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

Optimal. Leaf size=833 \[ -\frac{\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{3 h \left (c g^2-b h g+a h^2\right ) (g+h x)^3}-\frac{\left (2 c g \left (-\frac{10 f g^2}{h}+4 e g-d h\right )-6 a h (3 f g-e h)+b \left (17 f g^2-h (5 e g+d h)\right )+2 h \left (-\frac{5 c f g^2}{h}+2 c e g+3 b f g-2 c d h-3 a f h\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)^2}-\frac{\left (8 c^2 \left (10 f g^2-h (4 e g-d h)\right ) g^2-2 c h \left (3 b g \left (18 f g^2-6 e h g+d h^2\right )-2 a h \left (23 f g^2-8 e h g+2 d h^2\right )\right )+h^2 \left (\left (29 f g^2-h (5 e g+d h)\right ) b^2-6 a h (7 f g-e h) b+12 a^2 f h^2\right )+2 h \left (2 g \left (10 f g^2-h (4 e g-d h)\right ) c^2+h \left (6 a h (3 f g-e h)-b \left (22 f g^2-7 e h g+d h^2\right )\right ) c+3 b f h^2 (b g-a h)\right ) x\right ) \sqrt{c x^2+b x+a}}{8 h^5 \left (c g^2-b h g+a h^2\right ) (g+h x)}+\frac{\left (8 \left (10 f g^2-h (4 e g-d h)\right ) c^2-12 h (4 b f g-b e h-a f h) c+3 b^2 f h^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{8 \sqrt{c} h^6}-\frac{\left (16 c^3 \left (10 f g^2-h (4 e g-d h)\right ) g^3-24 c^2 h \left (b g \left (14 f g^2-5 e h g+d h^2\right )-a h \left (11 f g^2-4 e h g+d h^2\right )\right ) g-b h^3 \left (\left (35 f g^2-5 e h g-d h^2\right ) b^2-6 a h (10 f g-e h) b+24 a^2 f h^2\right )+6 c h^2 \left (g \left (35 f g^2-10 e h g+d h^2\right ) b^2-2 a h \left (25 f g^2-7 e h g+d h^2\right ) b+4 a^2 h^2 (4 f g-e h)\right )\right ) \tanh ^{-1}\left (\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right )}{16 h^6 \left (c g^2-b h g+a h^2\right )^{3/2}} \]

[Out]

-((8*c^2*g^2*(10*f*g^2 - h*(4*e*g - d*h)) - 2*c*h*(3*b*g*(18*f*g^2 - 6*e*g*h + d*h^2) - 2*a*h*(23*f*g^2 - 8*e*
g*h + 2*d*h^2)) + h^2*(12*a^2*f*h^2 - 6*a*b*h*(7*f*g - e*h) + b^2*(29*f*g^2 - h*(5*e*g + d*h))) + 2*h*(3*b*f*h
^2*(b*g - a*h) + 2*c^2*g*(10*f*g^2 - h*(4*e*g - d*h)) + c*h*(6*a*h*(3*f*g - e*h) - b*(22*f*g^2 - 7*e*g*h + d*h
^2)))*x)*Sqrt[a + b*x + c*x^2])/(8*h^5*(c*g^2 - b*g*h + a*h^2)*(g + h*x)) - ((2*c*g*(4*e*g - (10*f*g^2)/h - d*
h) - 6*a*h*(3*f*g - e*h) + b*(17*f*g^2 - h*(5*e*g + d*h)) + 2*h*(2*c*e*g + 3*b*f*g - (5*c*f*g^2)/h - 2*c*d*h -
3*a*f*h)*x)*(a + b*x + c*x^2)^(3/2))/(12*h^2*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) - ((f*g^2 - h*(e*g - d*h))*
(a + b*x + c*x^2)^(5/2))/(3*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^3) + ((3*b^2*f*h^2 - 12*c*h*(4*b*f*g - b*e*h -
a*f*h) + 8*c^2*(10*f*g^2 - h*(4*e*g - d*h)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[
c]*h^6) - ((16*c^3*g^3*(10*f*g^2 - h*(4*e*g - d*h)) - b*h^3*(24*a^2*f*h^2 - 6*a*b*h*(10*f*g - e*h) + b^2*(35*f
*g^2 - 5*e*g*h - d*h^2)) + 6*c*h^2*(4*a^2*h^2*(4*f*g - e*h) + b^2*g*(35*f*g^2 - 10*e*g*h + d*h^2) - 2*a*b*h*(2
5*f*g^2 - 7*e*g*h + d*h^2)) - 24*c^2*g*h*(b*g*(14*f*g^2 - 5*e*g*h + d*h^2) - a*h*(11*f*g^2 - 4*e*g*h + d*h^2))
)*ArcTanh[(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(16*h^6*(c*g
^2 - b*g*h + a*h^2)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 2.26438, antiderivative size = 829, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1650, 812, 843, 621, 206, 724} \[ -\frac{\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{3 h \left (c g^2-b h g+a h^2\right ) (g+h x)^3}-\frac{\left (17 b f g^2+2 c \left (-\frac{10 f g^2}{h}+4 e g-d h\right ) g-b h (5 e g+d h)-6 a h (3 f g-e h)+2 h \left (-\frac{5 c f g^2}{h}+2 c e g+3 b f g-2 c d h-3 a f h\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{12 h^2 \left (c g^2-b h g+a h^2\right ) (g+h x)^2}-\frac{\left (12 a^2 f h^3-6 a b (7 f g-e h) h^2+4 a c \left (23 f g^2-2 h (4 e g-d h)\right ) h+b^2 \left (29 f g^2-h (5 e g+d h)\right ) h-8 c^2 g^2 \left (-\frac{10 f g^2}{h}+4 e g-d h\right )-6 b c g \left (18 f g^2-h (6 e g-d h)\right )+2 \left (2 \left (10 f g^3-g h (4 e g-d h)\right ) c^2-h \left (22 b f g^2-b h (7 e g-d h)-6 a h (3 f g-e h)\right ) c+3 b f h^2 (b g-a h)\right ) x\right ) \sqrt{c x^2+b x+a}}{8 h^4 \left (c g^2-b h g+a h^2\right ) (g+h x)}+\frac{\left (8 \left (10 f g^2-h (4 e g-d h)\right ) c^2-12 h (4 b f g-b e h-a f h) c+3 b^2 f h^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{8 \sqrt{c} h^6}-\frac{\left (16 \left (10 f g^5-g^3 h (4 e g-d h)\right ) c^3-24 g h \left (b g \left (14 f g^2-5 e h g+d h^2\right )-a h \left (11 f g^2-4 e h g+d h^2\right )\right ) c^2+6 h^2 \left (g \left (35 f g^2-10 e h g+d h^2\right ) b^2-2 a h \left (25 f g^2-7 e h g+d h^2\right ) b+4 a^2 h^2 (4 f g-e h)\right ) c-b h^3 \left (\left (35 f g^2-5 e h g-d h^2\right ) b^2-6 a h (10 f g-e h) b+24 a^2 f h^2\right )\right ) \tanh ^{-1}\left (\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b h g+a h^2} \sqrt{c x^2+b x+a}}\right )}{16 h^6 \left (c g^2-b h g+a h^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((12*a^2*f*h^3 - 8*c^2*g^2*(4*e*g - (10*f*g^2)/h - d*h) - 6*a*b*h^2*(7*f*g - e*h) + 4*a*c*h*(23*f*g^2 - 2*h*(
4*e*g - d*h)) - 6*b*c*g*(18*f*g^2 - h*(6*e*g - d*h)) + b^2*h*(29*f*g^2 - h*(5*e*g + d*h)) + 2*(3*b*f*h^2*(b*g
- a*h) + 2*c^2*(10*f*g^3 - g*h*(4*e*g - d*h)) - c*h*(22*b*f*g^2 - b*h*(7*e*g - d*h) - 6*a*h*(3*f*g - e*h)))*x)
*Sqrt[a + b*x + c*x^2])/(8*h^4*(c*g^2 - b*g*h + a*h^2)*(g + h*x)) - ((17*b*f*g^2 + 2*c*g*(4*e*g - (10*f*g^2)/h
- d*h) - b*h*(5*e*g + d*h) - 6*a*h*(3*f*g - e*h) + 2*h*(2*c*e*g + 3*b*f*g - (5*c*f*g^2)/h - 2*c*d*h - 3*a*f*h
)*x)*(a + b*x + c*x^2)^(3/2))/(12*h^2*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) - ((f*g^2 - h*(e*g - d*h))*(a + b*x
+ c*x^2)^(5/2))/(3*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^3) + ((3*b^2*f*h^2 - 12*c*h*(4*b*f*g - b*e*h - a*f*h)
+ 8*c^2*(10*f*g^2 - h*(4*e*g - d*h)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*h^6)
- ((16*c^3*(10*f*g^5 - g^3*h*(4*e*g - d*h)) - b*h^3*(24*a^2*f*h^2 - 6*a*b*h*(10*f*g - e*h) + b^2*(35*f*g^2 - 5
*e*g*h - d*h^2)) + 6*c*h^2*(4*a^2*h^2*(4*f*g - e*h) + b^2*g*(35*f*g^2 - 10*e*g*h + d*h^2) - 2*a*b*h*(25*f*g^2
- 7*e*g*h + d*h^2)) - 24*c^2*g*h*(b*g*(14*f*g^2 - 5*e*g*h + d*h^2) - a*h*(11*f*g^2 - 4*e*g*h + d*h^2)))*ArcTan
h[(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(16*h^6*(c*g^2 - b*g
*h + a*h^2)^(3/2))

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx &=-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}-\frac{\int \frac{\left (\frac{1}{2} \left (-6 c d g+5 b e g+6 a f g-\frac{5 b f g^2}{h}+b d h-6 a e h\right )+\left (2 c e g+3 b f g-\frac{5 c f g^2}{h}-2 c d h-3 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(g+h x)^3} \, dx}{3 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac{\left (17 b f g^2+2 c g \left (4 e g-\frac{10 f g^2}{h}-d h\right )-b h (5 e g+d h)-6 a h (3 f g-e h)+2 h \left (2 c e g+3 b f g-\frac{5 c f g^2}{h}-2 c d h-3 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}+\frac{\int \frac{\left (b h \left (6 c d g-5 b e g-6 a f g+\frac{5 b f g^2}{h}-b d h+6 a e h\right )+(4 b g-4 a h) \left (2 c e g+3 b f g-\frac{5 c f g^2}{h}-2 c d h-3 a f h\right )-\frac{2 \left (3 b f h^2 (b g-a h)+2 c^2 \left (10 f g^3-g h (4 e g-d h)\right )-c h \left (22 b f g^2-b h (7 e g-d h)-6 a h (3 f g-e h)\right )\right ) x}{h}\right ) \sqrt{a+b x+c x^2}}{(g+h x)^2} \, dx}{8 h^2 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac{\left (12 a^2 f h^3-8 c^2 g^2 \left (4 e g-\frac{10 f g^2}{h}-d h\right )-6 a b h^2 (7 f g-e h)+4 a c h \left (23 f g^2-2 h (4 e g-d h)\right )-6 b c g \left (18 f g^2-h (6 e g-d h)\right )+b^2 h \left (29 f g^2-h (5 e g+d h)\right )+2 \left (3 b f h^2 (b g-a h)+2 c^2 \left (10 f g^3-g h (4 e g-d h)\right )-c h \left (22 b f g^2-b h (7 e g-d h)-6 a h (3 f g-e h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{8 h^4 \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac{\left (17 b f g^2+2 c g \left (4 e g-\frac{10 f g^2}{h}-d h\right )-b h (5 e g+d h)-6 a h (3 f g-e h)+2 h \left (2 c e g+3 b f g-\frac{5 c f g^2}{h}-2 c d h-3 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}-\frac{\int \frac{-\frac{b^3 h^2 \left (29 f g^2-h (5 e g+d h)\right )-8 a c h \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right )+4 b \left (6 a^2 f h^4+2 c^2 \left (10 f g^4-g^2 h (4 e g-d h)\right )+3 a c h^2 \left (15 f g^2-h (5 e g-d h)\right )\right )-6 b^2 \left (a h^3 (9 f g-e h)+c g h \left (18 f g^2-h (6 e g-d h)\right )\right )}{h}-\frac{2 \left (c g^2-b g h+a h^2\right ) \left (3 b^2 f h^2-12 c h (4 b f g-b e h-a f h)+8 c^2 \left (10 f g^2-h (4 e g-d h)\right )\right ) x}{h}}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{16 h^4 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac{\left (12 a^2 f h^3-8 c^2 g^2 \left (4 e g-\frac{10 f g^2}{h}-d h\right )-6 a b h^2 (7 f g-e h)+4 a c h \left (23 f g^2-2 h (4 e g-d h)\right )-6 b c g \left (18 f g^2-h (6 e g-d h)\right )+b^2 h \left (29 f g^2-h (5 e g+d h)\right )+2 \left (3 b f h^2 (b g-a h)+2 c^2 \left (10 f g^3-g h (4 e g-d h)\right )-c h \left (22 b f g^2-b h (7 e g-d h)-6 a h (3 f g-e h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{8 h^4 \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac{\left (17 b f g^2+2 c g \left (4 e g-\frac{10 f g^2}{h}-d h\right )-b h (5 e g+d h)-6 a h (3 f g-e h)+2 h \left (2 c e g+3 b f g-\frac{5 c f g^2}{h}-2 c d h-3 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}+\frac{\left (3 b^2 f h^2-12 c h (4 b f g-b e h-a f h)+8 c^2 \left (10 f g^2-h (4 e g-d h)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8 h^6}-\frac{\left (16 c^3 \left (10 f g^5-g^3 h (4 e g-d h)\right )-b h^3 \left (24 a^2 f h^2-6 a b h (10 f g-e h)+b^2 \left (35 f g^2-5 e g h-d h^2\right )\right )+6 c h^2 \left (4 a^2 h^2 (4 f g-e h)+b^2 g \left (35 f g^2-10 e g h+d h^2\right )-2 a b h \left (25 f g^2-7 e g h+d h^2\right )\right )-24 c^2 g h \left (b g \left (14 f g^2-5 e g h+d h^2\right )-a h \left (11 f g^2-4 e g h+d h^2\right )\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{16 h^6 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac{\left (12 a^2 f h^3-8 c^2 g^2 \left (4 e g-\frac{10 f g^2}{h}-d h\right )-6 a b h^2 (7 f g-e h)+4 a c h \left (23 f g^2-2 h (4 e g-d h)\right )-6 b c g \left (18 f g^2-h (6 e g-d h)\right )+b^2 h \left (29 f g^2-h (5 e g+d h)\right )+2 \left (3 b f h^2 (b g-a h)+2 c^2 \left (10 f g^3-g h (4 e g-d h)\right )-c h \left (22 b f g^2-b h (7 e g-d h)-6 a h (3 f g-e h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{8 h^4 \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac{\left (17 b f g^2+2 c g \left (4 e g-\frac{10 f g^2}{h}-d h\right )-b h (5 e g+d h)-6 a h (3 f g-e h)+2 h \left (2 c e g+3 b f g-\frac{5 c f g^2}{h}-2 c d h-3 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}+\frac{\left (3 b^2 f h^2-12 c h (4 b f g-b e h-a f h)+8 c^2 \left (10 f g^2-h (4 e g-d