\(\int \frac {x^2 (a+b x^2+c x^4)}{\sqrt {d-e x} \sqrt {d+e x}} \, dx\) [139]

Optimal result

Integrand size = 35, antiderivative size = 216 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \sqrt {d-e x} \sqrt {d+e x}}{16 e^6}-\frac {\left (5 c d^2+6 b e^2\right ) x^3 \sqrt {d-e x} \sqrt {d+e x}}{24 e^4}+\frac {c x^5 (-d+e x) \sqrt {d+e x}}{6 e^2 \sqrt {d-e x}}+\frac {d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt {d^2-e^2 x^2} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^7 \sqrt {d-e x} \sqrt {d+e x}} \]

[Out]

1/6*c*x^5*(e*x-d)*(e*x+d)^(1/2)/e^2/(-e*x+d)^(1/2)-1/16*(8*a*e^4+6*b*d^2*e^2+5*c*d^4)*x*(-e*x+d)^(1/2)*(e*x+d)
^(1/2)/e^6-1/24*(6*b*e^2+5*c*d^2)*x^3*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/e^4+1/16*d^2*(8*a*e^4+6*b*d^2*e^2+5*c*d^4)*
arctan(e*x/(-e^2*x^2+d^2)^(1/2))*(-e^2*x^2+d^2)^(1/2)/e^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {534, 1281, 470, 327, 223, 209} \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {d^2 \sqrt {d^2-e^2 x^2} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^7 \sqrt {d-e x} \sqrt {d+e x}}-\frac {x \left (d^2-e^2 x^2\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^6 \sqrt {d-e x} \sqrt {d+e x}}-\frac {x^3 \left (d^2-e^2 x^2\right ) \left (6 b e^2+5 c d^2\right )}{24 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt {d-e x} \sqrt {d+e x}} \]

[In]

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

[Out]

-1/16*((5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4)*x*(d^2 - e^2*x^2))/(e^6*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((5*c*d^2 + 6*
b*e^2)*x^3*(d^2 - e^2*x^2))/(24*e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - (c*x^5*(d^2 - e^2*x^2))/(6*e^2*Sqrt[d - e*x
]*Sqrt[d + e*x]) + (d^2*(5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]
])/(16*e^7*Sqrt[d - e*x]*Sqrt[d + e*x])

Rule 209

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

Rule 223

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

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 470

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

Rule 534

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*
a2 + b1*b2*x^n)^FracPart[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1281

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d^2-e^2 x^2} \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {d^2-e^2 x^2} \int \frac {x^2 \left (-6 a e^2-\left (5 c d^2+6 b e^2\right ) x^2\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{6 e^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt {d^2-e^2 x^2}\right ) \int \frac {x^2}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^4 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \left (d^2-e^2 x^2\right )}{16 e^6 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt {d^2-e^2 x^2}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{16 e^6 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \left (d^2-e^2 x^2\right )}{16 e^6 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt {d^2-e^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^6 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \left (d^2-e^2 x^2\right )}{16 e^6 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^7 \sqrt {d-e x} \sqrt {d+e x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.62 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {-e x \sqrt {d-e x} \sqrt {d+e x} \left (6 \left (3 b d^2 e^2+4 a e^4+2 b e^4 x^2\right )+c \left (15 d^4+10 d^2 e^2 x^2+8 e^4 x^4\right )\right )+6 d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \arctan \left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{48 e^7} \]

[In]

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

[Out]

(-(e*x*Sqrt[d - e*x]*Sqrt[d + e*x]*(6*(3*b*d^2*e^2 + 4*a*e^4 + 2*b*e^4*x^2) + c*(15*d^4 + 10*d^2*e^2*x^2 + 8*e
^4*x^4))) + 6*d^2*(5*c*d^4 + 6*b*d^2*e^2 + 8*a*e^4)*ArcTan[Sqrt[d + e*x]/Sqrt[d - e*x]])/(48*e^7)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.75

method	result	size
risch	\(-\frac {x \left (8 c \,x^{4} e^{4}+12 b \,e^{4} x^{2}+10 c \,d^{2} e^{2} x^{2}+24 e^{4} a +18 e^{2} d^{2} b +15 d^{4} c \right ) \sqrt {-e x +d}\, \sqrt {e x +d}}{48 e^{6}}+\frac {d^{2} \left (8 e^{4} a +6 e^{2} d^{2} b +5 d^{4} c \right ) \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{16 e^{6} \sqrt {e^{2}}\, \sqrt {e x +d}\, \sqrt {-e x +d}}\)	\(161\)
default	\(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (8 \,\operatorname {csgn}\left (e \right ) c \,e^{5} x^{5} \sqrt {-e^{2} x^{2}+d^{2}}+12 \,\operatorname {csgn}\left (e \right ) b \,e^{5} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}+10 \,\operatorname {csgn}\left (e \right ) c \,d^{2} e^{3} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}+24 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (e \right ) e^{5} a x +18 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (e \right ) e^{3} b \,d^{2} x +15 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (e \right ) e c \,d^{4} x -24 \arctan \left (\frac {\operatorname {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) a \,d^{2} e^{4}-18 \arctan \left (\frac {\operatorname {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) b \,d^{4} e^{2}-15 \arctan \left (\frac {\operatorname {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) c \,d^{6}\right ) \operatorname {csgn}\left (e \right )}{48 e^{7} \sqrt {-e^{2} x^{2}+d^{2}}}\)	\(273\)

