\(\int \frac {1}{(d+e x) (f+g x)^3 (a+b x+c x^2)^{3/2}} \, dx\) [885]

Optimal result

Integrand size = 29, antiderivative size = 1064 \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 e^3 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^3 \sqrt {a+b x+c x^2}}+\frac {2 e^2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^3 \left (c f^2-b f g+a g^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^2 \sqrt {a+b x+c x^2}}+\frac {2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt {a+b x+c x^2}}+\frac {g^2 \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)^2}+\frac {e g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^2 (f+g x)}+\frac {g^2 (2 c f-b g) \left (8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right ) \sqrt {a+b x+c x^2}}{4 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^3 (f+g x)}+\frac {e^5 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g)^3}-\frac {3 e g^3 (2 c f-b g) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{2 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^{5/2}}-\frac {e^2 g^3 \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{(e f-d g)^3 \left (c f^2-b f g+a g^2\right )^{3/2}}-\frac {3 g^3 \left (16 c^2 f^2+5 b^2 g^2-4 c g (4 b f+a g)\right ) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{8 (e f-d g) \left (c f^2-b f g+a g^2\right )^{7/2}} \]

[Out]

e^5*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^
(3/2)/(-d*g+e*f)^3-3/2*e*g^3*(-b*g+2*c*f)*arctanh(1/2*(b*f-2*a*g+(-b*g+2*c*f)*x)/(a*g^2-b*f*g+c*f^2)^(1/2)/(c*
x^2+b*x+a)^(1/2))/(-d*g+e*f)^2/(a*g^2-b*f*g+c*f^2)^(5/2)-e^2*g^3*arctanh(1/2*(b*f-2*a*g+(-b*g+2*c*f)*x)/(a*g^2
-b*f*g+c*f^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(-d*g+e*f)^3/(a*g^2-b*f*g+c*f^2)^(3/2)-3/8*g^3*(16*c^2*f^2+5*b^2*g^2-
4*c*g*(a*g+4*b*f))*arctanh(1/2*(b*f-2*a*g+(-b*g+2*c*f)*x)/(a*g^2-b*f*g+c*f^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(-d*g
+e*f)/(a*g^2-b*f*g+c*f^2)^(7/2)-2*e^3*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/
(-d*g+e*f)^3/(c*x^2+b*x+a)^(1/2)+2*e^2*g*(b*c*f-b^2*g+2*a*c*g+c*(-b*g+2*c*f)*x)/(-4*a*c+b^2)/(-d*g+e*f)^3/(a*g
^2-b*f*g+c*f^2)/(c*x^2+b*x+a)^(1/2)+2*g*(b*c*f-b^2*g+2*a*c*g+c*(-b*g+2*c*f)*x)/(-4*a*c+b^2)/(-d*g+e*f)/(a*g^2-
b*f*g+c*f^2)/(g*x+f)^2/(c*x^2+b*x+a)^(1/2)+2*e*g*(b*c*f-b^2*g+2*a*c*g+c*(-b*g+2*c*f)*x)/(-4*a*c+b^2)/(-d*g+e*f
)^2/(a*g^2-b*f*g+c*f^2)/(g*x+f)/(c*x^2+b*x+a)^(1/2)+1/2*g^2*(8*c^2*f^2+5*b^2*g^2-4*c*g*(3*a*g+2*b*f))*(c*x^2+b
*x+a)^(1/2)/(-4*a*c+b^2)/(-d*g+e*f)/(a*g^2-b*f*g+c*f^2)^2/(g*x+f)^2+e*g^2*(4*c^2*f^2+3*b^2*g^2-4*c*g*(2*a*g+b*
f))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(-d*g+e*f)^2/(a*g^2-b*f*g+c*f^2)^2/(g*x+f)+1/4*g^2*(-b*g+2*c*f)*(8*c^2*f^
2+15*b^2*g^2-4*c*g*(13*a*g+2*b*f))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(-d*g+e*f)/(a*g^2-b*f*g+c*f^2)^3/(g*x+f)

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 1064, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {974, 754, 12, 738, 212, 848, 820} \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right ) e^5}{\left (c d^2-b e d+a e^2\right )^{3/2} (e f-d g)^3}-\frac {2 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right ) e^3}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (e f-d g)^3 \sqrt {c x^2+b x+a}}-\frac {g^3 \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e^2}{(e f-d g)^3 \left (c f^2-b g f+a g^2\right )^{3/2}}+\frac {2 g \left (-g b^2+c f b+2 a c g+c (2 c f-b g) x\right ) e^2}{\left (b^2-4 a c\right ) (e f-d g)^3 \left (c f^2-b g f+a g^2\right ) \sqrt {c x^2+b x+a}}-\frac {3 g^3 (2 c f-b g) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e}{2 (e f-d g)^2 \left (c f^2-b g f+a g^2\right )^{5/2}}+\frac {g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt {c x^2+b x+a} e}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b g f+a g^2\right )^2 (f+g x)}+\frac {2 g \left (-g b^2+c f b+2 a c g+c (2 c f-b g) x\right ) e}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b g f+a g^2\right ) (f+g x) \sqrt {c x^2+b x+a}}-\frac {3 g^3 \left (16 c^2 f^2+5 b^2 g^2-4 c g (4 b f+a g)\right ) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right )}{8 (e f-d g) \left (c f^2-b g f+a g^2\right )^{7/2}}+\frac {g^2 (2 c f-b g) \left (8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right ) \sqrt {c x^2+b x+a}}{4 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b g f+a g^2\right )^3 (f+g x)}+\frac {g^2 \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) \sqrt {c x^2+b x+a}}{2 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b g f+a g^2\right )^2 (f+g x)^2}+\frac {2 g \left (-g b^2+c f b+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^2 \sqrt {c x^2+b x+a}} \]

[In]

Int[1/((d + e*x)*(f + g*x)^3*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*e^3*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^3*Sq
rt[a + b*x + c*x^2]) + (2*e^2*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)^3*(c
*f^2 - b*f*g + a*g^2)*Sqrt[a + b*x + c*x^2]) + (2*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a
*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^2*Sqrt[a + b*x + c*x^2]) + (2*e*g*(b*c*f - b^2*g + 2*a*c*g +
c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*(f + g*x)*Sqrt[a + b*x + c*x^2]) + (
g^2*(8*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(2*b*f + 3*a*g))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(e*f - d*g)*(c*f^2
- b*f*g + a*g^2)^2*(f + g*x)^2) + (e*g^2*(4*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(b*f + 2*a*g))*Sqrt[a + b*x + c*x^2])
/((b^2 - 4*a*c)*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)^2*(f + g*x)) + (g^2*(2*c*f - b*g)*(8*c^2*f^2 + 15*b^2*g^
2 - 4*c*g*(2*b*f + 13*a*g))*Sqrt[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^3*(f +
g*x)) + (e^5*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/
((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)^3) - (3*e*g^3*(2*c*f - b*g)*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)
/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(2*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)^(5/2)) - (e^
2*g^3*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/((e*f -
d*g)^3*(c*f^2 - b*f*g + a*g^2)^(3/2)) - (3*g^3*(16*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(4*b*f + a*g))*ArcTanh[(b*f - 2
*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(8*(e*f - d*g)*(c*f^2 - b*f*g
+ a*g^2)^(7/2))

Rule 12

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

Rule 212

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

Rule 738

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]

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(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*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 974