h)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{4 h^6}+\frac{\left (16 c^3 \left (10 f g^5-g^3 h (4 e g-d h)\right )-b h^3 \left (24 a^2 f h^2-6 a b h (10 f g-e h)+b^2 \left (35 f g^2-5 e g h-d h^2\right )\right )+6 c h^2 \left (4 a^2 h^2 (4 f g-e h)+b^2 g \left (35 f g^2-10 e g h+d h^2\right )-2 a b h \left (25 f g^2-7 e g h+d h^2\right )\right )-24 c^2 g h \left (b g \left (14 f g^2-5 e g h+d h^2\right )-a h \left (11 f g^2-4 e g h+d h^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c g^2-4 b g h+4 a h^2-x^2} \, dx,x,\frac{-b g+2 a h-(2 c g-b h) x}{\sqrt{a+b x+c x^2}}\right )}{8 h^6 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac{\left (12 a^2 f h^3-8 c^2 g^2 \left (4 e g-\frac{10 f g^2}{h}-d h\right )-6 a b h^2 (7 f g-e h)+4 a c h \left (23 f g^2-2 h (4 e g-d h)\right )-6 b c g \left (18 f g^2-h (6 e g-d h)\right )+b^2 h \left (29 f g^2-h (5 e g+d h)\right )+2 \left (3 b f h^2 (b g-a h)+2 c^2 \left (10 f g^3-g h (4 e g-d h)\right )-c h \left (22 b f g^2-b h (7 e g-d h)-6 a h (3 f g-e h)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{8 h^4 \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac{\left (17 b f g^2+2 c g \left (4 e g-\frac{10 f g^2}{h}-d h\right )-b h (5 e g+d h)-6 a h (3 f g-e h)+2 h \left (2 c e g+3 b f g-\frac{5 c f g^2}{h}-2 c d h-3 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}+\frac{\left (3 b^2 f h^2-12 c h (4 b f g-b e h-a f h)+8 c^2 \left (10 f g^2-h (4 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c} h^6}-\frac{\left (16 c^3 \left (10 f g^5-g^3 h (4 e g-d h)\right )-b h^3 \left (24 a^2 f h^2-6 a b h (10 f g-e h)+b^2 \left (35 f g^2-5 e g h-d h^2\right )\right )+6 c h^2 \left (4 a^2 h^2 (4 f g-e h)+b^2 g \left (35 f g^2-10 e g h+d h^2\right )-2 a b h \left (25 f g^2-7 e g h+d h^2\right )\right )-24 c^2 g h \left (b g \left (14 f g^2-5 e g h+d h^2\right )-a h \left (11 f g^2-4 e g h+d h^2\right )\right )\right ) \tanh ^{-1}\left (\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b g h+a h^2} \sqrt{a+b x+c x^2}}\right )}{16 h^6 \left (c g^2-b g h+a h^2\right )^{3/2}}\\ \end{align*}

Mathematica [B]  time = 6.50128, size = 7806, normalized size = 9.37 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B]  time = 0.299, size = 40092, normalized size = 48.1 \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+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^4,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+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^4,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+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^4,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+b*x+a)**(3/2)*(f*x**2+e*x+d)/(h*x+g)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 25.7356, size = 9881, normalized size = 11.86 \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+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^4,x, algorithm="giac")

[Out]

1/4*sqrt(c*x^2 + b*x + a)*(2*c*f*x/h^4 - (16*c^2*f*g*h^10 - 5*b*c*f*h^11 - 4*c^2*h^11*e)/(c*h^15)) - 1/8*(160*
c^3*f*g^5 - 336*b*c^2*f*g^4*h + 16*c^3*d*g^3*h^2 + 210*b^2*c*f*g^3*h^2 + 264*a*c^2*f*g^3*h^2 - 24*b*c^2*d*g^2*
h^3 - 35*b^3*f*g^2*h^3 - 300*a*b*c*f*g^2*h^3 + 6*b^2*c*d*g*h^4 + 24*a*c^2*d*g*h^4 + 60*a*b^2*f*g*h^4 + 96*a^2*
c*f*g*h^4 + b^3*d*h^5 - 12*a*b*c*d*h^5 - 24*a^2*b*f*h^5 - 64*c^3*g^4*h*e + 120*b*c^2*g^3*h^2*e - 60*b^2*c*g^2*
h^3*e - 96*a*c^2*g^2*h^3*e + 5*b^3*g*h^4*e + 84*a*b*c*g*h^4*e - 6*a*b^2*h^5*e - 24*a^2*c*h^5*e)*arctan(-((sqrt
(c)*x - sqrt(c*x^2 + b*x + a))*h + sqrt(c)*g)/sqrt(-c*g^2 + b*g*h - a*h^2))/((c*g^2*h^6 - b*g*h^7 + a*h^8)*sqr
