\(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx\) [861]

Optimal result

Integrand size = 29, antiderivative size = 933 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx=\frac {e^2 \sqrt {a+b x+c x^2}}{(e f-d g)^3 (f+g x)}-\frac {g (2 c f-b g) (b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{8 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)^2}-\frac {e g (b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{4 (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x)^2}+\frac {g^2 \left (a+b x+c x^2\right )^{3/2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^3}-\frac {e^2 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} (e f-d g)^4}+\frac {e^3 (2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} g (e f-d g)^4}-\frac {\sqrt {c} e^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{g (e f-d g)^3}+\frac {e^2 \sqrt {c d^2-b d e+a e^2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{(e f-d g)^4}+\frac {\left (b^2-4 a c\right ) g (2 c f-b g) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{16 (e f-d g) \left (c f^2-b f g+a g^2\right )^{5/2}}+\frac {\left (b^2-4 a c\right ) e g \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{8 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^{3/2}}+\frac {e^2 (2 c f-b g) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{2 g (e f-d g)^3 \sqrt {c f^2-b f g+a g^2}}-\frac {e^3 \sqrt {c f^2-b f g+a g^2} \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{g (e f-d g)^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {974, 748, 857, 635, 212, 738, 744, 734, 746} \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx=\frac {(2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^3}{2 \sqrt {c} g (e f-d g)^4}-\frac {\sqrt {c f^2-b g f+a g^2} \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e^3}{g (e f-d g)^4}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^2}{g (e f-d g)^3}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^2}{2 \sqrt {c} (e f-d g)^4}+\frac {\sqrt {c d^2-b e d+a e^2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right ) e^2}{(e f-d g)^4}+\frac {(2 c f-b g) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e^2}{2 g (e f-d g)^3 \sqrt {c f^2-b g f+a g^2}}+\frac {\sqrt {c x^2+b x+a} e^2}{(e f-d g)^3 (f+g x)}+\frac {\left (b^2-4 a c\right ) g \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right ) e}{8 (e f-d g)^2 \left (c f^2-b g f+a g^2\right )^{3/2}}-\frac {g (b f-2 a g+(2 c f-b g) x) \sqrt {c x^2+b x+a} e}{4 (e f-d g)^2 \left (c f^2-b g f+a g^2\right ) (f+g x)^2}+\frac {g^2 \left (c x^2+b x+a\right )^{3/2}}{3 (e f-d g) \left (c f^2-b g f+a g^2\right ) (f+g x)^3}+\frac {\left (b^2-4 a c\right ) g (2 c f-b g) \text {arctanh}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b g f+a g^2} \sqrt {c x^2+b x+a}}\right )}{16 (e f-d g) \left (c f^2-b g f+a g^2\right )^{5/2}}-\frac {g (2 c f-b g) (b f-2 a g+(2 c f-b g) x) \sqrt {c x^2+b x+a}}{8 (e f-d g) \left (c f^2-b g f+a g^2\right )^2 (f+g x)^2} \]

[In]

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

[Out]

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

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 635

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 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, 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] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 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 744

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(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[(2*c*d - b*e)/(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, m, p}, 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] && EqQ[m + 2*p + 3, 0]

Rule 746

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 1))), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*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] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 748

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*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] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 857

