\(\int \frac {\sqrt {d+e x}}{(a+b x+c x^2)^2} \, dx\) [2298]

Optimal result

Integrand size = 22, antiderivative size = 287 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {2} \sqrt {c} \left (4 c d-\left (2 b-\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \sqrt {c} \left (4 c d-\left (2 b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-(2*c*x+b)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)+arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+
b^2)^(1/2)))^(1/2))*2^(1/2)*c^(1/2)*(4*c*d-e*(2*b-(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(2*c*d-e*(b-(-4*a*c+
b^2)^(1/2)))^(1/2)-arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)*c^(1/
2)*(4*c*d-e*(2*b+(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {750, 840, 1180, 214} \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\sqrt {2} \sqrt {c} \left (4 c d-e \left (2 b-\sqrt {b^2-4 a c}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \sqrt {c} \left (4 c d-e \left (\sqrt {b^2-4 a c}+2 b\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[In]

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

[Out]

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

Rule 214

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

Rule 750

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

Rule 840

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

Rule 1180

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {-2 c d+\frac {b e}{2}-c e x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{-b^2+4 a c} \\ & = -\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 \text {Subst}\left (\int \frac {c d e+e \left (-2 c d+\frac {b e}{2}\right )-c e x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^2-4 a c} \\ & = -\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (c \left (4 c d-\left (2 b-\sqrt {b^2-4 a c}\right ) e\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (c \left (4 c d-\left (2 b+\sqrt {b^2-4 a c}\right ) e\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{\left (b^2-4 a c\right )^{3/2}} \\ & = -\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {2} \sqrt {c} \left (4 c d-\left (2 b-\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \sqrt {c} \left (4 c d-\left (2 b+\sqrt {b^2-4 a c}\right ) e\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.60 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {\frac {(b+2 c x) \sqrt {d+e x}}{a+x (b+c x)}+\frac {\sqrt {2} \sqrt {c} \left (-4 i c d+\left (2 i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (4 i c d+\left (-2 i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{b^2-4 a c} \]

[In]

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

[Out]

-((((b + 2*c*x)*Sqrt[d + e*x])/(a + x*(b + c*x)) + (Sqrt[2]*Sqrt[c]*((-4*I)*c*d + ((2*I)*b + Sqrt[-b^2 + 4*a*c
])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]
*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*((4*I)*c*d + ((-2*I)*b + Sqrt[-b^2 + 4*a*c])*
e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sq
rt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]))/(b^2 - 4*a*c))

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(32 e^{3} c^{2} \left (\frac {-\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{8 c \left (-e x -\frac {b e}{2 c}+\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}-\frac {\left (-b e +2 c d +\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 c \,e^{2} \left (4 a c -b^{2}\right ) \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}+\frac {-\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{8 c \left (-e x -\frac {b e}{2 c}-\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}+\frac {\left (b e -2 c d +\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 c \,e^{2} \left (4 a c -b^{2}\right ) \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}\right )\)	\(419\)
default	\(32 e^{3} c^{2} \left (\frac {-\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{8 c \left (-e x -\frac {b e}{2 c}+\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}-\frac {\left (-b e +2 c d +\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 c \,e^{2} \left (4 a c -b^{2}\right ) \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}+\frac {-\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{8 c \left (-e x -\frac {b e}{2 c}-\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}+\frac {\left (b e -2 c d +\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 c \,e^{2} \left (4 a c -b^{2}\right ) \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}\right )\)	\(419\)
pseudoelliptic	\(-\frac {e \left (\frac {\left (-e^{2} \left (2 c x +b \right )^{2} \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}-16 c^{3} d \,e^{2} x^{2}-16 \left (-\frac {b e \,x^{2}}{2}+d \left (b x +a \right )\right ) e^{2} c^{2}+8 b \,e^{3} \left (b x +a \right ) c +\left (-e^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{2}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )}{4}+\sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (-\frac {\sqrt {2}\, \left (-e^{2} \left (2 c x +b \right )^{2} \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+16 c^{3} d \,e^{2} x^{2}+16 \left (-\frac {b e \,x^{2}}{2}+d \left (b x +a \right )\right ) e^{2} c^{2}-8 b \,e^{3} \left (b x +a \right ) c +\left (-e^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}}\right )}{4}+\sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, e \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {e x +d}\, \left (2 c x +b \right )\right )\right ) c}{\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}\right ) c}\, \left (a c -\frac {b^{2}}{4}\right ) \left (\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}+e \left (2 c x +b \right )\right ) \left (\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) e^{2}}-e \left (2 c x +b \right )\right )}\)	\(544\)