t(-c*g^2 + b*g*h - a*h^2)) - 1/24*(480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^(7/2)*f*g^5*h^2 - 912*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^5*b*c^(5/2)*f*g^4*h^3 + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^(7/2)*d*g^3*h^4
+ 522*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^(3/2)*f*g^3*h^4 + 552*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5
*a*c^(5/2)*f*g^3*h^4 - 216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^(5/2)*d*g^2*h^5 - 87*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^5*b^3*sqrt(c)*f*g^2*h^5 - 540*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c^(3/2)*f*g^2*h^5 + 78
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^(3/2)*d*g*h^6 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^(5/
2)*d*g*h^6 + 108*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*sqrt(c)*f*g*h^6 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^5*a^2*c^(3/2)*f*g*h^6 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*sqrt(c)*d*h^7 - 60*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^5*a*b*c^(3/2)*d*h^7 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b*sqrt(c)*f*h^7 - 288*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^(7/2)*g^4*h^3*e + 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^(5/2)*
g^3*h^4*e - 252*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^(3/2)*g^2*h^5*e - 288*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^5*a*c^(5/2)*g^2*h^5*e + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*sqrt(c)*g*h^6*e + 228*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*a*b*c^(3/2)*g*h^6*e - 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*sqrt(c)*h^7*e
- 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^(3/2)*h^7*e + 1680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^4*
f*g^6*h - 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^3*f*g^5*h^2 + 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^4*c^4*d*g^4*h^3 + 1362*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^2*f*g^4*h^3 + 1464*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^4*a*c^3*f*g^4*h^3 - 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^3*d*g^3*h^4 - 147*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*b^3*c*f*g^3*h^4 - 876*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^2*f*g^3*h^4 + 54*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^2*d*g^2*h^5 + 216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^3*d*g^2*
h^5 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c*f*g^2*h^5 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a
^2*c^2*f*g^2*h^5 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*c*d*g*h^6 + 84*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^4*a*b*c^2*d*g*h^6 + 216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b*c*f*g*h^6 - 48*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^4*a*b^2*c*d*h^7 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^2*d*h^7 - 48*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^4*a^3*c*f*h^7 - 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^4*g^5*h^2*e + 1464*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*b*c^3*g^4*h^3*e - 540*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^2*g^3*h^4*e - 672*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^3*g^3*h^4*e + 21*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*c*g^2*h^5*
e + 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^2*g^2*h^5*e + 90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b