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^3}{(e f-d g)^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {g}{(e f-d g) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}-\frac {e g}{(e f-d g)^2 (f+g x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {e^2 g}{(e f-d g)^3 (f+g x) \left (a+b x+c x^2\right )^{3/2}}\right ) \, dx \\ & = \frac {e^3 \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx}{(e f-d g)^3}-\frac {\left (e^2 g\right ) \int \frac {1}{(f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx}{(e f-d g)^3}-\frac {(e g) \int \frac {1}{(f+g x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{(e f-d g)^2}-\frac {g \int \frac {1}{(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx}{e f-d g} \\ & = -\frac {2 e^3 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^3 \sqrt {a+b x+c x^2}}+\frac {2 e^2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^3 \left (c f^2-b f g+a g^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^2 \sqrt {a+b x+c x^2}}+\frac {2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt {a+b x+c x^2}}-\frac {\left (2 e^3\right ) \int -\frac {\left (b^2-4 a c\right ) e^2}{2 (d+e x) \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^3}+\frac {\left (2 e^2 g\right ) \int -\frac {\left (b^2-4 a c\right ) g^2}{2 (f+g x) \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) (e f-d g)^3 \left (c f^2-b f g+a g^2\right )}+\frac {(2 e g) \int \frac {\frac {1}{2} g \left (2 b c f-3 b^2 g+8 a c g\right )+c g (2 c f-b g) x}{(f+g x)^2 \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right )}+\frac {(2 g) \int \frac {\frac {1}{2} g \left (4 b c f-5 b^2 g+12 a c g\right )+2 c g (2 c f-b g) x}{(f+g x)^3 \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )} \\ & = -\frac {2 e^3 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^3 \sqrt {a+b x+c x^2}}+\frac {2 e^2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^3 \left (c f^2-b f g+a g^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^2 \sqrt {a+b x+c x^2}}+\frac {2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt {a+b x+c x^2}}+\frac {g^2 \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)^2}+\frac {e g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^2 (f+g x)}+\frac {e^5 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{\left (c d^2-b d e+a e^2\right ) (e f-d g)^3}-\frac {\left (3 e g^3 (2 c f-b g)\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{2 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^2}-\frac {g \int \frac {\frac {1}{4} g \left (28 b^2 c f g-80 a c^2 f g-15 b^3 g^2-4 b c \left (2 c f^2-13 a g^2\right )\right )-\frac {1}{2} c g \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) x}{(f+g x)^2 \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^2}-\frac {\left (e^2 g^3\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{(e f-d g)^3 \left (c f^2-b f g+a g^2\right )} \\ & = -\frac {2 e^3 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^3 \sqrt {a+b x+c x^2}}+\frac {2 e^2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^3 \left (c f^2-b f g+a g^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^2 \sqrt {a+b x+c x^2}}+\frac {2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt {a+b x+c x^2}}+\frac {g^2 \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)^2}+\frac {e g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^2 (f+g x)}+\frac {g^2 (2 c f-b g) \left (8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right ) \sqrt {a+b x+c x^2}}{4 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^3 (f+g x)}-\frac {\left (2 e^5\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right ) (e f-d g)^3}+\frac {\left (3 e g^3 (2 c f-b g)\right ) \text {Subst}\left (\int \frac {1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac {-b f+2 a g-(2 c f-b g) x}{\sqrt {a+b x+c x^2}}\right )}{(e f-d g)^2 \left (c f^2-b f g+a g^2\right )^2}+\frac {\left (2 e^2 g^3\right ) \text {Subst}\left (\int \frac {1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac {-b f+2 a g-(2 c f-b g) x}{\sqrt {a+b x+c x^2}}\right )}{(e f-d g)^3 \left (c f^2-b f g+a g^2\right )}-\frac {\left (3 g^3 \left (16 c^2 f^2+5 b^2 g^2-4 c g (4 b f+a g)\right )\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{8 (e f-d g) \left (c f^2-b f g+a g^2\right )^3} \\ & = -\frac {2 e^3 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^3 \sqrt {a+b x+c x^2}}+\frac {2 e^2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^3 \left (c f^2-b f g+a g^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^2 \sqrt {a+b x+c x^2}}+\frac {2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt {a+b x+c x^2}}+\frac {g^2 \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)^2}+\frac {e g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^2 (f+g x)}+\frac {g^2 (2 c f-b g) \left (8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right ) \sqrt {a+b x+c x^2}}{4 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^3 (f+g x)}+\frac {e^5 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g)^3}-\frac {3 e g^3 (2 c f-b g) \tanh ^{-1}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{2 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^{5/2}}-\frac {e^2 g^3 \tanh ^{-1}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{(e f-d g)^3 \left (c f^2-b f g+a g^2\right )^{3/2}}+\frac {\left (3 g^3 \left (16 c^2 f^2+5 b^2 g^2-4 c g (4 b f+a g)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac {-b f+2 a g-(2 c f-b g) x}{\sqrt {a+b x+c x^2}}\right )}{4 (e f-d g) \left (c f^2-b f g+a g^2\right )^3} \\ & = -\frac {2 e^3 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (e f-d g)^3 \sqrt {a+b x+c x^2}}+\frac {2 e^2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^3 \left (c f^2-b f g+a g^2\right ) \sqrt {a+b x+c x^2}}+\frac {2 g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^2 \sqrt {a+b x+c x^2}}+\frac {2 e g \left (b c f-b^2 g+2 a c g+c (2 c f-b g) x\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x) \sqrt {a+b x+c x^2}}+\frac {g^2 \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)^2}+\frac {e g^2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^2 (f+g x)}+\frac {g^2 (2 c f-b g) \left (8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right ) \sqrt {a+b x+c x^2}}{4 \left (b^2-4 a c\right ) (e f-d g) \left (c f^2-b f g+a g^2\right )^3 (f+g x)}+\frac {e^5 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g)^3}-\frac {3 e g^3 (2 c f-b g) \tanh ^{-1}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{2 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^{5/2}}-\frac {e^2 g^3 \tanh ^{-1}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{(e f-d g)^3 \left (c f^2-b f g+a g^2\right )^{3/2}}-\frac {3 g^3 \left (16 c^2 f^2+5 b^2 g^2-4 c g (4 b f+a g)\right ) \tanh ^{-1}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{8 (e f-d g) \left (c f^2-b f g+a g^2\right )^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 15.23 (sec) , antiderivative size = 1013, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 e^3 \left (b^2 e-2 c (a e+c d x)+b c (-d+e x)\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (e f-d g)^3 \sqrt {a+x (b+c x)}}-\frac {2 e^2 g \left (b^2 g-2 c (a g+c f x)+b c (-f+g x)\right )}{\left (b^2-4 a c\right ) (-e f+d g)^3 \left (-c f^2+g (b f-a g)\right ) \sqrt {a+x (b+c x)}}-\frac {2 g \left (b^2 g-2 c (a g+c f x)+b c (-f+g x)\right )}{\left (b^2-4 a c\right ) (-e f+d g) \left (-c f^2+g (b f-a g)\right ) (f+g x)^2 \sqrt {a+x (b+c x)}}+\frac {2 e g \left (b^2 g-2 c (a g+c f x)+b c (-f+g x)\right )}{\left (b^2-4 a c\right ) (e f-d g)^2 \left (-c f^2+g (b f-a g)\right ) (f+g x) \sqrt {a+x (b+c x)}}+\frac {e g^2 \left (\frac {2 \left (4 c^2 f^2+3 b^2 g^2-4 c g (b f+2 a g)\right ) \sqrt {a+x (b+c x)}}{\left (b^2-4 a c\right ) \left (c f^2+g (-b f+a g)\right )^2 (f+g x)}+\frac {3 g (2 c f-b g) \text {arctanh}\left (\frac {-b f+2 a g-2 c f x+b g x}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )}{\left (c f^2+g (-b f+a g)\right )^{5/2}}\right )}{2 (e f-d g)^2}-\frac {g^2 \left (\frac {4 \left (8 c^2 f^2+5 b^2 g^2-4 c g (2 b f+3 a g)\right ) \sqrt {a+x (b+c x)}}{(f+g x)^2}+\frac {2 (2 c f-b g) \left (8 c^2 f^2+15 b^2 g^2-4 c g (2 b f+13 a g)\right ) \sqrt {a+x (b+c x)}}{\left (c f^2+g (-b f+a g)\right ) (f+g x)}+\frac {3 \left (b^2-4 a c\right ) g \left (16 c^2 f^2+5 b^2 g^2-4 c g (4 b f+a g)\right ) \text {arctanh}\left (\frac {-b f+2 a g-2 c f x+b g x}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )}{\left (c f^2+g (-b f+a g)\right )^{3/2}}\right )}{8 \left (b^2-4 a c\right ) (-e f+d g) \left (c f^2+g (-b f+a g)\right )^2}-\frac {e^5 \text {arctanh}\left (\frac {-2 a e+2 c d x+b (d-e x)}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{\left (c d^2+e (-b d+a e)\right )^{3/2} (-e f+d g)^3}-\frac {e^2 g^3 \text {arctanh}\left (\frac {-2 a g+2 c f x+b (f-g x)}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )}{(e f-d g)^3 \left (c f^2+g (-b f+a g)\right )^{3/2}} \]

[In]

Integrate[1/((d + e*x)*(f + g*x)^3*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*e^3*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(e*f - d*g)^3*
Sqrt[a + x*(b + c*x)]) - (2*e^2*g*(b^2*g - 2*c*(a*g + c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(-(e*f) + d*g)^
3*(-(c*f^2) + g*(b*f - a*g))*Sqrt[a + x*(b + c*x)]) - (2*g*(b^2*g - 2*c*(a*g + c*f*x) + b*c*(-f + g*x)))/((b^2
- 4*a*c)*(-(e*f) + d*g)*(-(c*f^2) + g*(b*f - a*g))*(f + g*x)^2*Sqrt[a + x*(b + c*x)]) + (2*e*g*(b^2*g - 2*c*(
a*g + c*f*x) + b*c*(-f + g*x)))/((b^2 - 4*a*c)*(e*f - d*g)^2*(-(c*f^2) + g*(b*f - a*g))*(f + g*x)*Sqrt[a + x*(
b + c*x)]) + (e*g^2*((2*(4*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(b*f + 2*a*g))*Sqrt[a + x*(b + c*x)])/((b^2 - 4*a*c)*(c
*f^2 + g*(-(b*f) + a*g))^2*(f + g*x)) + (3*g*(2*c*f - b*g)*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[
c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(5/2)))/(2*(e*f - d*g)^2) - (g^2
*((4*(8*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(2*b*f + 3*a*g))*Sqrt[a + x*(b + c*x)])/(f + g*x)^2 + (2*(2*c*f - b*g)*(8*
c^2*f^2 + 15*b^2*g^2 - 4*c*g*(2*b*f + 13*a*g))*Sqrt[a + x*(b + c*x)])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)) +
(3*(b^2 - 4*a*c)*g*(16*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(4*b*f + a*g))*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/
(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(c*f^2 + g*(-(b*f) + a*g))^(3/2)))/(8*(b^2 - 4*a*c)
*(-(e*f) + d*g)*(c*f^2 + g*(-(b*f) + a*g))^2) - (e^5*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 +
e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/((c*d^2 + e*(-(b*d) + a*e))^(3/2)*(-(e*f) + d*g)^3) - (e^2*g^3*ArcT
anh[(-2*a*g + 2*c*f*x + b*(f - g*x))/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/((e*f - d*g)^3
*(c*f^2 + g*(-(b*f) + a*g))^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2684\) vs. \(2(1010)=2020\).