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 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^4 \sqrt {a+b x+c x^2}}{(e f-d g)^4 (d+e x)}-\frac {g \sqrt {a+b x+c x^2}}{(e f-d g) (f+g x)^4}-\frac {e g \sqrt {a+b x+c x^2}}{(e f-d g)^2 (f+g x)^3}-\frac {e^2 g \sqrt {a+b x+c x^2}}{(e f-d g)^3 (f+g x)^2}-\frac {e^3 g \sqrt {a+b x+c x^2}}{(e f-d g)^4 (f+g x)}\right ) \, dx \\ & = \frac {e^4 \int \frac {\sqrt {a+b x+c x^2}}{d+e x} \, dx}{(e f-d g)^4}-\frac {\left (e^3 g\right ) \int \frac {\sqrt {a+b x+c x^2}}{f+g x} \, dx}{(e f-d g)^4}-\frac {\left (e^2 g\right ) \int \frac {\sqrt {a+b x+c x^2}}{(f+g x)^2} \, dx}{(e f-d g)^3}-\frac {(e g) \int \frac {\sqrt {a+b x+c x^2}}{(f+g x)^3} \, dx}{(e f-d g)^2}-\frac {g \int \frac {\sqrt {a+b x+c x^2}}{(f+g x)^4} \, dx}{e f-d g} \\ & = \frac {e^2 \sqrt {a+b x+c x^2}}{(e f-d g)^3 (f+g x)}-\frac {e g (b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{4 (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x)^2}+\frac {g^2 \left (a+b x+c x^2\right )^{3/2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^3}-\frac {e^3 \int \frac {b d-2 a e+(2 c d-b e) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2 (e f-d g)^4}+\frac {e^3 \int \frac {b f-2 a g+(2 c f-b g) x}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{2 (e f-d g)^4}-\frac {e^2 \int \frac {b+2 c x}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{2 (e f-d g)^3}+\frac {\left (\left (b^2-4 a c\right ) e g\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{8 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )}-\frac {(g (2 c f-b g)) \int \frac {\sqrt {a+b x+c x^2}}{(f+g x)^3} \, dx}{2 (e f-d g) \left (c f^2-b f g+a g^2\right )} \\ & = \frac {e^2 \sqrt {a+b x+c x^2}}{(e f-d g)^3 (f+g x)}-\frac {g (2 c f-b g) (b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{8 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)^2}-\frac {e g (b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{4 (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x)^2}+\frac {g^2 \left (a+b x+c x^2\right )^{3/2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^3}-\frac {\left (e^2 (2 c d-b e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 (e f-d g)^4}+\frac {\left (e^2 \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{(e f-d g)^4}+\frac {\left (e^3 (2 c f-b g)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 g (e f-d g)^4}-\frac {\left (c e^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{g (e f-d g)^3}+\frac {\left (e^2 (2 c f-b g)\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{2 g (e f-d g)^3}+\frac {\left (\left (b^2-4 a c\right ) g (2 c f-b g)\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{16 (e f-d g) \left (c f^2-b f g+a g^2\right )^2}-\frac {\left (\left (b^2-4 a c\right ) e 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 )}{4 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )}-\frac {\left (e^3 \left (c f^2-b f g+a g^2\right )\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{g (e f-d g)^4} \\ & = \frac {e^2 \sqrt {a+b x+c x^2}}{(e f-d g)^3 (f+g x)}-\frac {g (2 c f-b g) (b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{8 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)^2}-\frac {e g (b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{4 (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x)^2}+\frac {g^2 \left (a+b x+c x^2\right )^{3/2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^3}+\frac {\left (b^2-4 a c\right ) e 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 )}{8 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^{3/2}}-\frac {\left (e^2 (2 c d-b e)\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{(e f-d g)^4}-\frac {\left (2 e^2 \left (c d^2-b d e+a e^2\right )\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 )}{(e f-d g)^4}+\frac {\left (e^3 (2 c f-b g)\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{g (e f-d g)^4}-\frac {\left (2 c e^2\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{g (e f-d g)^3}-\frac {\left (e^2 (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 )}{g (e f-d g)^3}-\frac {\left (\left (b^2-4 a c\right ) g (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 )}{8 (e f-d g) \left (c f^2-b f g+a g^2\right )^2}+\frac {\left (2 e^3 \left (c f^2-b f g+a g^2\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 )}{g (e f-d g)^4} \\ & = \frac {e^2 \sqrt {a+b x+c x^2}}{(e f-d g)^3 (f+g x)}-\frac {g (2 c f-b g) (b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{8 (e f-d g) \left (c f^2-b f g+a g^2\right )^2 (f+g x)^2}-\frac {e g (b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{4 (e f-d g)^2 \left (c f^2-b f g+a g^2\right ) (f+g x)^2}+\frac {g^2 \left (a+b x+c x^2\right )^{3/2}}{3 (e f-d g) \left (c f^2-b f g+a g^2\right ) (f+g x)^3}-\frac {e^2 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} (e f-d g)^4}+\frac {e^3 (2 c f-b g) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} g (e f-d g)^4}-\frac {\sqrt {c} e^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{g (e f-d g)^3}+\frac {e^2 \sqrt {c d^2-b d e+a e^2} \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 )}{(e f-d g)^4}+\frac {\left (b^2-4 a c\right ) g (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 )}{16 (e f-d g) \left (c f^2-b f g+a g^2\right )^{5/2}}+\frac {\left (b^2-4 a c\right ) e 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 )}{8 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )^{3/2}}+\frac {e^2 (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 g (e f-d g)^3 \sqrt {c f^2-b f g+a g^2}}-\frac {e^3 \sqrt {c f^2-b f g+a g^2} \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 )}{g (e f-d g)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 12.45 (sec) , antiderivative size = 858, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx=\frac {\frac {48 e^2 (e f-d g) \sqrt {a+x (b+c x)}}{f+g x}+\frac {12 e g (e f-d g)^2 (-b f+2 a g-2 c f x+b g x) \sqrt {a+x (b+c x)}}{\left (c f^2+g (-b f+a g)\right ) (f+g x)^2}-\frac {16 g^2 (-e f+d g)^3 (a+x (b+c x))^{3/2}}{\left (c f^2+g (-b f+a g)\right ) (f+g x)^3}+24 e^2 \left (\frac {(-2 c d+b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}+2 \sqrt {c d^2+e (-b d+a e)} \text {arctanh}\left (\frac {-2 a e+2 c d x+b (d-e x)}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )+\frac {6 \left (b^2-4 a c\right ) e g (e f-d g)^2 \text {arctanh}\left (\frac {-2 a g+2 c f x+b (f-g x)}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )}{\left (c f^2+g (-b f+a g)\right )^{3/2}}-\frac {3 g (2 c f-b g) (e f-d g)^3 \left (\frac {2 \sqrt {a+x (b+c x)} (-2 a g+2 c f x+b (f-g x))}{\left (c f^2+g (-b f+a g)\right ) (f+g x)^2}+\frac {\left (-b^2+4 a c\right ) \text {arctanh}\left (\frac {-2 a g+2 c f x+b (f-g x)}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )}{\left (c f^2+g (-b f+a g)\right )^{3/2}}\right )}{c f^2+g (-b f+a g)}-\frac {24 e^2 (e f-d g) \left (2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-\frac {(2 c f-b g) \text {arctanh}\left (\frac {-2 a g+2 c f x+b (f-g x)}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c f^2+g (-b f+a g)}}\right )}{g}+\frac {24 e^3 \left ((2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} \sqrt {c f^2+g (-b f+a g)} \text {arctanh}\left (\frac {-2 a g+2 c f x+b (f-g x)}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )\right )}{\sqrt {c} g}}{48 (e f-d g)^4} \]

