (6 c^2 d g^3-a c \left (f g^2+h (3 d h-8 e g)\right ) g+a^2 h^2 (8 f g-e h)\right ) \log \left (a h-c g x+\sqrt{c g^2+a h^2} \sqrt{c x^2+a}\right ) c^3}{16 \left (c g^2+a h^2\right )^{11/2}}-\frac{\sqrt{c x^2+a} \left (240 \left (f g^2+h (d h-e g)\right ) \left (c g^2+a h^2\right )^6-40 \left (29 c f g^3+c h (15 d h-22 e g) g-7 a h^2 (e h-2 f g)\right ) (g+h x) \left (c g^2+a h^2\right )^5+8 \left (42 a^2 f h^4+a c \left (314 f g^2+h (48 d h-139 e g)\right ) h^2+c^2 \left (275 f g^4+h (51 d h-142 e g) g^2\right )\right ) (g+h x)^2 \left (c g^2+a h^2\right )^4-2 c \left (7 a^2 (136 f g-35 e h) h^4+a c g \left (1979 f g^2+h (33 d h-544 e g)\right ) h^2+2 c^2 \left (500 f g^5+h (3 d h-136 e g) g^3\right )\right ) (g+h x)^3 \left (c g^2+a h^2\right )^3+2 c \left (336 a^3 f h^6+3 a^2 c \left (400 f g^2+h (8 d h-29 e g)\right ) h^4+a c^2 g^2 \left (1201 f g^2-h (32 e g+45 d h)\right ) h^2+c^3 \left (400 f g^6-2 g^4 h (4 e g+3 d h)\right )\right ) (g+h x)^4 \left (c g^2+a h^2\right )^2-c^2 \left (21 a^3 (24 f g-5 e h) h^6+3 a^2 c g \left (109 f g^2+h (94 e g-73 d h)\right ) h^4+2 a c^2 g^3 \left (89 f g^2+44 e h g+54 d h^2\right ) h^2+4 c^3 \left (10 f g^7+h (4 e g+3 d h) g^5\right )\right ) (g+h x)^5 \left (c g^2+a h^2\right )-c^2 \left (-336 a^4 f h^8+3 a^3 c \left (312 f g^2+h (32 d h-221 e g)\right ) h^6+a^2 c^2 g^2 \left (505 f g^2+h (370 e g-741 d h)\right ) h^4+2 a c^3 g^4 \left (109 f g^2+52 e h g+60 d h^2\right ) h^2+4 c^4 \left (10 f g^8+h (4 e g+3 d h) g^6\right )\right ) (g+h x)^6\right )}{1680 \left (a h^3+c g^2 h\right )^5 (g+h x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8,x]

[Out]

-(Sqrt[a + c*x^2]*(240*(c*g^2 + a*h^2)^6*(f*g^2 + h*(-(e*g) + d*h)) - 40*(c*g^2 + a*h^2)^5*(29*c*f*g^3 + c*g*h
*(-22*e*g + 15*d*h) - 7*a*h^2*(-2*f*g + e*h))*(g + h*x) + 8*(c*g^2 + a*h^2)^4*(42*a^2*f*h^4 + a*c*h^2*(314*f*g
^2 + h*(-139*e*g + 48*d*h)) + c^2*(275*f*g^4 + g^2*h*(-142*e*g + 51*d*h)))*(g + h*x)^2 - 2*c*(c*g^2 + a*h^2)^3
*(7*a^2*h^4*(136*f*g - 35*e*h) + 2*c^2*(500*f*g^5 + g^3*h*(-136*e*g + 3*d*h)) + a*c*g*h^2*(1979*f*g^2 + h*(-54
4*e*g + 33*d*h)))*(g + h*x)^3 + 2*c*(c*g^2 + a*h^2)^2*(336*a^3*f*h^6 + c^3*(400*f*g^6 - 2*g^4*h*(4*e*g + 3*d*h
)) + 3*a^2*c*h^4*(400*f*g^2 + h*(-29*e*g + 8*d*h)) + a*c^2*g^2*h^2*(1201*f*g^2 - h*(32*e*g + 45*d*h)))*(g + h*
x)^4 - c^2*(c*g^2 + a*h^2)*(21*a^3*h^6*(24*f*g - 5*e*h) + 2*a*c^2*g^3*h^2*(89*f*g^2 + 44*e*g*h + 54*d*h^2) + 3
*a^2*c*g*h^4*(109*f*g^2 + h*(94*e*g - 73*d*h)) + 4*c^3*(10*f*g^7 + g^5*h*(4*e*g + 3*d*h)))*(g + h*x)^5 - c^2*(
-336*a^4*f*h^8 + 2*a*c^3*g^4*h^2*(109*f*g^2 + 52*e*g*h + 60*d*h^2) + a^2*c^2*g^2*h^4*(505*f*g^2 + h*(370*e*g -
741*d*h)) + 4*c^4*(10*f*g^8 + g^6*h*(4*e*g + 3*d*h)) + 3*a^3*c*h^6*(312*f*g^2 + h*(-221*e*g + 32*d*h)))*(g +
h*x)^6))/(1680*(c*g^2*h + a*h^3)^5*(g + h*x)^7) + (a^2*c^3*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e*h) - a*c*g*(f*g^2
+ h*(-8*e*g + 3*d*h)))*Log[g + h*x])/(16*(c*g^2 + a*h^2)^(11/2)) - (a^2*c^3*(6*c^2*d*g^3 + a^2*h^2*(8*f*g - e
*h) - a*c*g*(f*g^2 + h*(-8*e*g + 3*d*h)))*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(16*(c*g^2 +
a*h^2)^(11/2))

________________________________________________________________________________________

Maple [B]  time = 0.307, size = 19093, normalized size = 35.9 \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)^8,x)

[Out]

result too large to display