Time = 1.14 (sec) , antiderivative size = 2685, normalized size of antiderivative = 2.52

method	result	size
default	\(\text {Expression too large to display}\)	\(2685\)

[In]

int(1/(e*x+d)/(g*x+f)^3/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/g^2/(d*g-e*f)*(-1/2/(a*g^2-b*f*g+c*f^2)*g^2/(x+f/g)^2/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)
/g^2)^(1/2)-5/4*(b*g-2*c*f)*g/(a*g^2-b*f*g+c*f^2)*(-1/(a*g^2-b*f*g+c*f^2)*g^2/(x+f/g)/((x+f/g)^2*c+(b*g-2*c*f)
/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)-3/2*(b*g-2*c*f)*g/(a*g^2-b*f*g+c*f^2)*(1/(a*g^2-b*f*g+c*f^2)*g^2/((x
+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)-(b*g-2*c*f)*g/(a*g^2-b*f*g+c*f^2)*(2*c*(x+f/g)+
(b*g-2*c*f)/g)/(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*c*f)^2/g^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g
+c*f^2)/g^2)^(1/2)-1/(a*g^2-b*f*g+c*f^2)*g^2/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2+(b*
g-2*c*f)/g*(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^
2)^(1/2))/(x+f/g)))-4*c/(a*g^2-b*f*g+c*f^2)*g^2*(2*c*(x+f/g)+(b*g-2*c*f)/g)/(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-
2*c*f)^2/g^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))-3/2*c/(a*g^2-b*f*g+c*f^2)*g^2
*(1/(a*g^2-b*f*g+c*f^2)*g^2/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)-(b*g-2*c*f)*g/(a
*g^2-b*f*g+c*f^2)*(2*c*(x+f/g)+(b*g-2*c*f)/g)/(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*c*f)^2/g^2)/((x+f/g)^2*c+(b*
g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)-1/(a*g^2-b*f*g+c*f^2)*g^2/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln
((2*(a*g^2-b*f*g+c*f^2)/g^2+(b*g-2*c*f)/g*(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)/g
*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))))-e^2/(d*g-e*f)^3*(1/(a*e^2-b*d*e+c*d^2)*e^2/((x+d/e)^2*c+(b
*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/
e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(
x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x
+d/e)))+e^2/(d*g-e*f)^3*(1/(a*g^2-b*f*g+c*f^2)*g^2/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)
^(1/2)-(b*g-2*c*f)*g/(a*g^2-b*f*g+c*f^2)*(2*c*(x+f/g)+(b*g-2*c*f)/g)/(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*c*f)^
2/g^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)-1/(a*g^2-b*f*g+c*f^2)*g^2/((a*g^2-b*f
*g+c*f^2)/g^2)^(1/2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2+(b*g-2*c*f)/g*(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x
+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g)))-1/g*e/(d*g-e*f)^2*(-1/(a*g^2-b*f*g+c
*f^2)*g^2/(x+f/g)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)-3/2*(b*g-2*c*f)*g/(a*g^2-b
*f*g+c*f^2)*(1/(a*g^2-b*f*g+c*f^2)*g^2/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)-(b*g-
2*c*f)*g/(a*g^2-b*f*g+c*f^2)*(2*c*(x+f/g)+(b*g-2*c*f)/g)/(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*c*f)^2/g^2)/((x+f
/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)-1/(a*g^2-b*f*g+c*f^2)*g^2/((a*g^2-b*f*g+c*f^2)/g^
2)^(1/2)*ln((2*(a*g^2-b*f*g+c*f^2)/g^2+(b*g-2*c*f)/g*(x+f/g)+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b
*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g)))-4*c/(a*g^2-b*f*g+c*f^2)*g^2*(2*c*(x+f/g)+(b*g-2*
c*f)/g)/(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*c*f)^2/g^2)/((x+f/g)^2*c+(b*g-2*c*f)/g*(x+f/g)+(a*g^2-b*f*g+c*f^2)
/g^2)^(1/2))

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(g*x+f)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(g*x+f)**3/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )} {\left (g x + f\right )}^{3}} \,d x } \]

[In]

integrate(1/(e*x+d)/(g*x+f)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)*(g*x + f)^3), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14979 vs. \(2 (1010) = 2020\).

Time = 4.14 (sec) , antiderivative size = 14979, normalized size of antiderivative = 14.08 \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)/(g*x+f)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