[In]

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

[Out]

((48*e^2*(e*f - d*g)*Sqrt[a + x*(b + c*x)])/(f + g*x) + (12*e*g*(e*f - d*g)^2*(-(b*f) + 2*a*g - 2*c*f*x + b*g*
x)*Sqrt[a + x*(b + c*x)])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)^2) - (16*g^2*(-(e*f) + d*g)^3*(a + x*(b + c*x)
)^(3/2))/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)^3) + 24*e^2*(((-2*c*d + b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqr
t[a + x*(b + c*x)])])/Sqrt[c] + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*S
qrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]) + (6*(b^2 - 4*a*c)*e*g*(e*f - d*g)^2*ArcTanh[(-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)])])/(c*f^2 + g*(-(b*f) + a*g))^(
3/2) - (3*g*(2*c*f - b*g)*(e*f - d*g)^3*((2*Sqrt[a + x*(b + c*x)]*(-2*a*g + 2*c*f*x + b*(f - g*x)))/((c*f^2 +
g*(-(b*f) + a*g))*(f + g*x)^2) + ((-b^2 + 4*a*c)*ArcTanh[(-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)])])/(c*f^2 + g*(-(b*f) + a*g))^(3/2)))/(c*f^2 + g*(-(b*f) + a*g)) - (24*e^2
*(e*f - d*g)*(2*Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - ((2*c*f - b*g)*ArcTanh[(-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)])])/Sqrt[c*f^2 + g*(-(b*f) +
a*g)]))/g + (24*e^3*((2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] - 2*Sqrt[c]*Sqrt[c*
f^2 + g*(-(b*f) + a*g)]*ArcTanh[(-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)])]))/(Sqrt[c]*g))/(48*(e*f - d*g)^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3776\) vs. \(2(845)=1690\).