________________________________________________________________________________________

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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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)^8,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((c*x**2+a)**(3/2)*(f*x**2+e*x+d)/(h*x+g)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.14222, size = 10714, normalized size = 20.14 \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)^8,x, algorithm="giac")

[Out]

-1/8*(6*a^2*c^5*d*g^3 - a^3*c^4*f*g^3 - 3*a^3*c^4*d*g*h^2 + 8*a^4*c^3*f*g*h^2 + 8*a^3*c^4*g^2*h*e - a^4*c^3*h^
3*e)*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c^5*g^10 + 5*a*c^4*g^8*h^2 +
10*a^2*c^3*g^6*h^4 + 10*a^3*c^2*g^4*h^6 + 5*a^4*c*g^2*h^8 + a^5*h^10)*sqrt(-c*g^2 - a*h^2)) - 1/840*(630*(sqr
t(c)*x - sqrt(c*x^2 + a))^13*a^2*c^5*d*g^3*h^12 - 105*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^3*c^4*f*g^3*h^12 - 31
5*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^3*c^4*d*g*h^14 + 840*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^4*c^3*f*g*h^14 +
840*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^3*c^4*g^2*h^13*e - 105*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^4*c^3*h^15*e
- 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^12*c^(15/2)*f*g^10*h^5 - 8400*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a*c^(13/2)
*f*g^8*h^7 - 16800*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^2*c^(11/2)*f*g^6*h^9 + 8190*(sqrt(c)*x - sqrt(c*x^2 + a)
)^12*a^2*c^(11/2)*d*g^4*h^11 - 18165*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^3*c^(9/2)*f*g^4*h^11 - 4095*(sqrt(c)*x
- sqrt(c*x^2 + a))^12*a^3*c^(9/2)*d*g^2*h^13 + 2520*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^4*c^(7/2)*f*g^2*h^13 -
1680*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^5*c^(5/2)*f*h^15 + 10920*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^3*c^(9/2)
*g^3*h^12*e - 1365*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^4*c^(7/2)*g*h^14*e - 5600*(sqrt(c)*x - sqrt(c*x^2 + a))^
11*c^8*f*g^11*h^4 - 28000*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^7*f*g^9*h^6 - 56000*(sqrt(c)*x - sqrt(c*x^2 + a
))^11*a^2*c^6*f*g^7*h^8 + 44940*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^6*d*g^5*h^10 - 63490*(sqrt(c)*x - sqrt(
c*x^2 + a))^11*a^3*c^5*f*g^5*h^10 - 26670*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^5*d*g^3*h^12 + 32620*(sqrt(c)
*x - sqrt(c*x^2 + a))^11*a^4*c^4*f*g^3*h^12 + 2100*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^4*c^4*d*g*h^14 - 11200*(
sqrt(c)*x - sqrt(c*x^2 + a))^11*a^5*c^3*f*g*h^14 - 2240*(sqrt(c)*x - sqrt(c*x^2 + a))^11*c^8*g^10*h^5*e - 1120
0*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^7*g^8*h^7*e - 22400*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^6*g^6*h^9*e
+ 37520*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^5*g^4*h^11*e - 24290*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^4*c^4*g
^2*h^13*e - 1540*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^5*c^3*h^15*e - 11200*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(1
7/2)*f*g^12*h^3 - 3360*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(17/2)*d*g^10*h^5 - 52640*(sqrt(c)*x - sqrt(c*x^2 +
a))^10*a*c^(15/2)*f*g^10*h^5 - 16800*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a*c^(15/2)*d*g^8*h^7 - 95200*(sqrt(c)*x
- sqrt(c*x^2 + a))^10*a^2*c^(13/2)*f*g^8*h^7 + 100380*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(13/2)*d*g^6*h^9
- 100730*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(11/2)*f*g^6*h^9 - 146790*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3
