\(\int \frac {1}{(a+\frac {c}{x^2}+\frac {b}{x}) (d+e x)^2} \, dx\) [73]

Optimal result

Integrand size = 22, antiderivative size = 194 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=-\frac {d^2}{e \left (a d^2-b d e+c e^2\right ) (d+e x)}-\frac {\left (b^2 d^2-2 b c d e-2 c \left (a d^2-c e^2\right )\right ) \text {arctanh}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {d (b d-2 c e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2}-\frac {d (b d-2 c e) \log \left (c+b x+a x^2\right )}{2 \left (a d^2-e (b d-c e)\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1459, 1642, 648, 632, 212, 642} \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=-\frac {\text {arctanh}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right ) \left (-2 c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right )}{\sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac {d^2}{e (d+e x) \left (a d^2-b d e+c e^2\right )}-\frac {d (b d-2 c e) \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac {d (b d-2 c e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2} \]

[In]

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

[Out]

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

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 632

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

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 1459

Int[((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Int[(
(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p)/x^(2*n*p), x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[mn, -n] && Eq
Q[mn2, 2*mn] && IntegerQ[p]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=\frac {-\frac {2 d^2 \left (a d^2+e (-b d+c e)\right )}{e (d+e x)}+\frac {2 \left (b^2 d^2-2 b c d e+2 c \left (-a d^2+c e^2\right )\right ) \arctan \left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+2 d (b d-2 c e) \log (d+e x)-d (b d-2 c e) \log (c+x (b+a x))}{2 \left (a d^2+e (-b d+c e)\right )^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {\frac {\left (-a b \,d^{2}+2 a c d e \right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (-d^{2} a c +e^{2} c^{2}-\frac {\left (-a b \,d^{2}+2 a c d e \right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-b d e +c \,e^{2}\right )^{2}}-\frac {d^{2}}{e \left (a \,d^{2}-b d e +c \,e^{2}\right ) \left (e x +d \right )}+\frac {d \left (b d -2 e c \right ) \ln \left (e x +d \right )}{\left (a \,d^{2}-b d e +c \,e^{2}\right )^{2}}\)	\(188\)
risch	\(\text {Expression too large to display}\)	\(14628\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (188) = 376\).