2*e^5*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c*d^2*e^3*f^
3 - b*d*e^4*f^3 + a*e^5*f^3 - 3*c*d^3*e^2*f^2*g + 3*b*d^2*e^3*f^2*g - 3*a*d*e^4*f^2*g + 3*c*d^4*e*f*g^2 - 3*b*
d^3*e^2*f*g^2 + 3*a*d^2*e^3*f*g^2 - c*d^5*g^3 + b*d^4*e*g^3 - a*d^3*e^2*g^3)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 2
*((2*c^9*d^3*f^9 - 3*b*c^8*d^2*e*f^9 + b^2*c^7*d*e^2*f^9 + 2*a*c^8*d*e^2*f^9 - a*b*c^7*e^3*f^9 - 9*b*c^8*d^3*f
^8*g + 15*b^2*c^7*d^2*e*f^8*g - 6*a*c^8*d^2*e*f^8*g - 6*b^3*c^6*d*e^2*f^8*g - 3*a*b*c^7*d*e^2*f^8*g + 6*a*b^2*
c^6*e^3*f^8*g - 6*a^2*c^7*e^3*f^8*g + 18*b^2*c^7*d^3*f^7*g^2 - 33*b^3*c^6*d^2*e*f^7*g^2 + 24*a*b*c^7*d^2*e*f^7
*g^2 + 15*b^4*c^5*d*e^2*f^7*g^2 - 6*a*b^2*c^6*d*e^2*f^7*g^2 - 15*a*b^3*c^5*e^3*f^7*g^2 + 24*a^2*b*c^6*e^3*f^7*
g^2 - 21*b^3*c^6*d^3*f^6*g^3 + 41*b^4*c^5*d^2*e*f^6*g^3 - 34*a*b^2*c^6*d^2*e*f^6*g^3 - 16*a^2*c^7*d^2*e*f^6*g^
3 - 20*b^5*c^4*d*e^2*f^6*g^3 + 13*a*b^3*c^5*d*e^2*f^6*g^3 + 16*a^2*b*c^6*d*e^2*f^6*g^3 + 20*a*b^4*c^4*e^3*f^6*
g^3 - 34*a^2*b^2*c^5*e^3*f^6*g^3 - 16*a^3*c^6*e^3*f^6*g^3 + 15*b^4*c^5*d^3*f^5*g^4 + 6*a*b^2*c^6*d^3*f^5*g^4 -
12*a^2*c^7*d^3*f^5*g^4 - 30*b^5*c^4*d^2*e*f^5*g^4 + 9*a*b^3*c^5*d^2*e*f^5*g^4 + 66*a^2*b*c^6*d^2*e*f^5*g^4 +
15*b^6*c^3*d*e^2*f^5*g^4 - 48*a^2*b^2*c^5*d*e^2*f^5*g^4 - 12*a^3*c^6*d*e^2*f^5*g^4 - 15*a*b^5*c^3*e^3*f^5*g^4
+ 15*a^2*b^3*c^4*e^3*f^5*g^4 + 54*a^3*b*c^5*e^3*f^5*g^4 - 6*b^5*c^4*d^3*f^4*g^5 - 15*a*b^3*c^5*d^3*f^4*g^5 + 3
0*a^2*b*c^6*d^3*f^4*g^5 + 12*b^6*c^3*d^2*e*f^4*g^5 + 24*a*b^4*c^4*d^2*e*f^4*g^5 - 96*a^2*b^2*c^5*d^2*e*f^4*g^5
- 12*a^3*c^6*d^2*e*f^4*g^5 - 6*b^7*c^2*d*e^2*f^4*g^5 - 15*a*b^5*c^3*d*e^2*f^4*g^5 + 51*a^2*b^3*c^4*d*e^2*f^4*
g^5 + 42*a^3*b*c^5*d*e^2*f^4*g^5 + 6*a*b^6*c^2*e^3*f^4*g^5 + 9*a^2*b^4*c^3*e^3*f^4*g^5 - 66*a^3*b^2*c^4*e^3*f^
4*g^5 - 12*a^4*c^5*e^3*f^4*g^5 + b^6*c^3*d^3*f^3*g^6 + 12*a*b^4*c^4*d^3*f^3*g^6 - 18*a^2*b^2*c^5*d^3*f^3*g^6 -
16*a^3*c^6*d^3*f^3*g^6 - 2*b^7*c^2*d^2*e*f^3*g^6 - 23*a*b^5*c^3*d^2*e*f^3*g^6 + 49*a^2*b^3*c^4*d^2*e*f^3*g^6
+ 48*a^3*b*c^5*d^2*e*f^3*g^6 + b^8*c*d*e^2*f^3*g^6 + 12*a*b^6*c^2*d*e^2*f^3*g^6 - 19*a^2*b^4*c^3*d*e^2*f^3*g^6
- 50*a^3*b^2*c^4*d*e^2*f^3*g^6 - 16*a^4*c^5*d*e^2*f^3*g^6 - a*b^7*c*e^3*f^3*g^6 - 11*a^2*b^5*c^2*e^3*f^3*g^6
+ 31*a^3*b^3*c^3*e^3*f^3*g^6 + 32*a^4*b*c^4*e^3*f^3*g^6 - 3*a*b^5*c^3*d^3*f^2*g^7 - 3*a^2*b^3*c^4*d^3*f^2*g^7
+ 24*a^3*b*c^5*d^3*f^2*g^7 + 6*a*b^6*c^2*d^2*e*f^2*g^7 + 3*a^2*b^4*c^3*d^2*e*f^2*g^7 - 54*a^3*b^2*c^4*d^2*e*f^
2*g^7 - 3*a*b^7*c*d*e^2*f^2*g^7 - 3*a^2*b^5*c^2*d*e^2*f^2*g^7 + 27*a^3*b^3*c^3*d*e^2*f^2*g^7 + 24*a^4*b*c^4*d*
e^2*f^2*g^7 + 3*a^2*b^6*c*e^3*f^2*g^7 - 30*a^4*b^2*c^3*e^3*f^2*g^7 + 3*a^2*b^4*c^3*d^3*f*g^8 - 6*a^3*b^2*c^4*d
^3*f*g^8 - 6*a^4*c^5*d^3*f*g^8 - 6*a^2*b^5*c^2*d^2*e*f*g^8 + 15*a^3*b^3*c^3*d^2*e*f*g^8 + 9*a^4*b*c^4*d^2*e*f*
g^8 + 3*a^2*b^6*c*d*e^2*f*g^8 - 6*a^3*b^4*c^2*d*e^2*f*g^8 - 9*a^4*b^2*c^3*d*e^2*f*g^8 - 6*a^5*c^4*d*e^2*f*g^8
- 3*a^3*b^5*c*e^3*f*g^8 + 9*a^4*b^3*c^2*e^3*f*g^8 + 3*a^5*b*c^3*e^3*f*g^8 - a^3*b^3*c^3*d^3*g^9 + 3*a^4*b*c^4*
d^3*g^9 + 2*a^3*b^4*c^2*d^2*e*g^9 - 7*a^4*b^2*c^3*d^2*e*g^9 + 2*a^5*c^4*d^2*e*g^9 - a^3*b^5*c*d*e^2*g^9 + 3*a^
4*b^3*c^2*d*e^2*g^9 + a^5*b*c^3*d*e^2*g^9 + a^4*b^4*c*e^3*g^9 - 4*a^5*b^2*c^2*e^3*g^9 + 2*a^6*c^3*e^3*g^9)*x/(
b^2*c^8*d^4*f^12 - 4*a*c^9*d^4*f^12 - 2*b^3*c^7*d^3*e*f^12 + 8*a*b*c^8*d^3*e*f^12 + b^4*c^6*d^2*e^2*f^12 - 2*a
*b^2*c^7*d^2*e^2*f^12 - 8*a^2*c^8*d^2*e^2*f^12 - 2*a*b^3*c^6*d*e^3*f^12 + 8*a^2*b*c^7*d*e^3*f^12 + a^2*b^2*c^6
*e^4*f^12 - 4*a^3*c^7*e^4*f^12 - 6*b^3*c^7*d^4*f^11*g + 24*a*b*c^8*d^4*f^11*g + 12*b^4*c^6*d^3*e*f^11*g - 48*a
*b^2*c^7*d^3*e*f^11*g - 6*b^5*c^5*d^2*e^2*f^11*g + 12*a*b^3*c^6*d^2*e^2*f^11*g + 48*a^2*b*c^7*d^2*e^2*f^11*g +
12*a*b^4*c^5*d*e^3*f^11*g - 48*a^2*b^2*c^6*d*e^3*f^11*g - 6*a^2*b^3*c^5*e^4*f^11*g + 24*a^3*b*c^6*e^4*f^11*g
+ 15*b^4*c^6*d^4*f^10*g^2 - 54*a*b^2*c^7*d^4*f^10*g^2 - 24*a^2*c^8*d^4*f^10*g^2 - 30*b^5*c^5*d^3*e*f^10*g^2 +
108*a*b^3*c^6*d^3*e*f^10*g^2 + 48*a^2*b*c^7*d^3*e*f^10*g^2 + 15*b^6*c^4*d^2*e^2*f^10*g^2 - 24*a*b^4*c^5*d^2*e^
2*f^10*g^2 - 132*a^2*b^2*c^6*d^2*e^2*f^10*g^2 - 48*a^3*c^7*d^2*e^2*f^10*g^2 - 30*a*b^5*c^4*d*e^3*f^10*g^2 + 10
8*a^2*b^3*c^5*d*e^3*f^10*g^2 + 48*a^3*b*c^6*d*e^3*f^10*g^2 + 15*a^2*b^4*c^4*e^4*f^10*g^2 - 54*a^3*b^2*c^5*e^4*
f^10*g^2 - 24*a^4*c^6*e^4*f^10*g^2 - 20*b^5*c^5*d^4*f^9*g^3 + 50*a*b^3*c^6*d^4*f^9*g^3 + 120*a^2*b*c^7*d^4*f^9
*g^3 + 40*b^6*c^4*d^3*e*f^9*g^3 - 100*a*b^4*c^5*d^3*e*f^9*g^3 - 240*a^2*b^2*c^6*d^3*e*f^9*g^3 - 20*b^7*c^3*d^2
*e^2*f^9*g^3 + 10*a*b^5*c^4*d^2*e^2*f^9*g^3 + 220*a^2*b^3*c^5*d^2*e^2*f^9*g^3 + 240*a^3*b*c^6*d^2*e^2*f^9*g^3
+ 40*a*b^6*c^3*d*e^3*f^9*g^3 - 100*a^2*b^4*c^4*d*e^3*f^9*g^3 - 240*a^3*b^2*c^5*d*e^3*f^9*g^3 - 20*a^2*b^5*c^3*
e^4*f^9*g^3 + 50*a^3*b^3*c^4*e^4*f^9*g^3 + 120*a^4*b*c^5*e^4*f^9*g^3 + 15*b^6*c^4*d^4*f^8*g^4 - 225*a^2*b^2*c^