*c^(11/2)*d*g^4*h^11 + 163940*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^4*c^(9/2)*f*g^4*h^11 + 6300*(sqrt(c)*x - sqrt
(c*x^2 + a))^10*a^4*c^(9/2)*d*g^2*h^13 - 56000*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^5*c^(7/2)*f*g^2*h^13 - 3360*
(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^5*c^(7/2)*d*h^15 + 3360*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^6*c^(5/2)*f*h^15
- 4480*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(17/2)*g^11*h^4*e - 22400*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a*c^(15/
2)*g^9*h^6*e - 44800*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(13/2)*g^7*h^8*e + 133840*(sqrt(c)*x - sqrt(c*x^2
+ a))^10*a^3*c^(11/2)*g^5*h^10*e - 106330*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^4*c^(9/2)*g^3*h^12*e + 3220*(sqrt
(c)*x - sqrt(c*x^2 + a))^10*a^5*c^(7/2)*g*h^14*e - 13440*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^9*f*g^13*h^2 - 4032
*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^9*d*g^11*h^4 - 50848*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^8*f*g^11*h^4 - 201
60*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^8*d*g^9*h^6 - 52640*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^7*f*g^9*h^6 +
191016*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^7*d*g^7*h^8 - 9436*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^6*f*g^7
*h^8 - 363216*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^6*d*g^5*h^10 + 439306*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*
c^5*f*g^5*h^10 + 95340*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^5*d*g^3*h^12 - 209965*(sqrt(c)*x - sqrt(c*x^2 + a
))^9*a^5*c^4*f*g^3*h^12 - 9975*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^5*c^4*d*g*h^14 + 32200*(sqrt(c)*x - sqrt(c*x^
2 + a))^9*a^6*c^3*f*g*h^14 - 5376*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^9*g^12*h^3*e - 25984*(sqrt(c)*x - sqrt(c*x
^2 + a))^9*a*c^8*g^10*h^5*e - 49280*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^7*g^8*h^7*e + 263648*(sqrt(c)*x - sq
rt(c*x^2 + a))^9*a^3*c^6*g^6*h^9*e - 332780*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^5*g^4*h^11*e + 49490*(sqrt(c
)*x - sqrt(c*x^2 + a))^9*a^5*c^4*g^2*h^13*e - 1085*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^6*c^3*h^15*e - 8960*(sqrt
(c)*x - sqrt(c*x^2 + a))^8*c^(19/2)*f*g^14*h - 2688*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(19/2)*d*g^12*h^3 - 1523
2*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(17/2)*f*g^12*h^3 - 16800*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(17/2)*d*g
^10*h^5 + 53200*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(15/2)*f*g^10*h^5 + 181104*(sqrt(c)*x - sqrt(c*x^2 + a))
^8*a^2*c^(15/2)*d*g^8*h^7 + 143416*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(13/2)*f*g^8*h^7 - 651924*(sqrt(c)*x
- sqrt(c*x^2 + a))^8*a^3*c^(13/2)*d*g^6*h^9 + 580034*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(11/2)*f*g^6*h^9 +
299460*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(11/2)*d*g^4*h^11 - 568085*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^5*c^
(9/2)*f*g^4*h^11 - 72975*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^5*c^(9/2)*d*g^2*h^13 + 147000*(sqrt(c)*x - sqrt(c*x
^2 + a))^8*a^6*c^(7/2)*f*g^2*h^13 - 3360*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^6*c^(7/2)*d*h^15 - 5040*(sqrt(c)*x
- sqrt(c*x^2 + a))^8*a^7*c^(5/2)*f*h^15 - 3584*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(19/2)*g^13*h^2*e - 9856*(sqr