Time = 1.16 (sec) , antiderivative size = 3777, normalized size of antiderivative = 4.05

method	result	size
default	\(\text {Expression too large to display}\)	\(3777\)

[In]

int((c*x^2+b*x+a)^(1/2)/(e*x+d)/(g*x+f)^4,x,method=_RETURNVERBOSE)

[Out]

e^3/(d*g-e*f)^4*(((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)+1/2*(b*e-2*c*d)/e*ln((1/2*(
b*e-2*c*d)/e+c*(x+d/e))/c^(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))/c^(1/2)-(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)))+1/g^
3/(d*g-e*f)*(-1/3/(a*g^2-b*f*g+c*f^2)*g^2/(x+f/g)^3*((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
)^(3/2)-1/2*(b*g-2*c*f)*g/(a*g^2-b*f*g+c*f^2)*(-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)^(3/2)-1/4*(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)^(3/2)+1/2*(b*g-2*c*f)*g/(a*g^2-b*f*g+c*f^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)+1/2*(b*g-2*c*f)/g*ln((1/2*(b*g-2*c*f)/g+c*(
x+f/g))/c^(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))/c^(1/2)-(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)))+2*c/(a*g^2-b*f*g+c*f
^2)*g^2*(1/4*(2*c*(x+f/g)+(b*g-2*c*f)/g)/c*((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
/8*(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*c*f)^2/g^2)/c^(3/2)*ln((1/2*(b*g-2*c*f)/g+c*(x+f/g))/c^(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))))+1/2*c/(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)+1/2*(b*g-2*c*f)/g*ln((1/2*(b*g-2*c*f)/g+c*(x+f/g))/c^(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))/c^(1/2)-(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^2/(d*g-e*f)^3*(-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)^(3/2)+1/2*(b*g-2*c*f)*g/(a*g^2-b*
f*g+c*f^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)+1/2*(b*g-2*c*f)/g*ln((1/2*(b*g-2
*c*f)/g+c*(x+f/g))/c^(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))/c^(1/2)-(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)))+2*c/(a*g^
2-b*f*g+c*f^2)*g^2*(1/4*(2*c*(x+f/g)+(b*g-2*c*f)/g)/c*((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)+1/8*(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*c*f)^2/g^2)/c^(3/2)*ln((1/2*(b*g-2*c*f)/g+c*(x+f/g))/c^(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))))-1/g^2*e/(d*g-e*f)^2*(-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)^(3/2)-1/4*(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)^
(3/2)+1/2*(b*g-2*c*f)*g/(a*g^2-b*f*g+c*f^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)
+1/2*(b*g-2*c*f)/g*ln((1/2*(b*g-2*c*f)/g+c*(x+f/g))/c^(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))/c^(1/2)-(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)))+2*c/(a*g^2-b*f*g+c*f^2)*g^2*(1/4*(2*c*(x+f/g)+(b*g-2*c*f)/g)/c*((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)+1/8*(4*c*(a*g^2-b*f*g+c*f^2)/g^2-(b*g-2*c*f)^2/g^2)/c^(3/2)*ln((1/2
*(b*g-2*c*f)/g+c*(x+f/g))/c^(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))))+1/2*c/(
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)+1/2*(b*g-2*c*f)/g*ln
((1/2*(b*g-2*c*f)/g+c*(x+f/g))/c^(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))/c^(1
/2)-(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^3/(d*g-e*f)^4*(((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)+1/2*(b*g-2*c*f)/g*ln((1
/2*(b*g-2*c*f)/g+c*(x+f/g))/c^(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))/c^(1/2)
-(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)))

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx=\text {Timed out} \]

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)/(g*x+f)^4,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)/(g*x+f)**4,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )} {\left (g x + f\right )}^{4}} \,d x } \]

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)/(g*x+f)^4,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)/((e*x + d)*(g*x + f)^4), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8076 vs. \(2 (845) = 1690\).