6*d^4*f^8*g^4 - 60*a^3*c^7*d^4*f^8*g^4 - 30*b^7*c^3*d^3*e*f^8*g^4 + 450*a^2*b^3*c^5*d^3*e*f^8*g^4 + 120*a^3*b*
c^6*d^3*e*f^8*g^4 + 15*b^8*c^2*d^2*e^2*f^8*g^4 + 30*a*b^6*c^3*d^2*e^2*f^8*g^4 - 225*a^2*b^4*c^4*d^2*e^2*f^8*g^
4 - 510*a^3*b^2*c^5*d^2*e^2*f^8*g^4 - 120*a^4*c^6*d^2*e^2*f^8*g^4 - 30*a*b^7*c^2*d*e^3*f^8*g^4 + 450*a^3*b^3*c
^4*d*e^3*f^8*g^4 + 120*a^4*b*c^5*d*e^3*f^8*g^4 + 15*a^2*b^6*c^2*e^4*f^8*g^4 - 225*a^4*b^2*c^4*e^4*f^8*g^4 - 60
*a^5*c^5*e^4*f^8*g^4 - 6*b^7*c^3*d^4*f^7*g^5 - 36*a*b^5*c^4*d^4*f^7*g^5 + 180*a^2*b^3*c^5*d^4*f^7*g^5 + 240*a^
3*b*c^6*d^4*f^7*g^5 + 12*b^8*c^2*d^3*e*f^7*g^5 + 72*a*b^6*c^3*d^3*e*f^7*g^5 - 360*a^2*b^4*c^4*d^3*e*f^7*g^5 -
480*a^3*b^2*c^5*d^3*e*f^7*g^5 - 6*b^9*c*d^2*e^2*f^7*g^5 - 48*a*b^7*c^2*d^2*e^2*f^7*g^5 + 108*a^2*b^5*c^3*d^2*e
^2*f^7*g^5 + 600*a^3*b^3*c^4*d^2*e^2*f^7*g^5 + 480*a^4*b*c^5*d^2*e^2*f^7*g^5 + 12*a*b^8*c*d*e^3*f^7*g^5 + 72*a
^2*b^6*c^2*d*e^3*f^7*g^5 - 360*a^3*b^4*c^3*d*e^3*f^7*g^5 - 480*a^4*b^2*c^4*d*e^3*f^7*g^5 - 6*a^2*b^7*c*e^4*f^7
*g^5 - 36*a^3*b^5*c^2*e^4*f^7*g^5 + 180*a^4*b^3*c^3*e^4*f^7*g^5 + 240*a^5*b*c^4*e^4*f^7*g^5 + b^8*c^2*d^4*f^6*
g^6 + 26*a*b^6*c^3*d^4*f^6*g^6 - 30*a^2*b^4*c^4*d^4*f^6*g^6 - 340*a^3*b^2*c^5*d^4*f^6*g^6 - 80*a^4*c^6*d^4*f^6
*g^6 - 2*b^9*c*d^3*e*f^6*g^6 - 52*a*b^7*c^2*d^3*e*f^6*g^6 + 60*a^2*b^5*c^3*d^3*e*f^6*g^6 + 680*a^3*b^3*c^4*d^3
*e*f^6*g^6 + 160*a^4*b*c^5*d^3*e*f^6*g^6 + b^10*d^2*e^2*f^6*g^6 + 28*a*b^8*c*d^2*e^2*f^6*g^6 + 22*a^2*b^6*c^2*
d^2*e^2*f^6*g^6 - 400*a^3*b^4*c^3*d^2*e^2*f^6*g^6 - 760*a^4*b^2*c^4*d^2*e^2*f^6*g^6 - 160*a^5*c^5*d^2*e^2*f^6*
g^6 - 2*a*b^9*d*e^3*f^6*g^6 - 52*a^2*b^7*c*d*e^3*f^6*g^6 + 60*a^3*b^5*c^2*d*e^3*f^6*g^6 + 680*a^4*b^3*c^3*d*e^
3*f^6*g^6 + 160*a^5*b*c^4*d*e^3*f^6*g^6 + a^2*b^8*e^4*f^6*g^6 + 26*a^3*b^6*c*e^4*f^6*g^6 - 30*a^4*b^4*c^2*e^4*
f^6*g^6 - 340*a^5*b^2*c^3*e^4*f^6*g^6 - 80*a^6*c^4*e^4*f^6*g^6 - 6*a*b^7*c^2*d^4*f^5*g^7 - 36*a^2*b^5*c^3*d^4*
f^5*g^7 + 180*a^3*b^3*c^4*d^4*f^5*g^7 + 240*a^4*b*c^5*d^4*f^5*g^7 + 12*a*b^8*c*d^3*e*f^5*g^7 + 72*a^2*b^6*c^2*
d^3*e*f^5*g^7 - 360*a^3*b^4*c^3*d^3*e*f^5*g^7 - 480*a^4*b^2*c^4*d^3*e*f^5*g^7 - 6*a*b^9*d^2*e^2*f^5*g^7 - 48*a
^2*b^7*c*d^2*e^2*f^5*g^7 + 108*a^3*b^5*c^2*d^2*e^2*f^5*g^7 + 600*a^4*b^3*c^3*d^2*e^2*f^5*g^7 + 480*a^5*b*c^4*d
^2*e^2*f^5*g^7 + 12*a^2*b^8*d*e^3*f^5*g^7 + 72*a^3*b^6*c*d*e^3*f^5*g^7 - 360*a^4*b^4*c^2*d*e^3*f^5*g^7 - 480*a
^5*b^2*c^3*d*e^3*f^5*g^7 - 6*a^3*b^7*e^4*f^5*g^7 - 36*a^4*b^5*c*e^4*f^5*g^7 + 180*a^5*b^3*c^2*e^4*f^5*g^7 + 24
0*a^6*b*c^3*e^4*f^5*g^7 + 15*a^2*b^6*c^2*d^4*f^4*g^8 - 225*a^4*b^2*c^4*d^4*f^4*g^8 - 60*a^5*c^5*d^4*f^4*g^8 -
30*a^2*b^7*c*d^3*e*f^4*g^8 + 450*a^4*b^3*c^3*d^3*e*f^4*g^8 + 120*a^5*b*c^4*d^3*e*f^4*g^8 + 15*a^2*b^8*d^2*e^2*
f^4*g^8 + 30*a^3*b^6*c*d^2*e^2*f^4*g^8 - 225*a^4*b^4*c^2*d^2*e^2*f^4*g^8 - 510*a^5*b^2*c^3*d^2*e^2*f^4*g^8 - 1
20*a^6*c^4*d^2*e^2*f^4*g^8 - 30*a^3*b^7*d*e^3*f^4*g^8 + 450*a^5*b^3*c^2*d*e^3*f^4*g^8 + 120*a^6*b*c^3*d*e^3*f^
4*g^8 + 15*a^4*b^6*e^4*f^4*g^8 - 225*a^6*b^2*c^2*e^4*f^4*g^8 - 60*a^7*c^3*e^4*f^4*g^8 - 20*a^3*b^5*c^2*d^4*f^3
*g^9 + 50*a^4*b^3*c^3*d^4*f^3*g^9 + 120*a^5*b*c^4*d^4*f^3*g^9 + 40*a^3*b^6*c*d^3*e*f^3*g^9 - 100*a^4*b^4*c^2*d
^3*e*f^3*g^9 - 240*a^5*b^2*c^3*d^3*e*f^3*g^9 - 20*a^3*b^7*d^2*e^2*f^3*g^9 + 10*a^4*b^5*c*d^2*e^2*f^3*g^9 + 220
*a^5*b^3*c^2*d^2*e^2*f^3*g^9 + 240*a^6*b*c^3*d^2*e^2*f^3*g^9 + 40*a^4*b^6*d*e^3*f^3*g^9 - 100*a^5*b^4*c*d*e^3*
f^3*g^9 - 240*a^6*b^2*c^2*d*e^3*f^3*g^9 - 20*a^5*b^5*e^4*f^3*g^9 + 50*a^6*b^3*c*e^4*f^3*g^9 + 120*a^7*b*c^2*e^
4*f^3*g^9 + 15*a^4*b^4*c^2*d^4*f^2*g^10 - 54*a^5*b^2*c^3*d^4*f^2*g^10 - 24*a^6*c^4*d^4*f^2*g^10 - 30*a^4*b^5*c
*d^3*e*f^2*g^10 + 108*a^5*b^3*c^2*d^3*e*f^2*g^10 + 48*a^6*b*c^3*d^3*e*f^2*g^10 + 15*a^4*b^6*d^2*e^2*f^2*g^10 -
24*a^5*b^4*c*d^2*e^2*f^2*g^10 - 132*a^6*b^2*c^2*d^2*e^2*f^2*g^10 - 48*a^7*c^3*d^2*e^2*f^2*g^10 - 30*a^5*b^5*d
*e^3*f^2*g^10 + 108*a^6*b^3*c*d*e^3*f^2*g^10 + 48*a^7*b*c^2*d*e^3*f^2*g^10 + 15*a^6*b^4*e^4*f^2*g^10 - 54*a^7*
b^2*c*e^4*f^2*g^10 - 24*a^8*c^2*e^4*f^2*g^10 - 6*a^5*b^3*c^2*d^4*f*g^11 + 24*a^6*b*c^3*d^4*f*g^11 + 12*a^5*b^4
*c*d^3*e*f*g^11 - 48*a^6*b^2*c^2*d^3*e*f*g^11 - 6*a^5*b^5*d^2*e^2*f*g^11 + 12*a^6*b^3*c*d^2*e^2*f*g^11 + 48*a^
7*b*c^2*d^2*e^2*f*g^11 + 12*a^6*b^4*d*e^3*f*g^11 - 48*a^7*b^2*c*d*e^3*f*g^11 - 6*a^7*b^3*e^4*f*g^11 + 24*a^8*b
*c*e^4*f*g^11 + a^6*b^2*c^2*d^4*g^12 - 4*a^7*c^3*d^4*g^12 - 2*a^6*b^3*c*d^3*e*g^12 + 8*a^7*b*c^2*d^3*e*g^12 +
a^6*b^4*d^2*e^2*g^12 - 2*a^7*b^2*c*d^2*e^2*g^12 - 8*a^8*c^2*d^2*e^2*g^12 - 2*a^7*b^3*d*e^3*g^12 + 8*a^8*b*c*d*
e^3*g^12 + a^8*b^2*e^4*g^12 - 4*a^9*c*e^4*g^12) + (b*c^8*d^3*f^9 - 2*b^2*c^7*d^2*e*f^9 + 2*a*c^8*d^2*e*f^9 + b
^3*c^6*d*e^2*f^9 - a*b*c^7*d*e^2*f^9 - a*b^2*c^6*e^3*f^9 + 2*a^2*c^7*e^3*f^9 - 6*b^2*c^7*d^3*f^8*g + 6*a*c^8*d
^3*f^8*g + 12*b^3*c^6*d^2*e*f^8*g - 21*a*b*c^7*d^2*e*f^8*g - 6*b^4*c^5*d*e^2*f^8*g + 9*a*b^2*c^6*d*e^2*f^8*g +
