3.2133 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx\)
Optimal. Leaf size=139 \[ -\frac {224967}{65219 \sqrt {1-2 x}}+\frac {33115}{1694 \sqrt {1-2 x} (5 x+3)}-\frac {505}{154 \sqrt {1-2 x} (5 x+3)^2}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}+\frac {5832}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {153825 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
[Out]
5832/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-153825/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/
2)-224967/65219/(1-2*x)^(1/2)-505/154/(3+5*x)^2/(1-2*x)^(1/2)+3/7/(2+3*x)/(3+5*x)^2/(1-2*x)^(1/2)+33115/1694/(
3+5*x)/(1-2*x)^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 139, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used =
{103, 151, 152, 156, 63, 206} \[ -\frac {224967}{65219 \sqrt {1-2 x}}+\frac {33115}{1694 \sqrt {1-2 x} (5 x+3)}-\frac {505}{154 \sqrt {1-2 x} (5 x+3)^2}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)^2}+\frac {5832}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {153825 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
Antiderivative was successfully verified.
[In]
Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^3),x]
[Out]
-224967/(65219*Sqrt[1 - 2*x]) - 505/(154*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 3/(7*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^2
) + 33115/(1694*Sqrt[1 - 2*x]*(3 + 5*x)) + (5832*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (153825*Sqrt
[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 103
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 151
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]
Rule 152
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]
Rule 156
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx &=\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^2}+\frac {1}{7} \int \frac {38-105 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {505}{154 \sqrt {1-2 x} (3+5 x)^2}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^2}-\frac {1}{154} \int \frac {2078-7575 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {505}{154 \sqrt {1-2 x} (3+5 x)^2}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^2}+\frac {33115}{1694 \sqrt {1-2 x} (3+5 x)}+\frac {\int \frac {36534-298035 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{1694}\\ &=-\frac {224967}{65219 \sqrt {1-2 x}}-\frac {505}{154 \sqrt {1-2 x} (3+5 x)^2}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^2}+\frac {33115}{1694 \sqrt {1-2 x} (3+5 x)}-\frac {\int \frac {-2756361+\frac {3374505 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{65219}\\ &=-\frac {224967}{65219 \sqrt {1-2 x}}-\frac {505}{154 \sqrt {1-2 x} (3+5 x)^2}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^2}+\frac {33115}{1694 \sqrt {1-2 x} (3+5 x)}-\frac {8748}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {769125 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{2662}\\ &=-\frac {224967}{65219 \sqrt {1-2 x}}-\frac {505}{154 \sqrt {1-2 x} (3+5 x)^2}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^2}+\frac {33115}{1694 \sqrt {1-2 x} (3+5 x)}+\frac {8748}{49} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {769125 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2662}\\ &=-\frac {224967}{65219 \sqrt {1-2 x}}-\frac {505}{154 \sqrt {1-2 x} (3+5 x)^2}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)^2}+\frac {33115}{1694 \sqrt {1-2 x} (3+5 x)}+\frac {5832}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {153825 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 78, normalized size = 0.56 \[ \frac {-15524784 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+15074850 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+\frac {77 \left (496725 x^2+612520 x+188306\right )}{(3 x+2) (5 x+3)^2}}{130438 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^3),x]
[Out]
((77*(188306 + 612520*x + 496725*x^2))/((2 + 3*x)*(3 + 5*x)^2) - 15524784*Hypergeometric2F1[-1/2, 1, 1/2, 3/7
- (6*x)/7] + 15074850*Hypergeometric2F1[-1/2, 1, 1/2, (-5*(-1 + 2*x))/11])/(130438*Sqrt[1 - 2*x])
________________________________________________________________________________________
fricas [A] time = 0.87, size = 162, normalized size = 1.17 \[ \frac {52761975 \, \sqrt {11} \sqrt {5} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 85386312 \, \sqrt {7} \sqrt {3} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (33745050 \, x^{3} + 24742935 \, x^{2} - 8019782 \, x - 6400750\right )} \sqrt {-2 \, x + 1}}{10043726 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^3,x, algorithm="fricas")
[Out]
1/10043726*(52761975*sqrt(11)*sqrt(5)*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*log((sqrt(11)*sqrt(5)*sqrt(-2*x