[In]

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

[Out]

32*e^3*c^2*(1/4/c/e^2/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*(-1/8/c*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)/(-
e*x-1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))-1/4*(-b*e+2*c*d+1/2*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2
*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))
*c)^(1/2)))+1/4/c/e^2/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*(-1/8/c*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)/(-
e*x-1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))+1/4*(b*e-2*c*d+1/2*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c
*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^
(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6393 vs. \(2 (241) = 482\).

Time = 0.45 (sec) , antiderivative size = 6393, normalized size of antiderivative = 22.28 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(d + e*x)/(a + b*x + c*x**2)**2, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (241) = 482\).

Time = 0.53 (sec) , antiderivative size = 1005, normalized size of antiderivative = 3.50 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c e - 2 \, \sqrt {e x + d} c d e + \sqrt {e x + d} b e^{2}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e + a e^{2}\right )} {\left (b^{2} - 4 \, a c\right )}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} e - 4 \, a c e\right )}^{2} e + {\left (2 \, \sqrt {b^{2} - 4 \, a c} c d e - \sqrt {b^{2} - 4 \, a c} b e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | b^{2} e - 4 \, a c e \right |} - 2 \, {\left (4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 4 \, a b^{2} c\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e + \sqrt {{\left (2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e\right )}^{2} - 4 \, {\left (b^{2} c d^{2} - 4 \, a c^{2} d^{2} - b^{3} d e + 4 \, a b c d e + a b^{2} e^{2} - 4 \, a^{2} c e^{2}\right )} {\left (b^{2} c - 4 \, a c^{2}\right )}}}{b^{2} c - 4 \, a c^{2}}}}\right )}{4 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} - 4 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} {\left | b^{2} e - 4 \, a c e \right |} {\left | c \right |}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} e - 4 \, a c e\right )}^{2} e - {\left (2 \, \sqrt {b^{2} - 4 \, a c} c d e - \sqrt {b^{2} - 4 \, a c} b e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | b^{2} e - 4 \, a c e \right |} - 2 \, {\left (4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 4 \, a b^{2} c\right )} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e - \sqrt {{\left (2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e\right )}^{2} - 4 \, {\left (b^{2} c d^{2} - 4 \, a c^{2} d^{2} - b^{3} d e + 4 \, a b c d e + a b^{2} e^{2} - 4 \, a^{2} c e^{2}\right )} {\left (b^{2} c - 4 \, a c^{2}\right )}}}{b^{2} c - 4 \, a c^{2}}}}\right )}{4 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} - 4 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} {\left | b^{2} e - 4 \, a c e \right |} {\left | c \right |}} \]

[In]

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

[Out]