6*a^2*c^7*d*e^2*f^8*g + 6*a*b^3*c^5*e^3*f^8*g - 15*a^2*b*c^6*e^3*f^8*g + 15*b^3*c^6*d^3*f^7*g^2 - 24*a*b*c^7*
d^3*f^7*g^2 - 30*b^4*c^5*d^2*e*f^7*g^2 + 66*a*b^2*c^6*d^2*e*f^7*g^2 + 15*b^5*c^4*d*e^2*f^7*g^2 - 27*a*b^3*c^5*
d*e^2*f^7*g^2 - 24*a^2*b*c^6*d*e^2*f^7*g^2 - 15*a*b^4*c^4*e^3*f^7*g^2 + 42*a^2*b^2*c^5*e^3*f^7*g^2 - 20*b^4*c^
5*d^3*f^6*g^3 + 34*a*b^2*c^6*d^3*f^6*g^3 + 16*a^2*c^7*d^3*f^6*g^3 + 40*b^5*c^4*d^2*e*f^6*g^3 - 89*a*b^3*c^5*d^
2*e*f^6*g^3 - 32*a^2*b*c^6*d^2*e*f^6*g^3 - 20*b^6*c^3*d*e^2*f^6*g^3 + 35*a*b^4*c^4*d*e^2*f^6*g^3 + 50*a^2*b^2*
c^5*d*e^2*f^6*g^3 + 16*a^3*c^6*d*e^2*f^6*g^3 + 20*a*b^5*c^3*e^3*f^6*g^3 - 55*a^2*b^3*c^4*e^3*f^6*g^3 - 16*a^3*
b*c^5*e^3*f^6*g^3 + 15*b^5*c^4*d^3*f^5*g^4 - 15*a*b^3*c^5*d^3*f^5*g^4 - 54*a^2*b*c^6*d^3*f^5*g^4 - 30*b^6*c^3*
d^2*e*f^5*g^4 + 45*a*b^4*c^4*d^2*e*f^5*g^4 + 114*a^2*b^2*c^5*d^2*e*f^5*g^4 - 12*a^3*c^6*d^2*e*f^5*g^4 + 15*b^7
*c^2*d*e^2*f^5*g^4 - 15*a*b^5*c^3*d*e^2*f^5*g^4 - 75*a^2*b^3*c^4*d*e^2*f^5*g^4 - 42*a^3*b*c^5*d*e^2*f^5*g^4 -
15*a*b^6*c^2*e^3*f^5*g^4 + 30*a^2*b^4*c^3*e^3*f^5*g^4 + 60*a^3*b^2*c^4*e^3*f^5*g^4 - 12*a^4*c^5*e^3*f^5*g^4 -
6*b^6*c^3*d^3*f^4*g^5 - 9*a*b^4*c^4*d^3*f^4*g^5 + 66*a^2*b^2*c^5*d^3*f^4*g^5 + 12*a^3*c^6*d^3*f^4*g^5 + 12*b^7
*c^2*d^2*e*f^4*g^5 + 12*a*b^5*c^3*d^2*e*f^4*g^5 - 147*a^2*b^3*c^4*d^2*e*f^4*g^5 + 6*a^3*b*c^5*d^2*e*f^4*g^5 -
6*b^8*c*d*e^2*f^4*g^5 - 9*a*b^6*c^2*d*e^2*f^4*g^5 + 72*a^2*b^4*c^3*d*e^2*f^4*g^5 + 48*a^3*b^2*c^4*d*e^2*f^4*g^
5 + 12*a^4*c^5*d*e^2*f^4*g^5 + 6*a*b^7*c*e^3*f^4*g^5 + 3*a^2*b^5*c^2*e^3*f^4*g^5 - 81*a^3*b^3*c^3*e^3*f^4*g^5
+ 18*a^4*b*c^4*e^3*f^4*g^5 + b^7*c^2*d^3*f^3*g^6 + 11*a*b^5*c^3*d^3*f^3*g^6 - 31*a^2*b^3*c^4*d^3*f^3*g^6 - 32*
a^3*b*c^5*d^3*f^3*g^6 - 2*b^8*c*d^2*e*f^3*g^6 - 21*a*b^6*c^2*d^2*e*f^3*g^6 + 74*a^2*b^4*c^3*d^2*e*f^3*g^6 + 46
*a^3*b^2*c^4*d^2*e*f^3*g^6 - 16*a^4*c^5*d^2*e*f^3*g^6 + b^9*d*e^2*f^3*g^6 + 11*a*b^7*c*d*e^2*f^3*g^6 - 32*a^2*
b^5*c^2*d*e^2*f^3*g^6 - 45*a^3*b^3*c^3*d*e^2*f^3*g^6 - 16*a^4*b*c^4*d*e^2*f^3*g^6 - a*b^8*e^3*f^3*g^6 - 10*a^2
*b^6*c*e^3*f^3*g^6 + 43*a^3*b^4*c^2*e^3*f^3*g^6 + 14*a^4*b^2*c^3*e^3*f^3*g^6 - 16*a^5*c^4*e^3*f^3*g^6 - 3*a*b^
6*c^2*d^3*f^2*g^7 + 30*a^3*b^2*c^4*d^3*f^2*g^7 + 6*a*b^7*c*d^2*e*f^2*g^7 - 3*a^2*b^5*c^2*d^2*e*f^2*g^7 - 63*a^
3*b^3*c^3*d^2*e*f^2*g^7 + 24*a^4*b*c^4*d^2*e*f^2*g^7 - 3*a*b^8*d*e^2*f^2*g^7 + 33*a^3*b^4*c^2*d*e^2*f^2*g^7 +
6*a^4*b^2*c^3*d*e^2*f^2*g^7 + 3*a^2*b^7*e^3*f^2*g^7 - 3*a^3*b^5*c*e^3*f^2*g^7 - 33*a^4*b^3*c^2*e^3*f^2*g^7 + 2
4*a^5*b*c^3*e^3*f^2*g^7 + 3*a^2*b^5*c^2*d^3*f*g^8 - 9*a^3*b^3*c^3*d^3*f*g^8 - 3*a^4*b*c^4*d^3*f*g^8 - 6*a^2*b^
6*c*d^2*e*f*g^8 + 21*a^3*b^4*c^2*d^2*e*f*g^8 - 6*a^5*c^4*d^2*e*f*g^8 + 3*a^2*b^7*d*e^2*f*g^8 - 9*a^3*b^5*c*d*e
^2*f*g^8 - 6*a^4*b^3*c^2*d*e^2*f*g^8 + 3*a^5*b*c^3*d*e^2*f*g^8 - 3*a^3*b^6*e^3*f*g^8 + 12*a^4*b^4*c*e^3*f*g^8
- 3*a^5*b^2*c^2*e^3*f*g^8 - 6*a^6*c^3*e^3*f*g^8 - a^3*b^4*c^2*d^3*g^9 + 4*a^4*b^2*c^3*d^3*g^9 - 2*a^5*c^4*d^3*
g^9 + 2*a^3*b^5*c*d^2*e*g^9 - 9*a^4*b^3*c^2*d^2*e*g^9 + 7*a^5*b*c^3*d^2*e*g^9 - a^3*b^6*d*e^2*g^9 + 4*a^4*b^4*
c*d*e^2*g^9 - a^5*b^2*c^2*d*e^2*g^9 - 2*a^6*c^3*d*e^2*g^9 + a^4*b^5*e^3*g^9 - 5*a^5*b^3*c*e^3*g^9 + 5*a^6*b*c^
2*e^3*g^9)/(b^2*c^8*d^4*f^12 - 4*a*c^9*d^4*f^12 - 2*b^3*c^7*d^3*e*f^12 + 8*a*b*c^8*d^3*e*f^12 + b^4*c^6*d^2*e^
2*f^12 - 2*a*b^2*c^7*d^2*e^2*f^12 - 8*a^2*c^8*d^2*e^2*f^12 - 2*a*b^3*c^6*d*e^3*f^12 + 8*a^2*b*c^7*d*e^3*f^12 +
a^2*b^2*c^6*e^4*f^12 - 4*a^3*c^7*e^4*f^12 - 6*b^3*c^7*d^4*f^11*g + 24*a*b*c^8*d^4*f^11*g + 12*b^4*c^6*d^3*e*f
^11*g - 48*a*b^2*c^7*d^3*e*f^11*g - 6*b^5*c^5*d^2*e^2*f^11*g + 12*a*b^3*c^6*d^2*e^2*f^11*g + 48*a^2*b*c^7*d^2*
e^2*f^11*g + 12*a*b^4*c^5*d*e^3*f^11*g - 48*a^2*b^2*c^6*d*e^3*f^11*g - 6*a^2*b^3*c^5*e^4*f^11*g + 24*a^3*b*c^6
*e^4*f^11*g + 15*b^4*c^6*d^4*f^10*g^2 - 54*a*b^2*c^7*d^4*f^10*g^2 - 24*a^2*c^8*d^4*f^10*g^2 - 30*b^5*c^5*d^3*e
*f^10*g^2 + 108*a*b^3*c^6*d^3*e*f^10*g^2 + 48*a^2*b*c^7*d^3*e*f^10*g^2 + 15*b^6*c^4*d^2*e^2*f^10*g^2 - 24*a*b^
4*c^5*d^2*e^2*f^10*g^2 - 132*a^2*b^2*c^6*d^2*e^2*f^10*g^2 - 48*a^3*c^7*d^2*e^2*f^10*g^2 - 30*a*b^5*c^4*d*e^3*f
^10*g^2 + 108*a^2*b^3*c^5*d*e^3*f^10*g^2 + 48*a^3*b*c^6*d*e^3*f^10*g^2 + 15*a^2*b^4*c^4*e^4*f^10*g^2 - 54*a^3*
b^2*c^5*e^4*f^10*g^2 - 24*a^4*c^6*e^4*f^10*g^2 - 20*b^5*c^5*d^4*f^9*g^3 + 50*a*b^3*c^6*d^4*f^9*g^3 + 120*a^2*b
*c^7*d^4*f^9*g^3 + 40*b^6*c^4*d^3*e*f^9*g^3 - 100*a*b^4*c^5*d^3*e*f^9*g^3 - 240*a^2*b^2*c^6*d^3*e*f^9*g^3 - 20
*b^7*c^3*d^2*e^2*f^9*g^3 + 10*a*b^5*c^4*d^2*e^2*f^9*g^3 + 220*a^2*b^3*c^5*d^2*e^2*f^9*g^3 + 240*a^3*b*c^6*d^2*
e^2*f^9*g^3 + 40*a*b^6*c^3*d*e^3*f^9*g^3 - 100*a^2*b^4*c^4*d*e^3*f^9*g^3 - 240*a^3*b^2*c^5*d*e^3*f^9*g^3 - 20*
a^2*b^5*c^3*e^4*f^9*g^3 + 50*a^3*b^3*c^4*e^4*f^9*g^3 + 120*a^4*b*c^5*e^4*f^9*g^3 + 15*b^6*c^4*d^4*f^8*g^4 - 22
5*a^2*b^2*c^6*d^4*f^8*g^4 - 60*a^3*c^7*d^4*f^8*g^4 - 30*b^7*c^3*d^3*e*f^8*g^4 + 450*a^2*b^3*c^5*d^3*e*f^8*g^4