+ 1) + 5*x - 8)/(5*x + 3)) + 85386312*sqrt(7)*sqrt(3)*(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*log(-(sqrt(7)*
sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(33745050*x^3 + 24742935*x^2 - 8019782*x - 6400750)*sqrt(-2*
x + 1))/(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)
________________________________________________________________________________________
giac [A] time = 1.32, size = 135, normalized size = 0.97 \[ \frac {153825}{29282} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2916}{343} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (215526 \, x - 107875\right )}}{65219 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} - \frac {125 \, {\left (625 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1353 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^3,x, algorithm="giac")
[Out]
153825/29282*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2916/343*s
qrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/65219*(215526*x - 10787
5)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1)) - 125/5324*(625*(-2*x + 1)^(3/2) - 1353*sqrt(-2*x + 1))/(5*x + 3)^2
________________________________________________________________________________________
maple [A] time = 0.02, size = 91, normalized size = 0.65 \[ \frac {5832 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{343}-\frac {153825 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{14641}+\frac {32}{65219 \sqrt {-2 x +1}}+\frac {-\frac {78125 \left (-2 x +1\right )^{\frac {3}{2}}}{1331}+\frac {15375 \sqrt {-2 x +1}}{121}}{\left (-10 x -6\right )^{2}}-\frac {54 \sqrt {-2 x +1}}{49 \left (-2 x -\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(-2*x+1)^(3/2)/(3*x+2)^2/(5*x+3)^3,x)
[Out]
32/65219/(-2*x+1)^(1/2)+31250/1331*(-5/2*(-2*x+1)^(3/2)+1353/250*(-2*x+1)^(1/2))/(-10*x-6)^2-153825/14641*arct
anh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)-54/49*(-2*x+1)^(1/2)/(-2*x-4/3)+5832/343*arctanh(1/7*21^(1/2)*(-2*x
+1)^(1/2))*21^(1/2)
________________________________________________________________________________________
maxima [A] time = 1.19, size = 137, normalized size = 0.99 \[ \frac {153825}{29282} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2916}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {16872525 \, {\left (2 \, x - 1\right )}^{3} + 75360510 \, {\left (2 \, x - 1\right )}^{2} + 168127762 \, x - 84090985}{65219 \, {\left (75 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 505 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 1133 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 847 \, \sqrt {-2 \, x + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^3,x, algorithm="maxima")
[Out]
153825/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2916/343*sqrt(21)*lo
g(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/65219*(16872525*(2*x - 1)^3 + 75360510*(2*
x - 1)^2 + 168127762*x - 84090985)/(75*(-2*x + 1)^(7/2) - 505*(-2*x + 1)^(5/2) + 1133*(-2*x + 1)^(3/2) - 847*s
qrt(-2*x + 1))
________________________________________________________________________________________
mupad [B] time = 0.11, size = 101, normalized size = 0.73 \[ \frac {5832\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {15284342\,x}{444675}+\frac {5024034\,{\left (2\,x-1\right )}^2}{326095}+\frac {224967\,{\left (2\,x-1\right )}^3}{65219}-\frac {1528927}{88935}}{\frac {847\,\sqrt {1-2\,x}}{75}-\frac {1133\,{\left (1-2\,x\right )}^{3/2}}{75}+\frac {101\,{\left (1-2\,x\right )}^{5/2}}{15}-{\left (1-2\,x\right )}^{7/2}}-\frac {153825\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((1 - 2*x)^(3/2)*(3*x + 2)^2*(5*x + 3)^3),x)
[Out]
(5832*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - ((15284342*x)/444675 + (5024034*(2*x - 1)^2)/326095
+ (224967*(2*x - 1)^3)/65219 - 1528927/88935)/((847*(1 - 2*x)^(1/2))/75 - (1133*(1 - 2*x)^(3/2))/75 + (101*(1
- 2*x)^(5/2))/15 - (1 - 2*x)^(7/2)) - (153825*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641
________________________________________________________________________________________
sympy [C] time = 22.09, size = 2222, normalized size = 15.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2*x)**(3/2)/(2+3*x)**2/(3+5*x)**3,x)
[Out]
1039347540000*sqrt(2)*I*(x - 1/2)**(13/2)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653
600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 460
7668296000*sqrt(2)*I*(x - 1/2)**(11/2)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600
*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 765862
2448400*sqrt(2)*I*(x - 1/2)**(9/2)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x
- 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 5656411074
160*sqrt(2)*I*(x - 1/2)**(7/2)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/
2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 1565987216794*