Time = 5.56 (sec) , antiderivative size = 8076, normalized size of antiderivative = 8.66 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)/(g*x+f)^4,x, algorithm="giac")

[Out]

2*(c*d^2*e^2 - b*d*e^3 + a*e^4)*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))/((e^4*f^4 - 4*d*e^3*f^3*g + 6*d^2*e^2*f^2*g^2 - 4*d^3*e*f*g^3 + d^4*g^4)*sqrt(-c*d^2 + b*d*e - a*e
^2)) - 1/8*(16*c^3*d*e^2*f^5 - 8*b*c^2*e^3*f^5 - 40*b*c^2*d*e^2*f^4*g + 12*b^2*c*e^3*f^4*g + 32*a*c^2*e^3*f^4*
g + 42*b^2*c*d*e^2*f^3*g^2 - 8*a*c^2*d*e^2*f^3*g^2 - 5*b^3*e^3*f^3*g^2 - 60*a*b*c*e^3*f^3*g^2 - 8*b^2*c*d^2*e*
f^2*g^3 + 32*a*c^2*d^2*e*f^2*g^3 - 15*b^3*d*e^2*f^2*g^3 - 20*a*b*c*d*e^2*f^2*g^3 + 30*a*b^2*e^3*f^2*g^3 + 40*a
^2*c*e^3*f^2*g^3 + 2*b^2*c*d^3*f*g^4 - 8*a*c^2*d^3*f*g^4 + 5*b^3*d^2*e*f*g^4 - 20*a*b*c*d^2*e*f*g^4 + 20*a*b^2
*d*e^2*f*g^4 - 40*a^2*b*e^3*f*g^4 - b^3*d^3*g^5 + 4*a*b*c*d^3*g^5 - 2*a*b^2*d^2*e*g^5 + 8*a^2*c*d^2*e*g^5 - 8*
a^2*b*d*e^2*g^5 + 16*a^3*e^3*g^5)*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^2*e^4*f^8 - 4*c^2*d*e^3*f^7*g - 2*b*c*e^4*f^7*g + 6*c^2*d^2*e^2*f^6*g^2 + 8*b*c*d*e^3*f^6*g^
2 + b^2*e^4*f^6*g^2 + 2*a*c*e^4*f^6*g^2 - 4*c^2*d^3*e*f^5*g^3 - 12*b*c*d^2*e^2*f^5*g^3 - 4*b^2*d*e^3*f^5*g^3 -
8*a*c*d*e^3*f^5*g^3 - 2*a*b*e^4*f^5*g^3 + c^2*d^4*f^4*g^4 + 8*b*c*d^3*e*f^4*g^4 + 6*b^2*d^2*e^2*f^4*g^4 + 12*
a*c*d^2*e^2*f^4*g^4 + 8*a*b*d*e^3*f^4*g^4 + a^2*e^4*f^4*g^4 - 2*b*c*d^4*f^3*g^5 - 4*b^2*d^3*e*f^3*g^5 - 8*a*c*
d^3*e*f^3*g^5 - 12*a*b*d^2*e^2*f^3*g^5 - 4*a^2*d*e^3*f^3*g^5 + b^2*d^4*f^2*g^6 + 2*a*c*d^4*f^2*g^6 + 8*a*b*d^3
*e*f^2*g^6 + 6*a^2*d^2*e^2*f^2*g^6 - 2*a*b*d^4*f*g^7 - 4*a^2*d^3*e*f*g^7 + a^2*d^4*g^8)*sqrt(-c*f^2 + b*f*g -
a*g^2)) + 1/24*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^3*d*e*f^4*g^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^5*b*c^2*e^2*f^4*g^3 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^2*d*e*f^3*g^4 + 36*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^5*b^2*c*e^2*f^3*g^4 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^2*e^2*f^3*g^4 + 66*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^5*b^2*c*d*e*f^2*g^5 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^2*d*e*f^2*g^5 -
15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*e^2*f^2*g^5 - 84*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c*e^2
*f^2*g^5 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c*d^2*f*g^6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*
a*c^2*d^2*f*g^6 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*d*e*f*g^6 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^5*a*b*c*d*e*f*g^6 + 42*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*e^2*f*g^6 + 24*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*a^2*c*e^2*f*g^6 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*d^2*g^7 - 12*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*a*b*c*d^2*g^7 + 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*d*e*g^7 + 24*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^5*a^2*c*d*e*g^7 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b*e^2*g^7 + 240*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*c^(7/2)*d*e*f^5*g^2 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(5/2)*e^2*f^5*g^2
- 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(7/2)*d^2*f^4*g^3 - 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*
c^(5/2)*d*e*f^4*g^3 + 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(3/2)*e^2*f^4*g^3 + 192*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^4*a*c^(5/2)*e^2*f^4*g^3 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(5/2)*d^2*f^3*g^4 +
234*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(3/2)*d*e*f^3*g^4 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4
*a*c^(5/2)*d*e*f^3*g^4 - 75*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*sqrt(c)*e^2*f^3*g^4 - 324*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^4*a*b*c^(3/2)*e^2*f^3*g^4 - 78*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(3/2)*d^2*f^2