t(c)*x - sqrt(c*x^2 + a))^8*a*c^(17/2)*g^11*h^4*e + 4480*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(15/2)*g^9*h^6*
e + 344512*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(13/2)*g^7*h^8*e - 613480*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4
*c^(11/2)*g^5*h^10*e + 259210*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^5*c^(9/2)*g^3*h^12*e - 9765*(sqrt(c)*x - sqrt(
c*x^2 + a))^8*a^6*c^(7/2)*g*h^14*e - 2560*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^10*f*g^15 - 768*(sqrt(c)*x - sqrt(
c*x^2 + a))^7*c^10*d*g^13*h^2 + 12928*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^9*f*g^13*h^2 + 384*(sqrt(c)*x - sqrt
(c*x^2 + a))^7*a*c^9*d*g^11*h^4 + 80576*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^8*f*g^11*h^4 + 117984*(sqrt(c)*x
- sqrt(c*x^2 + a))^7*a^2*c^8*d*g^9*h^6 + 101936*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^7*f*g^9*h^6 - 603216*(s
qrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^7*d*g^7*h^8 + 256816*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^6*f*g^7*h^8 + 7
03752*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^6*d*g^5*h^10 - 941332*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^5*f*g^
5*h^10 - 184380*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^5*d*g^3*h^12 + 413280*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^
6*c^4*f*g^3*h^12 + 13440*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^6*c^4*d*g*h^14 - 47040*(sqrt(c)*x - sqrt(c*x^2 + a)
)^7*a^7*c^3*f*g*h^14 - 1024*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^10*g^14*h*e + 4096*(sqrt(c)*x - sqrt(c*x^2 + a))
^7*a*c^9*g^12*h^3*e + 32768*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^8*g^10*h^5*e + 205952*(sqrt(c)*x - sqrt(c*x^
2 + a))^7*a^3*c^7*g^8*h^7*e - 741776*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^6*g^6*h^9*e + 608720*(sqrt(c)*x - s
qrt(c*x^2 + a))^7*a^5*c^5*g^4*h^11*e - 92820*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^6*c^4*g^2*h^13*e + 8960*(sqrt(c
)*x - sqrt(c*x^2 + a))^6*a*c^(19/2)*f*g^14*h + 2688*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(19/2)*d*g^12*h^3 + 15
232*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(17/2)*f*g^12*h^3 + 16800*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(17/
2)*d*g^10*h^5 - 53200*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(15/2)*f*g^10*h^5 - 342384*(sqrt(c)*x - sqrt(c*x^2
+ a))^6*a^3*c^(15/2)*d*g^8*h^7 - 103936*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(13/2)*f*g^8*h^7 + 736344*(sqrt
(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(13/2)*d*g^6*h^9 - 726404*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(11/2)*f*g^6*
h^9 - 488460*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(11/2)*d*g^4*h^11 + 764960*(sqrt(c)*x - sqrt(c*x^2 + a))^6*
a^6*c^(9/2)*f*g^4*h^11 + 33600*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^6*c^(9/2)*d*g^2*h^13 - 168000*(sqrt(c)*x - sq
rt(c*x^2 + a))^6*a^7*c^(7/2)*f*g^2*h^13 - 6720*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^7*c^(7/2)*d*h^15 + 6720*(sqrt
(c)*x - sqrt(c*x^2 + a))^6*a^8*c^(5/2)*f*h^15 + 3584*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(19/2)*g^13*h^2*e + 9
856*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(17/2)*g^11*h^4*e + 8960*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(15/2
)*g^9*h^6*e - 487312*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(13/2)*g^7*h^8*e + 807520*(sqrt(c)*x - sqrt(c*x^2 +