^2*c*g*h^6*e + 168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^2*g*h^6*e - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^4*a^2*b*c*h^7*e + 1504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^(9/2)*f*g^7 - 1072*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*b*c^(7/2)*f*g^6*h + 352*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^(9/2)*d*g^5*h^2 - 1308*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^3*b^2*c^(5/2)*f*g^5*h^2 - 656*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^(7/2)*f*g^5
*h^2 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^(7/2)*d*g^4*h^3 + 1042*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^3*b^3*c^(3/2)*f*g^4*h^3 + 4056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^(5/2)*f*g^4*h^3 - 420*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^3*b^2*c^(5/2)*d*g^3*h^4 - 272*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^(7/2)*d*g^3*h^
4 - 136*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*sqrt(c)*f*g^3*h^4 - 2712*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^3*a*b^2*c^(3/2)*f*g^3*h^4 - 2208*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^(5/2)*f*g^3*h^4 + 106*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*b^3*c^(3/2)*d*g^2*h^5 + 840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^(5/2)*d*g^
2*h^5 + 328*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*sqrt(c)*f*g^2*h^5 + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*a^2*b*c^(3/2)*f*g^2*h^5 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*sqrt(c)*d*g*h^6 - 144*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c^(3/2)*d*g*h^6 - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^(5/2)*d*g
*h^6 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*sqrt(c)*f*g*h^6 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^3*a^3*c^(3/2)*f*g*h^6 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*sqrt(c)*d*h^7 + 48*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^3*a^3*b*sqrt(c)*f*h^7 - 832*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^(9/2)*g^6*h*e + 400*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^(7/2)*g^5*h^2*e + 840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^(5/2
)*g^4*h^3*e + 512*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^(7/2)*g^4*h^3*e - 478*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^3*b^3*c^(3/2)*g^3*h^4*e - 2232*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^(5/2)*g^3*h^4*e + 40*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*sqrt(c)*g^2*h^5*e + 1092*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c^(3/
2)*g^2*h^5*e + 1104*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^(5/2)*g^2*h^5*e - 88*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^3*a*b^3*sqrt(c)*g*h^6*e - 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c^(3/2)*g*h^6*e + 48*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*sqrt(c)*h^7*e + 2256*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^4*f*g
^7 - 3420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^3*f*g^6*h - 2832*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a
*c^4*f*g^6*h + 528*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^4*d*g^5*h^2 + 1218*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*b^3*c^2*f*g^5*h^2 + 5976*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^3*f*g^5*h^2 - 516*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*b^2*c^3*d*g^4*h^3 - 624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^4*d*g^4*h^3 - 24*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*c*f*g^4*h^3 - 1944*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^2*f*g^4