*g^5 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(5/2)*d^2*f^2*g^5 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^4*b^3*sqrt(c)*d*e*f^2*g^5 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(3/2)*d*e*f^2*g^5 + 162*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^4*a*b^2*sqrt(c)*e^2*f^2*g^5 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(3/2)*
e^2*f^2*g^5 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*sqrt(c)*d^2*f*g^6 + 36*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^4*a*b*c^(3/2)*d^2*f*g^6 - 66*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*sqrt(c)*d*e*f*g^6 + 120*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(3/2)*d*e*f*g^6 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b*sqrt(c
)*e^2*f*g^6 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(3/2)*d^2*g^7 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*a^2*b*sqrt(c)*d*e*g^7 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*sqrt(c)*e^2*g^7 + 64*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^3*c^4*e^2*f^7 + 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^4*d*e*f^6*g - 304*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^3*b*c^3*e^2*f^6*g - 32*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^4*d^2*f^5*g^2 + 32*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^3*d*e*f^5*g^2 + 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^2*e^
2*f^5*g^2 + 256*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^3*e^2*f^5*g^2 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^3*b*c^3*d^2*f^4*g^3 - 396*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^2*d*e*f^4*g^3 - 272*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^3*a*c^3*d*e*f^4*g^3 - 14*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c*e^2*f^4*g^3 - 120*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^2*e^2*f^4*g^3 + 84*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^2*d^2*f
^3*g^4 + 112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^3*d^2*f^3*g^4 + 280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^3*b^3*c*d*e*f^3*g^4 + 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^2*d*e*f^3*g^4 - 40*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^3*b^4*e^2*f^3*g^4 - 204*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c*e^2*f^3*g^4 - 336*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^2*e^2*f^3*g^4 - 74*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c*d^2*f^2*g
^5 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^2*d^2*f^2*g^5 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*
b^4*d*e*f^2*g^5 - 612*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c*d*e*f^2*g^5 + 48*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^3*a^2*c^2*d*e*f^2*g^5 + 136*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*e^2*f^2*g^5 + 528*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c*e^2*f^2*g^5 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*d^2*f*g^6 + 144
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c*d^2*f*g^6 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^2*d^
2*f*g^6 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*d*e*f*g^6 + 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3
*a^2*b*c*d*e*f*g^6 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*e^2*f*g^6 - 288*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*a^3*c*e^2*f*g^6 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*d^2*g^7 - 48*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^3*a^2*b*c*d^2*g^7 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b*e^2*g^7 + 96*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*b*c^(7/2)*e^2*f^7 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(7/2)*d*e*f^6*g -
408*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(5/2)*e^2*f^6*g - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*