a))^6*a^5*c^(11/2)*g^5*h^10*e - 310660*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^6*c^(9/2)*g^3*h^12*e + 13440*(sqrt(c
)*x - sqrt(c*x^2 + a))^6*a^7*c^(7/2)*g*h^14*e - 13440*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^9*f*g^13*h^2 - 403
2*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^9*d*g^11*h^4 - 50848*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^8*f*g^11*h^
4 - 47040*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^8*d*g^9*h^6 - 50960*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^7*f*
g^9*h^6 + 438816*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^7*d*g^7*h^8 - 99736*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5
*c^6*f*g^7*h^8 - 556416*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^6*d*g^5*h^10 + 728756*(sqrt(c)*x - sqrt(c*x^2 +
a))^5*a^6*c^5*f*g^5*h^10 + 167790*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^5*d*g^3*h^12 - 362915*(sqrt(c)*x - sqr
t(c*x^2 + a))^5*a^7*c^4*f*g^3*h^12 - 10185*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^7*c^4*d*g*h^14 + 38360*(sqrt(c)*x
- sqrt(c*x^2 + a))^5*a^8*c^3*f*g*h^14 - 5376*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^9*g^12*h^3*e - 25984*(sqrt
(c)*x - sqrt(c*x^2 + a))^5*a^3*c^8*g^10*h^5*e - 86240*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^7*g^8*h^7*e + 5744
48*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^6*g^6*h^9*e - 487480*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^5*g^4*h^11
*e + 89740*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^7*c^4*g^2*h^13*e + 1085*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^8*c^3*h
^15*e + 11200*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(17/2)*f*g^12*h^3 + 3360*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a
^3*c^(17/2)*d*g^10*h^5 + 52640*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(15/2)*f*g^10*h^5 + 45360*(sqrt(c)*x - sq
rt(c*x^2 + a))^4*a^4*c^(15/2)*d*g^8*h^7 + 96880*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(13/2)*f*g^8*h^7 - 36472
8*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(13/2)*d*g^6*h^9 + 215908*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(11/2)
*f*g^6*h^9 + 220710*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(11/2)*d*g^4*h^11 - 406735*(sqrt(c)*x - sqrt(c*x^2 +
a))^4*a^7*c^(9/2)*f*g^4*h^11 - 49581*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^7*c^(9/2)*d*g^2*h^13 + 104776*(sqrt(c)
*x - sqrt(c*x^2 + a))^4*a^8*c^(7/2)*f*g^2*h^13 - 1344*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^8*c^(7/2)*d*h^15 - 369
6*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^9*c^(5/2)*f*h^15 + 4480*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(17/2)*g^11*
h^4*e + 29120*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(15/2)*g^9*h^6*e + 119056*(sqrt(c)*x - sqrt(c*x^2 + a))^4*
a^5*c^(13/2)*g^7*h^8*e - 390656*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(11/2)*g^5*h^10*e + 179900*(sqrt(c)*x -
sqrt(c*x^2 + a))^4*a^7*c^(9/2)*g^3*h^12*e - 10703*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^8*c^(7/2)*g*h^14*e - 5600*
(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^8*f*g^11*h^4 - 3360*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^8*d*g^9*h^6 -
29680*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^7*f*g^9*h^6 - 32592*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^7*d*g^7*
h^8 - 67088*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^6*f*g^7*h^8 + 172620*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^6