Time = 3.64 (sec) , antiderivative size = 1120, normalized size of antiderivative = 5.77 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/2*(2*(a*b^2 - 4*a^2*c)*d^4 - 2*(b^3 - 4*a*b*c)*d^3*e + 2*(b^2*c - 4*a*c^2)*d^2*e^2 + (2*b*c*d^2*e^2 - 2*c^
2*d*e^3 - (b^2 - 2*a*c)*d^3*e + (2*b*c*d*e^3 - 2*c^2*e^4 - (b^2 - 2*a*c)*d^2*e^2)*x)*sqrt(b^2 - 4*a*c)*log((2*
a^2*x^2 + 2*a*b*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*a*x + b))/(a*x^2 + b*x + c)) + ((b^3 - 4*a*b*c)*d^3*e -
2*(b^2*c - 4*a*c^2)*d^2*e^2 + ((b^3 - 4*a*b*c)*d^2*e^2 - 2*(b^2*c - 4*a*c^2)*d*e^3)*x)*log(a*x^2 + b*x + c) -
2*((b^3 - 4*a*b*c)*d^3*e - 2*(b^2*c - 4*a*c^2)*d^2*e^2 + ((b^3 - 4*a*b*c)*d^2*e^2 - 2*(b^2*c - 4*a*c^2)*d*e^3
)*x)*log(e*x + d))/((a^2*b^2 - 4*a^3*c)*d^5*e - 2*(a*b^3 - 4*a^2*b*c)*d^4*e^2 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*
d^3*e^3 - 2*(b^3*c - 4*a*b*c^2)*d^2*e^4 + (b^2*c^2 - 4*a*c^3)*d*e^5 + ((a^2*b^2 - 4*a^3*c)*d^4*e^2 - 2*(a*b^3
- 4*a^2*b*c)*d^3*e^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^4 - 2*(b^3*c - 4*a*b*c^2)*d*e^5 + (b^2*c^2 - 4*a*c^
3)*e^6)*x), -1/2*(2*(a*b^2 - 4*a^2*c)*d^4 - 2*(b^3 - 4*a*b*c)*d^3*e + 2*(b^2*c - 4*a*c^2)*d^2*e^2 - 2*(2*b*c*d
^2*e^2 - 2*c^2*d*e^3 - (b^2 - 2*a*c)*d^3*e + (2*b*c*d*e^3 - 2*c^2*e^4 - (b^2 - 2*a*c)*d^2*e^2)*x)*sqrt(-b^2 +
4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*a*x + b)/(b^2 - 4*a*c)) + ((b^3 - 4*a*b*c)*d^3*e - 2*(b^2*c - 4*a*c^2)*d^
2*e^2 + ((b^3 - 4*a*b*c)*d^2*e^2 - 2*(b^2*c - 4*a*c^2)*d*e^3)*x)*log(a*x^2 + b*x + c) - 2*((b^3 - 4*a*b*c)*d^3
*e - 2*(b^2*c - 4*a*c^2)*d^2*e^2 + ((b^3 - 4*a*b*c)*d^2*e^2 - 2*(b^2*c - 4*a*c^2)*d*e^3)*x)*log(e*x + d))/((a^
2*b^2 - 4*a^3*c)*d^5*e - 2*(a*b^3 - 4*a^2*b*c)*d^4*e^2 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^3*e^3 - 2*(b^3*c - 4*
a*b*c^2)*d^2*e^4 + (b^2*c^2 - 4*a*c^3)*d*e^5 + ((a^2*b^2 - 4*a^3*c)*d^4*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d^3*e^3 +
(b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^4 - 2*(b^3*c - 4*a*b*c^2)*d*e^5 + (b^2*c^2 - 4*a*c^3)*e^6)*x)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=-\frac {d^{2} e}{{\left (a d^{2} e^{2} - b d e^{3} + c e^{4}\right )} {\left (e x + d\right )}} - \frac {{\left (b d^{2} - 2 \, c d e\right )} \log \left (-a + \frac {2 \, a d}{e x + d} - \frac {a d^{2}}{{\left (e x + d\right )}^{2}} - \frac {b e}{e x + d} + \frac {b d e}{{\left (e x + d\right )}^{2}} - \frac {c e^{2}}{{\left (e x + d\right )}^{2}}\right )}{2 \, {\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + c^{2} e^{4}\right )}} - \frac {{\left (b^{2} d^{2} e^{2} - 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + 2 \, c^{2} e^{4}\right )} \arctan \left (-\frac {2 \, a d - \frac {2 \, a d^{2}}{e x + d} - b e + \frac {2 \, b d e}{e x + d} - \frac {2 \, c e^{2}}{e x + d}}{\sqrt {-b^{2} + 4 \, a c} e}\right )}{{\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + c^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c} e^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.25 (sec) , antiderivative size = 1585, normalized size of antiderivative = 8.17 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(log(2*a*b^3*d^4 + b*c^3*e^4 - c^3*e^4*(b^2 - 4*a*c)^(1/2) + 16*a^2*c^2*d^3*e + 2*b^2*c^2*d*e^3 - b^3*c*d^2*e^