-(2*(e*x + d)^(3/2)*c*e - 2*sqrt(e*x + d)*c*d*e + sqrt(e*x + d)*b*e^2)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d
^2 + (e*x + d)*b*e - b*d*e + a*e^2)*(b^2 - 4*a*c)) - 1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^
2*e - 4*a*c*e)^2*e + (2*sqrt(b^2 - 4*a*c)*c*d*e - sqrt(b^2 - 4*a*c)*b*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 -
4*a*c)*c)*e)*abs(b^2*e - 4*a*c*e) - 2*(4*(b^2*c^2 - 4*a*c^3)*d^2*e - 4*(b^3*c - 4*a*b*c^2)*d*e^2 + (b^4 - 4*a
*b^2*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^2*c
*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e + sqrt((2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c
^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*e^2)*(b^2*c - 4*a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c - 4
*a*c^2)*sqrt(b^2 - 4*a*c)*d^2 - (b^3 - 4*a*b*c)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2 - 4*a^2*c)*sqrt(b^2 - 4*a*c)*e^
2)*abs(b^2*e - 4*a*c*e)*abs(c)) + 1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(b^2*e - 4*a*c*e)^2*e
- (2*sqrt(b^2 - 4*a*c)*c*d*e - sqrt(b^2 - 4*a*c)*b*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(b
^2*e - 4*a*c*e) - 2*(4*(b^2*c^2 - 4*a*c^3)*d^2*e - 4*(b^3*c - 4*a*b*c^2)*d*e^2 + (b^4 - 4*a*b^2*c)*e^3)*sqrt(-
4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^2*c*d - 8*a*c^2*d - b^
3*e + 4*a*b*c*e - sqrt((2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e +
4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*e^2)*(b^2*c - 4*a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c - 4*a*c^2)*sqrt(b^2 -
4*a*c)*d^2 - (b^3 - 4*a*b*c)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2 - 4*a^2*c)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*
c*e)*abs(c))

Mupad [B] (verification not implemented)