c^(7/2)*e^2*f^6*g - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(7/2)*d^2*f^5*g^2 - 252*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2*b^2*c^(5/2)*d*e*f^5*g^2 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(7/2)*d*e*f^5*g^2 + 4
02*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(3/2)*e^2*f^5*g^2 + 1080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
a*b*c^(5/2)*e^2*f^5*g^2 + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(5/2)*d^2*f^4*g^3 + 48*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^2*a*c^(7/2)*d^2*f^4*g^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(3/2)*d*e*f^4*g
^3 + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^(5/2)*d*e*f^4*g^3 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^2*b^4*sqrt(c)*e^2*f^4*g^3 - 1068*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(3/2)*e^2*f^4*g^3 - 816*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(5/2)*e^2*f^4*g^3 + 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(3/2
)*d^2*f^3*g^4 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^(5/2)*d^2*f^3*g^4 + 96*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*b^4*sqrt(c)*d*e*f^3*g^4 + 156*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(3/2)*d*e*f^3*g^4 +
336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(5/2)*d*e*f^3*g^4 + 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
a*b^3*sqrt(c)*e^2*f^3*g^4 + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^(3/2)*e^2*f^3*g^4 - 24*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^2*b^4*sqrt(c)*d^2*f^2*g^5 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(3/2)*
d^2*f^2*g^5 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(5/2)*d^2*f^2*g^5 - 288*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*a*b^3*sqrt(c)*d*e*f^2*g^5 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^(3/2)*d*e*f^2*g^5
- 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*sqrt(c)*e^2*f^2*g^5 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^2*a^3*c^(3/2)*e^2*f^2*g^5 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3*sqrt(c)*d^2*f*g^6 + 48*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^(3/2)*d^2*f*g^6 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*sqrt(
c)*d*e*f*g^6 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(3/2)*d*e*f*g^6 - 192*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*a^3*b*sqrt(c)*e^2*f*g^6 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*sqrt(c)*d^2*g^7 - 48*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b*sqrt(c)*d*e*g^7 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*sqrt(
c)*e^2*g^7 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c^3*e^2*f^7 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*
b^2*c^3*d*e*f^6*g - 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c^2*e^2*f^6*g - 192*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a*b*c^3*e^2*f^6*g - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c^3*d^2*f^5*g^2 - 156*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*b^3*c^2*d*e*f^5*g^2 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^3*d*e*f^5*g^2 + 150*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c*e^2*f^5*g^2 + 840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c^2*e^2*f
^5*g^2 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c^3*e^2*f^5*g^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b
^3*c^2*d^2*f^4*g^3 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^3*d^2*f^4*g^3 + 54*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*b^4*c*d*e*f^4*g^3 + 252*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c^2*d*e*f^4*g^3 + 192*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*a^2*c^3*d*e*f^4*g^3 - 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*e^2*f^4*g^3 - 714*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c*e^2*f^4*g^3 - 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^2*e^2
*f^4*g^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c*d^2*f^3*g^4 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a