+ 120*a^3*b*c^6*d^3*e*f^8*g^4 + 15*b^8*c^2*d^2*e^2*f^8*g^4 + 30*a*b^6*c^3*d^2*e^2*f^8*g^4 - 225*a^2*b^4*c^4*d^
2*e^2*f^8*g^4 - 510*a^3*b^2*c^5*d^2*e^2*f^8*g^4 - 120*a^4*c^6*d^2*e^2*f^8*g^4 - 30*a*b^7*c^2*d*e^3*f^8*g^4 + 4
50*a^3*b^3*c^4*d*e^3*f^8*g^4 + 120*a^4*b*c^5*d*e^3*f^8*g^4 + 15*a^2*b^6*c^2*e^4*f^8*g^4 - 225*a^4*b^2*c^4*e^4*
f^8*g^4 - 60*a^5*c^5*e^4*f^8*g^4 - 6*b^7*c^3*d^4*f^7*g^5 - 36*a*b^5*c^4*d^4*f^7*g^5 + 180*a^2*b^3*c^5*d^4*f^7*
g^5 + 240*a^3*b*c^6*d^4*f^7*g^5 + 12*b^8*c^2*d^3*e*f^7*g^5 + 72*a*b^6*c^3*d^3*e*f^7*g^5 - 360*a^2*b^4*c^4*d^3*
e*f^7*g^5 - 480*a^3*b^2*c^5*d^3*e*f^7*g^5 - 6*b^9*c*d^2*e^2*f^7*g^5 - 48*a*b^7*c^2*d^2*e^2*f^7*g^5 + 108*a^2*b
^5*c^3*d^2*e^2*f^7*g^5 + 600*a^3*b^3*c^4*d^2*e^2*f^7*g^5 + 480*a^4*b*c^5*d^2*e^2*f^7*g^5 + 12*a*b^8*c*d*e^3*f^
7*g^5 + 72*a^2*b^6*c^2*d*e^3*f^7*g^5 - 360*a^3*b^4*c^3*d*e^3*f^7*g^5 - 480*a^4*b^2*c^4*d*e^3*f^7*g^5 - 6*a^2*b
^7*c*e^4*f^7*g^5 - 36*a^3*b^5*c^2*e^4*f^7*g^5 + 180*a^4*b^3*c^3*e^4*f^7*g^5 + 240*a^5*b*c^4*e^4*f^7*g^5 + b^8*
c^2*d^4*f^6*g^6 + 26*a*b^6*c^3*d^4*f^6*g^6 - 30*a^2*b^4*c^4*d^4*f^6*g^6 - 340*a^3*b^2*c^5*d^4*f^6*g^6 - 80*a^4
*c^6*d^4*f^6*g^6 - 2*b^9*c*d^3*e*f^6*g^6 - 52*a*b^7*c^2*d^3*e*f^6*g^6 + 60*a^2*b^5*c^3*d^3*e*f^6*g^6 + 680*a^3
*b^3*c^4*d^3*e*f^6*g^6 + 160*a^4*b*c^5*d^3*e*f^6*g^6 + b^10*d^2*e^2*f^6*g^6 + 28*a*b^8*c*d^2*e^2*f^6*g^6 + 22*
a^2*b^6*c^2*d^2*e^2*f^6*g^6 - 400*a^3*b^4*c^3*d^2*e^2*f^6*g^6 - 760*a^4*b^2*c^4*d^2*e^2*f^6*g^6 - 160*a^5*c^5*
d^2*e^2*f^6*g^6 - 2*a*b^9*d*e^3*f^6*g^6 - 52*a^2*b^7*c*d*e^3*f^6*g^6 + 60*a^3*b^5*c^2*d*e^3*f^6*g^6 + 680*a^4*
b^3*c^3*d*e^3*f^6*g^6 + 160*a^5*b*c^4*d*e^3*f^6*g^6 + a^2*b^8*e^4*f^6*g^6 + 26*a^3*b^6*c*e^4*f^6*g^6 - 30*a^4*
b^4*c^2*e^4*f^6*g^6 - 340*a^5*b^2*c^3*e^4*f^6*g^6 - 80*a^6*c^4*e^4*f^6*g^6 - 6*a*b^7*c^2*d^4*f^5*g^7 - 36*a^2*
b^5*c^3*d^4*f^5*g^7 + 180*a^3*b^3*c^4*d^4*f^5*g^7 + 240*a^4*b*c^5*d^4*f^5*g^7 + 12*a*b^8*c*d^3*e*f^5*g^7 + 72*
a^2*b^6*c^2*d^3*e*f^5*g^7 - 360*a^3*b^4*c^3*d^3*e*f^5*g^7 - 480*a^4*b^2*c^4*d^3*e*f^5*g^7 - 6*a*b^9*d^2*e^2*f^
5*g^7 - 48*a^2*b^7*c*d^2*e^2*f^5*g^7 + 108*a^3*b^5*c^2*d^2*e^2*f^5*g^7 + 600*a^4*b^3*c^3*d^2*e^2*f^5*g^7 + 480
*a^5*b*c^4*d^2*e^2*f^5*g^7 + 12*a^2*b^8*d*e^3*f^5*g^7 + 72*a^3*b^6*c*d*e^3*f^5*g^7 - 360*a^4*b^4*c^2*d*e^3*f^5
*g^7 - 480*a^5*b^2*c^3*d*e^3*f^5*g^7 - 6*a^3*b^7*e^4*f^5*g^7 - 36*a^4*b^5*c*e^4*f^5*g^7 + 180*a^5*b^3*c^2*e^4*
f^5*g^7 + 240*a^6*b*c^3*e^4*f^5*g^7 + 15*a^2*b^6*c^2*d^4*f^4*g^8 - 225*a^4*b^2*c^4*d^4*f^4*g^8 - 60*a^5*c^5*d^
4*f^4*g^8 - 30*a^2*b^7*c*d^3*e*f^4*g^8 + 450*a^4*b^3*c^3*d^3*e*f^4*g^8 + 120*a^5*b*c^4*d^3*e*f^4*g^8 + 15*a^2*
b^8*d^2*e^2*f^4*g^8 + 30*a^3*b^6*c*d^2*e^2*f^4*g^8 - 225*a^4*b^4*c^2*d^2*e^2*f^4*g^8 - 510*a^5*b^2*c^3*d^2*e^2
*f^4*g^8 - 120*a^6*c^4*d^2*e^2*f^4*g^8 - 30*a^3*b^7*d*e^3*f^4*g^8 + 450*a^5*b^3*c^2*d*e^3*f^4*g^8 + 120*a^6*b*
c^3*d*e^3*f^4*g^8 + 15*a^4*b^6*e^4*f^4*g^8 - 225*a^6*b^2*c^2*e^4*f^4*g^8 - 60*a^7*c^3*e^4*f^4*g^8 - 20*a^3*b^5
*c^2*d^4*f^3*g^9 + 50*a^4*b^3*c^3*d^4*f^3*g^9 + 120*a^5*b*c^4*d^4*f^3*g^9 + 40*a^3*b^6*c*d^3*e*f^3*g^9 - 100*a
^4*b^4*c^2*d^3*e*f^3*g^9 - 240*a^5*b^2*c^3*d^3*e*f^3*g^9 - 20*a^3*b^7*d^2*e^2*f^3*g^9 + 10*a^4*b^5*c*d^2*e^2*f
^3*g^9 + 220*a^5*b^3*c^2*d^2*e^2*f^3*g^9 + 240*a^6*b*c^3*d^2*e^2*f^3*g^9 + 40*a^4*b^6*d*e^3*f^3*g^9 - 100*a^5*
b^4*c*d*e^3*f^3*g^9 - 240*a^6*b^2*c^2*d*e^3*f^3*g^9 - 20*a^5*b^5*e^4*f^3*g^9 + 50*a^6*b^3*c*e^4*f^3*g^9 + 120*
a^7*b*c^2*e^4*f^3*g^9 + 15*a^4*b^4*c^2*d^4*f^2*g^10 - 54*a^5*b^2*c^3*d^4*f^2*g^10 - 24*a^6*c^4*d^4*f^2*g^10 -
30*a^4*b^5*c*d^3*e*f^2*g^10 + 108*a^5*b^3*c^2*d^3*e*f^2*g^10 + 48*a^6*b*c^3*d^3*e*f^2*g^10 + 15*a^4*b^6*d^2*e^
2*f^2*g^10 - 24*a^5*b^4*c*d^2*e^2*f^2*g^10 - 132*a^6*b^2*c^2*d^2*e^2*f^2*g^10 - 48*a^7*c^3*d^2*e^2*f^2*g^10 -
30*a^5*b^5*d*e^3*f^2*g^10 + 108*a^6*b^3*c*d*e^3*f^2*g^10 + 48*a^7*b*c^2*d*e^3*f^2*g^10 + 15*a^6*b^4*e^4*f^2*g^
10 - 54*a^7*b^2*c*e^4*f^2*g^10 - 24*a^8*c^2*e^4*f^2*g^10 - 6*a^5*b^3*c^2*d^4*f*g^11 + 24*a^6*b*c^3*d^4*f*g^11
+ 12*a^5*b^4*c*d^3*e*f*g^11 - 48*a^6*b^2*c^2*d^3*e*f*g^11 - 6*a^5*b^5*d^2*e^2*f*g^11 + 12*a^6*b^3*c*d^2*e^2*f*
g^11 + 48*a^7*b*c^2*d^2*e^2*f*g^11 + 12*a^6*b^4*d*e^3*f*g^11 - 48*a^7*b^2*c*d*e^3*f*g^11 - 6*a^7*b^3*e^4*f*g^1
1 + 24*a^8*b*c*e^4*f*g^11 + a^6*b^2*c^2*d^4*g^12 - 4*a^7*c^3*d^4*g^12 - 2*a^6*b^3*c*d^3*e*g^12 + 8*a^7*b*c^2*d
^3*e*g^12 + a^6*b^4*d^2*e^2*g^12 - 2*a^7*b^2*c*d^2*e^2*g^12 - 8*a^8*c^2*d^2*e^2*g^12 - 2*a^7*b^3*d*e^3*g^12 +
8*a^8*b*c*d*e^3*g^12 + a^8*b^2*e^4*g^12 - 4*a^9*c*e^4*g^12))/sqrt(c*x^2 + b*x + a) - 1/4*(80*c^2*e^2*f^4*g^3 -
120*c^2*d*e*f^3*g^4 - 100*b*c*e^2*f^3*g^4 + 48*c^2*d^2*f^2*g^5 + 132*b*c*d*e*f^2*g^5 + 35*b^2*e^2*f^2*g^5 + 2
8*a*c*e^2*f^2*g^5 - 48*b*c*d^2*f*g^6 - 42*b^2*d*e*f*g^6 - 28*a*b*e^2*f*g^6 + 15*b^2*d^2*g^7 - 12*a*c*d^2*g^7 +