Time = 16.60 (sec) , antiderivative size = 5740, normalized size of antiderivative = 20.00 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*c*e*(d + e*x)^(3/2))/(4*a*c - b^2) + (e*(b*e - 2*c*d)*(d + e*x)^(1/2))/(4*a*c - b^2))/((b*e - 2*c*d)*(d +
e*x) + c*(d + e*x)^2 + a*e^2 + c*d^2 - b*d*e) - log(((4*c^2*e^3*(b*e - 2*c*d) - 8*c^2*e^2*(4*a*c - b^2)*(b*e -
2*c*d)*(d + e*x)^(1/2)*(((e^3*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 4
8*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^
2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e
- 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(((e^3*(-(4
*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3
- 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8
*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b
^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2) + (4*c^3*e^2*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^
2*d^2 - 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^2)*(((e^3*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c
^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b
^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d
*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b
*d*e)))^(1/2) - (2*c^3*e^3*(3*b^2*e^2 + 16*c^2*d^2 + 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^3)*(((e^3*(-(4*a*c
- b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 19
2*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d
*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c
^3*d*e^2)/(a*b^12*e^2 + b^12*c*d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a
^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240
*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 2
40*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e))^(1
/2) - log(((4*c^2*e^3*(b*e - 2*c*d) - 8*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-((b^9*e^3)/8 + (
e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*
a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e
^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3
*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 +
256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e -
192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*
b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 +
c*d^2 - b*d*e)))^(1/2) + (4*c^3*e^2*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c
- b^2)^2)*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 256*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d
^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 192*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^
3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^
3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2) - (2*c^3*e^3*(3*b^2*e^2 +
16*c^2*d^2 + 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^3)*(-((b^9*e^3)/8 + (e^3*(-(4*a*c - b^2)^9)^(1/2))/8 + 25
6*a^3*c^6*d^3 - 4*b^6*c^3*d^3 + 48*a*b^4*c^4*d^3 - 96*a^4*b*c^4*e^3 + 192*a^4*c^5*d*e^2 + 6*b^7*c^2*d^2*e - 19
2*a^2*b^2*c^5*d^3 - 12*a^2*b^5*c^2*e^3 + 64*a^3*b^3*c^3*e^3 - (9*b^8*c*d*e^2)/4 - 72*a*b^5*c^3*d^2*e + 24*a*b^
6*c^2*d*e^2 - 384*a^3*b*c^5*d^2*e + 288*a^2*b^3*c^4*d^2*e - 72*a^2*b^4*c^3*d*e^2)/(a*b^12*e^2 + b^12*c*d^2 + 4
096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d^2 -
1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c^3*e^
2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7*c^3*
d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e))^(1/2) + log((2^(1/2)*((2^(1/2)*(4*c^2*e^
3*(b*e - 2*c*d) + 2*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^9*e^3 + e^3*(-(4*a*c - b^
2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 +
48*b^7*c^2*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b
^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/((
4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 -
32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2
*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e
^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 -
b*d*e)))^(1/2))/4 - (4*c^3*e^2*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2
)^2)*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a
^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3
*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^
4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))/4 - (2*c^3*e^3*(3*b^2*e^2 +
16*c^2*d^2 + 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^3)*(-(b^9*e^3 + e^3*(-(4*a*c - b^2)^9)^(1/2) + 2048*a^3*c
^6*d^3 - 32*b^6*c^3*d^3 + 384*a*b^4*c^4*d^3 - 768*a^4*b*c^4*e^3 + 1536*a^4*c^5*d*e^2 + 48*b^7*c^2*d^2*e - 1536
*a^2*b^2*c^5*d^3 - 96*a^2*b^5*c^2*e^3 + 512*a^3*b^3*c^3*e^3 - 18*b^8*c*d*e^2 - 576*a*b^5*c^3*d^2*e + 192*a*b^6
*c^2*d*e^2 - 3072*a^3*b*c^5*d^2*e + 2304*a^2*b^3*c^4*d^2*e - 576*a^2*b^4*c^3*d*e^2)/(8*(a*b^12*e^2 + b^12*c*d^
2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3*d
^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6*c
^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b^7
*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2) + log((2^(1/2)*((2^(1/2)*(4*
c^2*e^3*(b*e - 2*c*d) + 2*2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*((e^3*(-(4*a*c - b^2)^9)
^(1/2) - b^9*e^3 - 2048*a^3*c^6*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*
e^2 - 48*b^7*c^2*d^2*e + 1536*a^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 57
6*a*b^5*c^3*d^2*e - 192*a*b^6*c^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^
2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3*c^6*d
^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 1536*a^2
*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b^6*c^2
*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^
2 - b*d*e)))^(1/2))/4 - (4*c^3*e^2*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c -
b^2)^2)*((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3*c^6*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 76
8*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 1536*a^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*
b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b^6*c^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3
*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))/4 - (2*c^3*e^3*(3*b^2*e^
2 + 16*c^2*d^2 + 4*a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^3)*((e^3*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^3 - 2048*a^3
*c^6*d^3 + 32*b^6*c^3*d^3 - 384*a*b^4*c^4*d^3 + 768*a^4*b*c^4*e^3 - 1536*a^4*c^5*d*e^2 - 48*b^7*c^2*d^2*e + 15
36*a^2*b^2*c^5*d^3 + 96*a^2*b^5*c^2*e^3 - 512*a^3*b^3*c^3*e^3 + 18*b^8*c*d*e^2 + 576*a*b^5*c^3*d^2*e - 192*a*b
^6*c^2*d*e^2 + 3072*a^3*b*c^5*d^2*e - 2304*a^2*b^3*c^4*d^2*e + 576*a^2*b^4*c^3*d*e^2)/(8*(a*b^12*e^2 + b^12*c*
d^2 + 4096*a^6*c^7*d^2 + 4096*a^7*c^6*e^2 - b^13*d*e - 24*a*b^10*c^2*d^2 - 24*a^2*b^10*c*e^2 + 240*a^2*b^8*c^3
*d^2 - 1280*a^3*b^6*c^4*d^2 + 3840*a^4*b^4*c^5*d^2 - 6144*a^5*b^2*c^6*d^2 + 240*a^3*b^8*c^2*e^2 - 1280*a^4*b^6
*c^3*e^2 + 3840*a^5*b^4*c^4*e^2 - 6144*a^6*b^2*c^5*e^2 - 4096*a^6*b*c^6*d*e - 240*a^2*b^9*c^2*d*e + 1280*a^3*b
^7*c^3*d*e - 3840*a^4*b^5*c^4*d*e + 6144*a^5*b^3*c^5*d*e + 24*a*b^11*c*d*e)))^(1/2)