*h^3 - 2208*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^3*f*g^4*h^3 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
b^3*c^2*d*g^3*h^4 + 840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^3*d*g^3*h^4 - 264*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*a*b^3*c*f*g^3*h^4 + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^2*f*g^3*h^4 + 24*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^2*b^4*c*d*g^2*h^5 + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^2*d*g^2*h^5 -
288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^3*d*g^2*h^5 + 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b
^2*c*f*g^2*h^5 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^2*f*g^2*h^5 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^2*a*b^3*c*d*g*h^6 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^2*d*g*h^6 - 528*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^2*a^3*b*c*f*g*h^6 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^2*d*h^7 + 96*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*a^4*c*f*h^7 - 1248*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^4*g^6*h*e + 1656*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^3*g^5*h^2*e + 1536*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^4*g^5*h^2
*e - 414*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^2*g^4*h^3*e - 2760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
a*b*c^3*g^4*h^3*e - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*c*g^3*h^4*e + 420*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^2*a*b^2*c^2*g^3*h^4*e + 912*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^3*g^3*h^4*e + 168*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*a*b^3*c*g^2*h^5*e + 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^2*g^2*h^5*e -
288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c*g*h^6*e - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c
^2*g*h^6*e + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b*c*h^7*e + 1128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*b^2*c^(7/2)*f*g^7 - 1776*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c^(5/2)*f*g^6*h - 2832*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))*a*b*c^(7/2)*f*g^6*h + 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c^(7/2)*d*g^5*h^2 + 720*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c^(3/2)*f*g^5*h^2 + 5580*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c^(5/2)
*f*g^5*h^2 + 1776*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c^(7/2)*f*g^5*h^2 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))*b^3*c^(5/2)*d*g^4*h^3 - 624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^(7/2)*d*g^4*h^3 - 57*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*b^5*sqrt(c)*f*g^4*h^3 - 2514*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c^(3/2)*f*g^4*
h^3 - 5688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^(5/2)*f*g^4*h^3 + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*b^4*c^(3/2)*d*g^3*h^4 + 852*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c^(5/2)*d*g^3*h^4 + 384*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*a^2*c^(7/2)*d*g^3*h^4 + 198*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*sqrt(c)*f*g^3*h^4