sqrt(2)*I*(x - 1/2)**(5/2)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**
5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) - 252527968*sqrt(2)*
I*(x - 1/2)**(3/2)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312
589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) - 6331437000000*sqrt(55)*I*(
x - 1/2)**7*atan(sqrt(110)*sqrt(x - 1/2)/11)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514
653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) +
10246357440000*sqrt(21)*I*(x - 1/2)**7*atan(sqrt(42)*sqrt(x - 1/2)/7)/(602623560000*(x - 1/2)**7 + 33546044840
00*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 10293
51346562*(x - 1/2)**2) - 5123178720000*sqrt(21)*I*pi*(x - 1/2)**7/(602623560000*(x - 1/2)**7 + 3354604484000*(
x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 102935134
6562*(x - 1/2)**2) + 3165718500000*sqrt(55)*I*pi*(x - 1/2)**7/(602623560000*(x - 1/2)**7 + 3354604484000*(x -
1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562
*(x - 1/2)**2) - 35244999300000*sqrt(55)*I*(x - 1/2)**6*atan(sqrt(110)*sqrt(x - 1/2)/11)/(602623560000*(x - 1/
2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(
x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 57038056416000*sqrt(21)*I*(x - 1/2)**6*atan(sqrt(42)*sqrt(x - 1/2)
/7)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x -
1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) - 28519028208000*sqrt(21)*I*pi*(x - 1/2)**6
/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2
)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 17622499650000*sqrt(55)*I*pi*(x - 1/2)**6/(6
02623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**
4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) - 78467609220000*sqrt(55)*I*(x - 1/2)**5*atan(sqr
t(110)*sqrt(x - 1/2)/11)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5
+ 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 126986523206400*sqrt
(21)*I*(x - 1/2)**5*atan(sqrt(42)*sqrt(x - 1/2)/7)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7
468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)*
*2) - 63493261603200*sqrt(21)*I*pi*(x - 1/2)**5/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468
514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2)
+ 39233804610000*sqrt(55)*I*pi*(x - 1/2)**5/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514
653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) -
87335841978000*sqrt(55)*I*(x - 1/2)**4*atan(sqrt(110)*sqrt(x - 1/2)/11)/(602623560000*(x - 1/2)**7 + 335460448
4000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 102
9351346562*(x - 1/2)**2) + 141338254527360*sqrt(21)*I*(x - 1/2)**4*atan(sqrt(42)*sqrt(x - 1/2)/7)/(60262356000
0*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 462539
6959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) - 70669127263680*sqrt(21)*I*pi*(x - 1/2)**4/(602623560000*(
x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 462539695
9876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 43667920989000*sqrt(55)*I*pi*(x - 1/2)**4/(602623560000*(x -
1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 462539695987
6*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) - 48596522597700*sqrt(55)*I*(x - 1/2)**3*atan(sqrt(110)*sqrt(x -
1/2)/11)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*
(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 78645233440224*sqrt(21)*I*(x - 1/2)*
*3*atan(sqrt(42)*sqrt(x - 1/2)/7)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x -
1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) - 39322616720
112*sqrt(21)*I*pi*(x - 1/2)**3/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/
2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) + 24298261298850
*sqrt(55)*I*pi*(x - 1/2)**3/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)*
*5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1/2)**2) - 10814833063650*sq
rt(55)*I*(x - 1/2)**2*atan(sqrt(110)*sqrt(x - 1/2)/11)/(602623560000*(x - 1/2)**7 + 3354604484000*(x - 1/2)**6
+ 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 1029351346562*(x - 1
/2)**2) + 17501973915888*sqrt(21)*I*(x - 1/2)**2*atan(sqrt(42)*sqrt(x - 1/2)/7)/(602623560000*(x - 1/2)**7 + 3
354604484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)*
*3 + 1029351346562*(x - 1/2)**2) - 8750986957944*sqrt(21)*I*pi*(x - 1/2)**2/(602623560000*(x - 1/2)**7 + 33546
04484000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 +
1029351346562*(x - 1/2)**2) + 5407416531825*sqrt(55)*I*pi*(x - 1/2)**2/(602623560000*(x - 1/2)**7 + 335460448
4000*(x - 1/2)**6 + 7468514653600*(x - 1/2)**5 + 8312589386640*(x - 1/2)**4 + 4625396959876*(x - 1/2)**3 + 102
9351346562*(x - 1/2)**2)
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