*d*g^5*h^10 - 156170*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^5*f*g^5*h^10 - 62454*(sqrt(c)*x - sqrt(c*x^2 + a))^
3*a^7*c^5*d*g^3*h^12 + 140084*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^8*c^4*f*g^3*h^12 + 5964*(sqrt(c)*x - sqrt(c*x^
2 + a))^3*a^8*c^4*d*g*h^14 - 17024*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^9*c^3*f*g*h^14 - 2240*(sqrt(c)*x - sqrt(c
*x^2 + a))^3*a^4*c^8*g^10*h^5*e - 16576*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^7*g^8*h^7*e - 72464*(sqrt(c)*x -
sqrt(c*x^2 + a))^3*a^6*c^6*g^6*h^9*e + 179200*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^5*g^4*h^11*e - 31402*(sqr
t(c)*x - sqrt(c*x^2 + a))^3*a^8*c^4*g^2*h^13*e + 1540*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^9*c^3*h^15*e + 1680*(s
qrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(15/2)*f*g^10*h^5 + 1008*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(15/2)*d*g^
8*h^7 + 9632*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(13/2)*f*g^8*h^7 + 9996*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6
*c^(13/2)*d*g^6*h^9 + 24094*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(11/2)*f*g^6*h^9 - 54894*(sqrt(c)*x - sqrt(c
*x^2 + a))^2*a^7*c^(11/2)*d*g^4*h^11 + 56924*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^8*c^(9/2)*f*g^4*h^11 + 9156*(sq
rt(c)*x - sqrt(c*x^2 + a))^2*a^8*c^(9/2)*d*g^2*h^13 - 32256*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^9*c^(7/2)*f*g^2*
h^13 - 672*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^9*c^(7/2)*d*h^15 + 672*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^10*c^(5/
2)*f*h^15 + 1344*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(15/2)*g^9*h^6*e + 8624*(sqrt(c)*x - sqrt(c*x^2 + a))^2
*a^6*c^(13/2)*g^7*h^8*e + 30352*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(11/2)*g^5*h^10*e - 47362*(sqrt(c)*x - s
qrt(c*x^2 + a))^2*a^8*c^(9/2)*g^3*h^12*e + 3276*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^9*c^(7/2)*g*h^14*e - 560*(sq
rt(c)*x - sqrt(c*x^2 + a))*a^6*c^7*f*g^9*h^6 - 168*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^7*d*g^7*h^8 - 3052*(sqr
t(c)*x - sqrt(c*x^2 + a))*a^7*c^6*f*g^7*h^8 - 1680*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^6*d*g^5*h^10 - 7070*(sq
rt(c)*x - sqrt(c*x^2 + a))*a^8*c^5*f*g^5*h^10 + 9744*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^5*d*g^3*h^12 - 12999*
(sqrt(c)*x - sqrt(c*x^2 + a))*a^9*c^4*f*g^3*h^12 - 1029*(sqrt(c)*x - sqrt(c*x^2 + a))*a^9*c^4*d*g*h^14 + 3864*
(sqrt(c)*x - sqrt(c*x^2 + a))*a^10*c^3*f*g*h^14 - 224*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^7*g^8*h^7*e - 1456*(
sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^6*g^6*h^9*e - 5180*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^5*g^4*h^11*e + 8442*
(sqrt(c)*x - sqrt(c*x^2 + a))*a^9*c^4*g^2*h^13*e + 105*(sqrt(c)*x - sqrt(c*x^2 + a))*a^10*c^3*h^15*e + 40*a^7*
c^(13/2)*f*g^8*h^7 + 12*a^7*c^(13/2)*d*g^6*h^9 + 218*a^8*c^(11/2)*f*g^6*h^9 + 120*a^8*c^(11/2)*d*g^4*h^11 + 50
5*a^9*c^(9/2)*f*g^4*h^11 - 741*a^9*c^(9/2)*d*g^2*h^13 + 936*a^10*c^(7/2)*f*g^2*h^13 + 96*a^10*c^(7/2)*d*h^15 -
336*a^11*c^(5/2)*f*h^15 + 16*a^7*c^(13/2)*g^7*h^8*e + 104*a^8*c^(11/2)*g^5*h^10*e + 370*a^9*c^(9/2)*g^3*h^12*
e - 663*a^10*c^(7/2)*g*h^14*e)/((c^5*g^10*h^6 + 5*a*c^4*g^8*h^8 + 10*a^2*c^3*g^6*h^10 + 10*a^3*c^2*g^4*h^12 +
5*a^4*c*g^2*h^14 + a^5*h^16)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*h + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*g -
a*h)^7)