[In]

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

[Out]

-1/48*x*(8*c*e^4*x^4+12*b*e^4*x^2+10*c*d^2*e^2*x^2+24*a*e^4+18*b*d^2*e^2+15*c*d^4)/e^6*(-e*x+d)^(1/2)*(e*x+d)^
(1/2)+1/16*d^2*(8*a*e^4+6*b*d^2*e^2+5*c*d^4)/e^6/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))*((e*x+
d)*(-e*x+d))^(1/2)/(e*x+d)^(1/2)/(-e*x+d)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.62 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (8 \, c e^{5} x^{5} + 2 \, {\left (5 \, c d^{2} e^{3} + 6 \, b e^{5}\right )} x^{3} + 3 \, {\left (5 \, c d^{4} e + 6 \, b d^{2} e^{3} + 8 \, a e^{5}\right )} x\right )} \sqrt {e x + d} \sqrt {-e x + d} + 6 \, {\left (5 \, c d^{6} + 6 \, b d^{4} e^{2} + 8 \, a d^{2} e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{e x}\right )}{48 \, e^{7}} \]

[In]

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

[Out]

-1/48*((8*c*e^5*x^5 + 2*(5*c*d^2*e^3 + 6*b*e^5)*x^3 + 3*(5*c*d^4*e + 6*b*d^2*e^3 + 8*a*e^5)*x)*sqrt(e*x + d)*s
qrt(-e*x + d) + 6*(5*c*d^6 + 6*b*d^4*e^2 + 8*a*d^2*e^4)*arctan((sqrt(e*x + d)*sqrt(-e*x + d) - d)/(e*x)))/e^7

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.05 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{5}}{6 \, e^{2}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2} x^{3}}{24 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b x^{3}}{4 \, e^{2}} + \frac {5 \, c d^{6} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{16 \, \sqrt {e^{2}} e^{6}} + \frac {3 \, b d^{4} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{8 \, \sqrt {e^{2}} e^{4}} + \frac {a d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}} e^{2}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{4} x}{16 \, e^{6}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{2} x}{8 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a x}{2 \, e^{2}} \]

[In]

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

[Out]

-1/6*sqrt(-e^2*x^2 + d^2)*c*x^5/e^2 - 5/24*sqrt(-e^2*x^2 + d^2)*c*d^2*x^3/e^4 - 1/4*sqrt(-e^2*x^2 + d^2)*b*x^3
/e^2 + 5/16*c*d^6*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^6) + 3/8*b*d^4*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^
2)*e^4) + 1/2*a*d^2*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^2) - 5/16*sqrt(-e^2*x^2 + d^2)*c*d^4*x/e^6 - 3/8*
sqrt(-e^2*x^2 + d^2)*b*d^2*x/e^4 - 1/2*sqrt(-e^2*x^2 + d^2)*a*x/e^2

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {{\left (33 \, c d^{5} + 30 \, b d^{3} e^{2} + 24 \, a d e^{4} - {\left (85 \, c d^{4} + 54 \, b d^{2} e^{2} + 24 \, a e^{4} - 2 \, {\left (55 \, c d^{3} + 18 \, b d e^{2} - {\left (45 \, c d^{2} + 6 \, b e^{2} + 4 \, {\left ({\left (e x + d\right )} c - 5 \, c d\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} \sqrt {e x + d} \sqrt {-e x + d} + 6 \, {\left (5 \, c d^{6} + 6 \, b d^{4} e^{2} + 8 \, a d^{2} e^{4}\right )} \arcsin \left (\frac {\sqrt {2} \sqrt {e x + d}}{2 \, \sqrt {d}}\right )}{48 \, e^{7}} \]

[In]

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

[Out]

1/48*((33*c*d^5 + 30*b*d^3*e^2 + 24*a*d*e^4 - (85*c*d^4 + 54*b*d^2*e^2 + 24*a*e^4 - 2*(55*c*d^3 + 18*b*d*e^2 -
(45*c*d^2 + 6*b*e^2 + 4*((e*x + d)*c - 5*c*d)*(e*x + d))*(e*x + d))*(e*x + d))*(e*x + d))*sqrt(e*x + d)*sqrt(
-e*x + d) + 6*(5*c*d^6 + 6*b*d^4*e^2 + 8*a*d^2*e^4)*arcsin(1/2*sqrt(2)*sqrt(e*x + d)/sqrt(d)))/e^7