+ 3078*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c^(3/2)*f*g^3*h^4 + 1848*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*a^3*c^(5/2)*f*g^3*h^4 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*sqrt(c)*d*g^2*h^5 - 90*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*a*b^3*c^(3/2)*d*g^2*h^5 - 864*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^(5/2)*d*g^2*h^5 - 2
49*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*sqrt(c)*f*g^2*h^5 - 1476*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^
3*b*c^(3/2)*f*g^2*h^5 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*sqrt(c)*d*g*h^6 + 90*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*a^2*b^2*c^(3/2)*d*g*h^6 + 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c^(5/2)*d*g*h^6 + 132*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^2*sqrt(c)*f*g*h^6 + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*c^(3/2)
*f*g*h^6 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*sqrt(c)*d*h^7 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*a^3*b*c^(3/2)*d*h^7 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b*sqrt(c)*f*h^7 - 624*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*b^2*c^(7/2)*g^6*h*e + 876*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c^(5/2)*g^5*h^2*e + 1536*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^(7/2)*g^5*h^2*e - 282*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c^(3/2)*g^
4*h^3*e - 2664*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c^(5/2)*g^4*h^3*e - 960*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*a^2*c^(7/2)*g^4*h^3*e + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*sqrt(c)*g^3*h^4*e + 894*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a*b^3*c^(3/2)*g^3*h^4*e + 2640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^(5/2)*g^3*h
^4*e - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*sqrt(c)*g^2*h^5*e - 936*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*a^2*b^2*c^(3/2)*g^2*h^5*e - 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c^(5/2)*g^2*h^5*e + 51*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a^2*b^3*sqrt(c)*g*h^6*e + 300*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c^(3/2)*g*h^6*e
- 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^2*sqrt(c)*h^7*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*c
^(3/2)*h^7*e + 188*b^3*c^3*f*g^7 - 272*b^4*c^2*f*g^6*h - 708*a*b^2*c^3*f*g^6*h + 44*b^3*c^3*d*g^5*h^2 + 87*b^5
*c*f*g^5*h^2 + 1214*a*b^3*c^2*f*g^5*h^2 + 888*a^2*b*c^3*f*g^5*h^2 - 44*b^4*c^2*d*g^4*h^3 - 156*a*b^2*c^3*d*g^4
*h^3 - 426*a*b^4*c*f*g^4*h^3 - 2010*a^2*b^2*c^2*f*g^4*h^3 - 376*a^3*c^3*f*g^4*h^3 + 3*b^5*c*d*g^3*h^4 + 182*a*
b^3*c^2*d*g^3*h^4 + 192*a^2*b*c^3*d*g^3*h^4 + 807*a^2*b^3*c*f*g^3*h^4 + 1468*a^3*b*c^2*f*g^3*h^4 - 6*a*b^4*c*d
*g^2*h^5 - 294*a^2*b^2*c^2*d*g^2*h^5 - 88*a^3*c^3*d*g^2*h^5 - 732*a^3*b^2*c*f*g^2*h^5 - 400*a^4*c^2*f*g^2*h^5
+ 3*a^2*b^3*c*d*g*h^6 + 220*a^3*b*c^2*d*g*h^6 + 312*a^4*b*c*f*g*h^6 - 64*a^4*c^2*d*h^7 - 48*a^5*c*f*h^7 - 104*
b^3*c^3*g^6*h*e + 134*b^4*c^2*g^5*h^2*e + 384*a*b^2*c^3*g^5*h^2*e - 33*b^5*c*g^4*h^3*e - 578*a*b^3*c^2*g^4*h^3
*e - 480*a^2*b*c^3*g^4*h^3*e + 144*a*b^4*c*g^3*h^4*e + 936*a^2*b^2*c^2*g^3*h^4*e + 208*a^3*c^3*g^3*h^4*e - 237
*a^2*b^3*c*g^2*h^5*e - 676*a^3*b*c^2*g^2*h^5*e + 174*a^3*b^2*c*g*h^6*e + 184*a^4*c^2*g*h^6*e - 48*a^4*b*c*h^7*
e)/((c^(3/2)*g^2*h^6 - b*sqrt(c)*g*h^7 + a*sqrt(c)*h^8)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*h + 2*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))*sqrt(c)*g + b*g - a*h)^3) - 1/8*(80*c^2*f*g^2 - 48*b*c*f*g*h + 8*c^2*d*h^2 + 3*b^2*
f*h^2 + 12*a*c*f*h^2 - 32*c^2*g*h*e + 12*b*c*h^2*e)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b)
)/(sqrt(c)*h^6)