12*a*b*d*e*g^7 + 8*a^2*e^2*g^7)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*g + sqrt(c)*f)/sqrt(-c*f^2 + b*f
*g - a*g^2))/((c^3*e^3*f^9 - 3*c^3*d*e^2*f^8*g - 3*b*c^2*e^3*f^8*g + 3*c^3*d^2*e*f^7*g^2 + 9*b*c^2*d*e^2*f^7*g
^2 + 3*b^2*c*e^3*f^7*g^2 + 3*a*c^2*e^3*f^7*g^2 - c^3*d^3*f^6*g^3 - 9*b*c^2*d^2*e*f^6*g^3 - 9*b^2*c*d*e^2*f^6*g
^3 - 9*a*c^2*d*e^2*f^6*g^3 - b^3*e^3*f^6*g^3 - 6*a*b*c*e^3*f^6*g^3 + 3*b*c^2*d^3*f^5*g^4 + 9*b^2*c*d^2*e*f^5*g
^4 + 9*a*c^2*d^2*e*f^5*g^4 + 3*b^3*d*e^2*f^5*g^4 + 18*a*b*c*d*e^2*f^5*g^4 + 3*a*b^2*e^3*f^5*g^4 + 3*a^2*c*e^3*
f^5*g^4 - 3*b^2*c*d^3*f^4*g^5 - 3*a*c^2*d^3*f^4*g^5 - 3*b^3*d^2*e*f^4*g^5 - 18*a*b*c*d^2*e*f^4*g^5 - 9*a*b^2*d
*e^2*f^4*g^5 - 9*a^2*c*d*e^2*f^4*g^5 - 3*a^2*b*e^3*f^4*g^5 + b^3*d^3*f^3*g^6 + 6*a*b*c*d^3*f^3*g^6 + 9*a*b^2*d
^2*e*f^3*g^6 + 9*a^2*c*d^2*e*f^3*g^6 + 9*a^2*b*d*e^2*f^3*g^6 + a^3*e^3*f^3*g^6 - 3*a*b^2*d^3*f^2*g^7 - 3*a^2*c
*d^3*f^2*g^7 - 9*a^2*b*d^2*e*f^2*g^7 - 3*a^3*d*e^2*f^2*g^7 + 3*a^2*b*d^3*f*g^8 + 3*a^3*d^2*e*f*g^8 - a^3*d^3*g
^9)*sqrt(-c*f^2 + b*f*g - a*g^2)) + 1/4*(32*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*e*f^3*g^4 - 24*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^3*c^2*d*f^2*g^5 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*e*f^2*g^5 + 24*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d*f*g^6 + 11*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*e*f*g^6 + 4*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*e*f*g^6 - 7*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*d*g^7 + 4*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*a*c*d*g^7 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*e*g^7 + 72*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*c^(5/2)*e*f^4*g^3 - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*d*f^3*g^4 - 68*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*e*f^3*g^4 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*d*
f^2*g^5 + 17*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*sqrt(c)*e*f^2*g^5 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*a*c^(3/2)*e*f^2*g^5 - 13*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*sqrt(c)*d*f*g^6 + 28*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^2*a*c^(3/2)*d*f*g^6 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*sqrt(c)*e*f*g^6 - 8*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*sqrt(c)*d*g^7 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*sqrt(c)*e*g
^7 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*e*f^4*g^3 - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*d*f
^3*g^4 - 64*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c*e*f^3*g^4 - 112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*c^
2*e*f^3*g^4 + 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c*d*f^2*g^5 + 88*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*
a*c^2*d*f^2*g^5 + 13*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*e*f^2*g^5 + 104*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))*a*b*c*e*f^2*g^5 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*d*f*g^6 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*a*b*c*d*f*g^6 - 17*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*e*f*g^6 - 28*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*a^2*c*e*f*g^6 + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*d*g^7 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^
2*c*d*g^7 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*e*g^7 + 18*b^2*c^(3/2)*e*f^4*g^3 - 14*b^2*c^(3/2)*d*f^
3*g^4 - 11*b^3*sqrt(c)*e*f^3*g^4 - 56*a*b*c^(3/2)*e*f^3*g^4 + 7*b^3*sqrt(c)*d*f^2*g^5 + 44*a*b*c^(3/2)*d*f^2*g
^5 + 39*a*b^2*sqrt(c)*e*f^2*g^5 + 36*a^2*c^(3/2)*e*f^2*g^5 - 23*a*b^2*sqrt(c)*d*f*g^6 - 28*a^2*c^(3/2)*d*f*g^6
- 36*a^2*b*sqrt(c)*e*f*g^6 + 16*a^2*b*sqrt(c)*d*g^7 + 8*a^3*sqrt(c)*e*g^7)/((c^3*e^2*f^8 - 2*c^3*d*e*f^7*g -
3*b*c^2*e^2*f^7*g + c^3*d^2*f^6*g^2 + 6*b*c^2*d*e*f^6*g^2 + 3*b^2*c*e^2*f^6*g^2 + 3*a*c^2*e^2*f^6*g^2 - 3*b*c^
2*d^2*f^5*g^3 - 6*b^2*c*d*e*f^5*g^3 - 6*a*c^2*d*e*f^5*g^3 - b^3*e^2*f^5*g^3 - 6*a*b*c*e^2*f^5*g^3 + 3*b^2*c*d^
2*f^4*g^4 + 3*a*c^2*d^2*f^4*g^4 + 2*b^3*d*e*f^4*g^4 + 12*a*b*c*d*e*f^4*g^4 + 3*a*b^2*e^2*f^4*g^4 + 3*a^2*c*e^2
*f^4*g^4 - b^3*d^2*f^3*g^5 - 6*a*b*c*d^2*f^3*g^5 - 6*a*b^2*d*e*f^3*g^5 - 6*a^2*c*d*e*f^3*g^5 - 3*a^2*b*e^2*f^3
*g^5 + 3*a*b^2*d^2*f^2*g^6 + 3*a^2*c*d^2*f^2*g^6 + 6*a^2*b*d*e*f^2*g^6 + a^3*e^2*f^2*g^6 - 3*a^2*b*d^2*f*g^7 -
2*a^3*d*e*f*g^7 + a^3*d^2*g^8)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*g + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))*sqrt(c)*f + b*f - a*g)^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^3\,\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]

[In]

int(1/((f + g*x)^3*(d + e*x)*(a + b*x + c*x^2)^(3/2)),x)

[Out]

int(1/((f + g*x)^3*(d + e*x)*(a + b*x + c*x^2)^(3/2)), x)