2 + a^2*b^2*d^4*x + b^2*c^2*e^4*x - b^4*d^2*e^2*x - 7*a^2*b*c*d^4 - 16*a*c^3*d*e^3 - 2*a^3*c*d^4*x - 2*a*c^3*e
^4*x + 2*a*b^2*d^4*(b^2 - 4*a*c)^(1/2) - a^2*c*d^4*(b^2 - 4*a*c)^(1/2) - 6*a*b^2*c*d^3*e + 2*a*b^3*d^3*e*x + 2
*b^3*c*d*e^3*x - 2*b*c^2*d*e^3*(b^2 - 4*a*c)^(1/2) + 3*a^2*b*d^4*x*(b^2 - 4*a*c)^(1/2) - b*c^2*e^4*x*(b^2 - 4*
a*c)^(1/2) + 10*a*b*c^2*d^2*e^2 + 14*a*c^2*d^2*e^2*(b^2 - 4*a*c)^(1/2) + b^2*c*d^2*e^2*(b^2 - 4*a*c)^(1/2) + b
^3*d^2*e^2*x*(b^2 - 4*a*c)^(1/2) + 28*a^2*c^2*d^2*e^2*x - 10*a*b*c*d^3*e*(b^2 - 4*a*c)^(1/2) - 12*a*b*c^2*d*e^
3*x - 12*a^2*b*c*d^3*e*x - 2*a*b^2*d^3*e*x*(b^2 - 4*a*c)^(1/2) + 8*a*c^2*d*e^3*x*(b^2 - 4*a*c)^(1/2) - 8*a^2*c
*d^3*e*x*(b^2 - 4*a*c)^(1/2) - 2*b^2*c*d*e^3*x*(b^2 - 4*a*c)^(1/2) + 2*a*b*c*d^2*e^2*x*(b^2 - 4*a*c)^(1/2))*(d
^2*(b^3/2 + (b^2*(b^2 - 4*a*c)^(1/2))/2) - c*(d^2*(2*a*b + a*(b^2 - 4*a*c)^(1/2)) + d*(b^2*e + b*e*(b^2 - 4*a*
c)^(1/2))) + c^2*(e^2*(b^2 - 4*a*c)^(1/2) + 4*a*d*e)))/(4*a^3*c*d^4 + 4*a*c^3*e^4 - a^2*b^2*d^4 - b^2*c^2*e^4
- b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d^3*e + 2*b^3*c*d*e^3 - 8*a*b*c^2*d*e^3 - 8*a^2*b*c*d^3*e + 2*a*b^
2*c*d^2*e^2) - (log(2*a*b^3*d^4 + b*c^3*e^4 + c^3*e^4*(b^2 - 4*a*c)^(1/2) + 16*a^2*c^2*d^3*e + 2*b^2*c^2*d*e^3
- b^3*c*d^2*e^2 + a^2*b^2*d^4*x + b^2*c^2*e^4*x - b^4*d^2*e^2*x - 7*a^2*b*c*d^4 - 16*a*c^3*d*e^3 - 2*a^3*c*d^
4*x - 2*a*c^3*e^4*x - 2*a*b^2*d^4*(b^2 - 4*a*c)^(1/2) + a^2*c*d^4*(b^2 - 4*a*c)^(1/2) - 6*a*b^2*c*d^3*e + 2*a*
b^3*d^3*e*x + 2*b^3*c*d*e^3*x + 2*b*c^2*d*e^3*(b^2 - 4*a*c)^(1/2) - 3*a^2*b*d^4*x*(b^2 - 4*a*c)^(1/2) + b*c^2*
e^4*x*(b^2 - 4*a*c)^(1/2) + 10*a*b*c^2*d^2*e^2 - 14*a*c^2*d^2*e^2*(b^2 - 4*a*c)^(1/2) - b^2*c*d^2*e^2*(b^2 - 4
*a*c)^(1/2) - b^3*d^2*e^2*x*(b^2 - 4*a*c)^(1/2) + 28*a^2*c^2*d^2*e^2*x + 10*a*b*c*d^3*e*(b^2 - 4*a*c)^(1/2) -
12*a*b*c^2*d*e^3*x - 12*a^2*b*c*d^3*e*x + 2*a*b^2*d^3*e*x*(b^2 - 4*a*c)^(1/2) - 8*a*c^2*d*e^3*x*(b^2 - 4*a*c)^
(1/2) + 8*a^2*c*d^3*e*x*(b^2 - 4*a*c)^(1/2) + 2*b^2*c*d*e^3*x*(b^2 - 4*a*c)^(1/2) - 2*a*b*c*d^2*e^2*x*(b^2 - 4
*a*c)^(1/2))*(c*(d^2*(2*a*b - a*(b^2 - 4*a*c)^(1/2)) + d*(b^2*e - b*e*(b^2 - 4*a*c)^(1/2))) - d^2*(b^3/2 - (b^
2*(b^2 - 4*a*c)^(1/2))/2) + c^2*(e^2*(b^2 - 4*a*c)^(1/2) - 4*a*d*e)))/(4*a^3*c*d^4 + 4*a*c^3*e^4 - a^2*b^2*d^4
- b^2*c^2*e^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d^3*e + 2*b^3*c*d*e^3 - 8*a*b*c^2*d*e^3 - 8*a^2*b*c
*d^3*e + 2*a*b^2*c*d^2*e^2) + (log(d + e*x)*(b*d^2 - 2*c*d*e))/(a^2*d^4 + c^2*e^4 + b^2*d^2*e^2 - 2*a*b*d^3*e
- 2*b*c*d*e^3 + 2*a*c*d^2*e^2) - d^2/(e*(d + e*x)*(a*d^2 + c*e^2 - b*d*e))