*b^2*c^2*d^2*f^3*g^4 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c^3*d^2*f^3*g^4 + 12*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*b^5*d*e*f^3*g^4 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c*d*e*f^3*g^4 + 120*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a*b^4*e^2*f^3*g^4 + 1272*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c*e^2*f^3*g^4 + 48
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c^2*e^2*f^3*g^4 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*d^2*f^2*
g^5 + 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c*d^2*f^2*g^5 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2
*b*c^2*d^2*f^2*g^5 - 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*d*e*f^2*g^5 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))*a^2*b^2*c*d*e*f^2*g^5 - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c^2*d*e*f^2*g^5 - 165*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a^2*b^3*e^2*f^2*g^5 - 972*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c*e^2*f^2*g^5 + 6*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*d^2*f*g^6 + 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c*d^2*f*g^
6 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c^2*d^2*f*g^6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*
d*e*f*g^6 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c*d*e*f*g^6 + 102*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*a^3*b^2*e^2*f*g^6 + 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*c*e^2*f*g^6 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*a^2*b^3*d^2*g^7 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c*d^2*g^7 - 6*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a^3*b^2*d*e*g^7 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*c*d*e*g^7 - 24*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*a^4*b*e^2*g^7 + 8*b^3*c^(5/2)*e^2*f^7 + 20*b^3*c^(5/2)*d*e*f^6*g - 26*b^4*c^(3/2)*e^2*f^6*g - 48*a
*b^2*c^(5/2)*e^2*f^6*g - 4*b^3*c^(5/2)*d^2*f^5*g^2 - 26*b^4*c^(3/2)*d*e*f^5*g^2 - 60*a*b^2*c^(5/2)*d*e*f^5*g^2
+ 15*b^5*sqrt(c)*e^2*f^5*g^2 + 182*a*b^3*c^(3/2)*e^2*f^5*g^2 + 48*a^2*b*c^(5/2)*e^2*f^5*g^2 + 4*b^4*c^(3/2)*d
^2*f^4*g^3 + 12*a*b^2*c^(5/2)*d^2*f^4*g^3 + 12*b^5*sqrt(c)*d*e*f^4*g^3 + 56*a*b^3*c^(3/2)*d*e*f^4*g^3 + 96*a^2
*b*c^(5/2)*d*e*f^4*g^3 - 120*a*b^4*sqrt(c)*e^2*f^4*g^3 - 360*a^2*b^2*c^(3/2)*e^2*f^4*g^3 - 16*a^3*c^(5/2)*e^2*
f^4*g^3 - 3*b^5*sqrt(c)*d^2*f^3*g^4 + 2*a*b^3*c^(3/2)*d^2*f^3*g^4 - 24*a^2*b*c^(5/2)*d^2*f^3*g^4 - 30*a*b^4*sq
rt(c)*d*e*f^3*g^4 - 54*a^2*b^2*c^(3/2)*d*e*f^3*g^4 - 40*a^3*c^(5/2)*d*e*f^3*g^4 + 315*a^2*b^3*sqrt(c)*e^2*f^3*
g^4 + 292*a^3*b*c^(3/2)*e^2*f^3*g^4 + 6*a*b^4*sqrt(c)*d^2*f^2*g^5 - 18*a^2*b^2*c^(3/2)*d^2*f^2*g^5 + 8*a^3*c^(
5/2)*d^2*f^2*g^5 + 24*a^2*b^3*sqrt(c)*d*e*f^2*g^5 + 16*a^3*b*c^(3/2)*d*e*f^2*g^5 - 378*a^3*b^2*sqrt(c)*e^2*f^2
*g^5 - 88*a^4*c^(3/2)*e^2*f^2*g^5 - 3*a^2*b^3*sqrt(c)*d^2*f*g^6 + 28*a^3*b*c^(3/2)*d^2*f*g^6 - 6*a^3*b^2*sqrt(
c)*d*e*f*g^6 + 8*a^4*c^(3/2)*d*e*f*g^6 + 216*a^4*b*sqrt(c)*e^2*f*g^6 - 16*a^4*c^(3/2)*d^2*g^7 - 48*a^5*sqrt(c)
*e^2*g^7)/((c^2*e^3*f^7*g - 3*c^2*d*e^2*f^6*g^2 - 2*b*c*e^3*f^6*g^2 + 3*c^2*d^2*e*f^5*g^3 + 6*b*c*d*e^2*f^5*g^
3 + b^2*e^3*f^5*g^3 + 2*a*c*e^3*f^5*g^3 - c^2*d^3*f^4*g^4 - 6*b*c*d^2*e*f^4*g^4 - 3*b^2*d*e^2*f^4*g^4 - 6*a*c*
d*e^2*f^4*g^4 - 2*a*b*e^3*f^4*g^4 + 2*b*c*d^3*f^3*g^5 + 3*b^2*d^2*e*f^3*g^5 + 6*a*c*d^2*e*f^3*g^5 + 6*a*b*d*e^
2*f^3*g^5 + a^2*e^3*f^3*g^5 - b^2*d^3*f^2*g^6 - 2*a*c*d^3*f^2*g^6 - 6*a*b*d^2*e*f^2*g^6 - 3*a^2*d*e^2*f^2*g^6
+ 2*a*b*d^3*f*g^7 + 3*a^2*d^2*e*f*g^7 - a^2*d^3*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)^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x) (f+g x)^4} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (f+g\,x\right )}^4\,\left (d+e\,x\right )} \,d x \]

[In]

int((a + b*x + c*x^2)^(1/2)/((f + g*x)^4*(d + e*x)),x)

[Out]

int((a + b*x + c*x^2)^(1/2)/((f + g*x)^4*(d + e*x)), x)