Mupad [B] (verification not implemented)

Time = 27.91 (sec) , antiderivative size = 1132, normalized size of antiderivative = 5.24 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((14*a*d^2*((d + e*x)^(1/2) - d^(1/2))^3)/((d - e*x)^(1/2) - d^(1/2))^3 - (14*a*d^2*((d + e*x)^(1/2) - d^(1/2)
)^5)/((d - e*x)^(1/2) - d^(1/2))^5 + (2*a*d^2*((d + e*x)^(1/2) - d^(1/2))^7)/((d - e*x)^(1/2) - d^(1/2))^7 - (
2*a*d^2*((d + e*x)^(1/2) - d^(1/2)))/((d - e*x)^(1/2) - d^(1/2)))/(e^3*(((d + e*x)^(1/2) - d^(1/2))^2/((d - e*
x)^(1/2) - d^(1/2))^2 + 1)^4) - ((175*c*d^6*((d + e*x)^(1/2) - d^(1/2))^3)/(12*((d - e*x)^(1/2) - d^(1/2))^3)
+ (311*c*d^6*((d + e*x)^(1/2) - d^(1/2))^5)/(4*((d - e*x)^(1/2) - d^(1/2))^5) - (8361*c*d^6*((d + e*x)^(1/2) -
d^(1/2))^7)/(4*((d - e*x)^(1/2) - d^(1/2))^7) + (42259*c*d^6*((d + e*x)^(1/2) - d^(1/2))^9)/(6*((d - e*x)^(1/
2) - d^(1/2))^9) - (25295*c*d^6*((d + e*x)^(1/2) - d^(1/2))^11)/(2*((d - e*x)^(1/2) - d^(1/2))^11) + (25295*c*
d^6*((d + e*x)^(1/2) - d^(1/2))^13)/(2*((d - e*x)^(1/2) - d^(1/2))^13) - (42259*c*d^6*((d + e*x)^(1/2) - d^(1/
2))^15)/(6*((d - e*x)^(1/2) - d^(1/2))^15) + (8361*c*d^6*((d + e*x)^(1/2) - d^(1/2))^17)/(4*((d - e*x)^(1/2) -
d^(1/2))^17) - (311*c*d^6*((d + e*x)^(1/2) - d^(1/2))^19)/(4*((d - e*x)^(1/2) - d^(1/2))^19) - (175*c*d^6*((d
+ e*x)^(1/2) - d^(1/2))^21)/(12*((d - e*x)^(1/2) - d^(1/2))^21) - (5*c*d^6*((d + e*x)^(1/2) - d^(1/2))^23)/(4
*((d - e*x)^(1/2) - d^(1/2))^23) + (5*c*d^6*((d + e*x)^(1/2) - d^(1/2)))/(4*((d - e*x)^(1/2) - d^(1/2))))/(e^7
*(((d + e*x)^(1/2) - d^(1/2))^2/((d - e*x)^(1/2) - d^(1/2))^2 + 1)^12) - ((23*b*d^4*((d + e*x)^(1/2) - d^(1/2)
)^3)/(2*((d - e*x)^(1/2) - d^(1/2))^3) - (333*b*d^4*((d + e*x)^(1/2) - d^(1/2))^5)/(2*((d - e*x)^(1/2) - d^(1/
2))^5) + (671*b*d^4*((d + e*x)^(1/2) - d^(1/2))^7)/(2*((d - e*x)^(1/2) - d^(1/2))^7) - (671*b*d^4*((d + e*x)^(
1/2) - d^(1/2))^9)/(2*((d - e*x)^(1/2) - d^(1/2))^9) + (333*b*d^4*((d + e*x)^(1/2) - d^(1/2))^11)/(2*((d - e*x
)^(1/2) - d^(1/2))^11) - (23*b*d^4*((d + e*x)^(1/2) - d^(1/2))^13)/(2*((d - e*x)^(1/2) - d^(1/2))^13) - (3*b*d
^4*((d + e*x)^(1/2) - d^(1/2))^15)/(2*((d - e*x)^(1/2) - d^(1/2))^15) + (3*b*d^4*((d + e*x)^(1/2) - d^(1/2)))/
(2*((d - e*x)^(1/2) - d^(1/2))))/(e^5*(((d + e*x)^(1/2) - d^(1/2))^2/((d - e*x)^(1/2) - d^(1/2))^2 + 1)^8) + (
2*a*d^2*atan(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2) - d^(1/2))))/e^3 + (3*b*d^4*atan(((d + e*x)^(1/2) -
d^(1/2))/((d - e*x)^(1/2) - d^(1/2))))/(2*e^5) + (5*c*d^6*atan(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2) -
d^(1/2))))/(4*e^7)