3.20.26 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=97 \[ \frac {315 \sqrt {1-2 x}}{242 (5 x+3)}-\frac {5 \sqrt {1-2 x}}{22 (5 x+3)^2}+18 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2115}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {103, 151, 156, 63, 206} \begin {gather*} \frac {315 \sqrt {1-2 x}}{242 (5 x+3)}-\frac {5 \sqrt {1-2 x}}{22 (5 x+3)^2}+18 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2115}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^3),x]

[Out]

(-5*Sqrt[1 - 2*x])/(22*(3 + 5*x)^2) + (315*Sqrt[1 - 2*x])/(242*(3 + 5*x)) + 18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]] - (2115*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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 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}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx &=-\frac {5 \sqrt {1-2 x}}{22 (3+5 x)^2}-\frac {1}{22} \int \frac {36-45 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {5 \sqrt {1-2 x}}{22 (3+5 x)^2}+\frac {315 \sqrt {1-2 x}}{242 (3+5 x)}+\frac {1}{242} \int \frac {1548-945 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {5 \sqrt {1-2 x}}{22 (3+5 x)^2}+\frac {315 \sqrt {1-2 x}}{242 (3+5 x)}-27 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {10575}{242} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {5 \sqrt {1-2 x}}{22 (3+5 x)^2}+\frac {315 \sqrt {1-2 x}}{242 (3+5 x)}+27 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {10575}{242} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {5 \sqrt {1-2 x}}{22 (3+5 x)^2}+\frac {315 \sqrt {1-2 x}}{242 (3+5 x)}+18 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2115}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 81, normalized size = 0.84 \begin {gather*} 18 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {5 \left (\frac {11 \sqrt {1-2 x} (315 x+178)}{(5 x+3)^2}-846 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )}{2662} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^3),x]

[Out]

18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + (5*((11*Sqrt[1 - 2*x]*(178 + 315*x))/(3 + 5*x)^2 - 846*Sqrt[55
]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/2662

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IntegrateAlgebraic [A]  time = 0.27, size = 90, normalized size = 0.93 \begin {gather*} -\frac {5 \sqrt {1-2 x} (315 (1-2 x)-671)}{121 (5 (1-2 x)-11)^2}+18 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2115}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^3),x]

[Out]

(-5*(-671 + 315*(1 - 2*x))*Sqrt[1 - 2*x])/(121*(-11 + 5*(1 - 2*x))^2) + 18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]] - (2115*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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fricas [A]  time = 1.65, size = 122, normalized size = 1.26 \begin {gather*} \frac {14805 \, \sqrt {11} \sqrt {5} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 23958 \, \sqrt {7} \sqrt {3} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 385 \, {\left (315 \, x + 178\right )} \sqrt {-2 \, x + 1}}{18634 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/18634*(14805*sqrt(11)*sqrt(5)*(25*x^2 + 30*x + 9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3))
+ 23958*sqrt(7)*sqrt(3)*(25*x^2 + 30*x + 9)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 385*
(315*x + 178)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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giac [A]  time = 1.24, size = 107, normalized size = 1.10 \begin {gather*} \frac {2115}{2662} \, \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 {9}{7} \, \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 {5 \, {\left (315 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 671 \, \sqrt {-2 \, x + 1}\right )}}{484 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

2115/2662*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 9/7*sqrt(21)*
log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 5/484*(315*(-2*x + 1)^(3/2) - 671
*sqrt(-2*x + 1))/(5*x + 3)^2

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maple [A]  time = 0.01, size = 66, normalized size = 0.68 \begin {gather*} \frac {18 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{7}-\frac {2115 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1331}+\frac {-\frac {1575 \left (-2 x +1\right )^{\frac {3}{2}}}{121}+\frac {305 \sqrt {-2 x +1}}{11}}{\left (-10 x -6\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x+2)/(5*x+3)^3/(-2*x+1)^(1/2),x)

[Out]

250*(-63/1210*(-2*x+1)^(3/2)+61/550*(-2*x+1)^(1/2))/(-10*x-6)^2-2115/1331*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2)
)*55^(1/2)+18/7*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.41, size = 110, normalized size = 1.13 \begin {gather*} \frac {2115}{2662} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {9}{7} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {5 \, {\left (315 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 671 \, \sqrt {-2 \, x + 1}\right )}}{121 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

2115/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 9/7*sqrt(21)*log(-(sqrt
(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 5/121*(315*(-2*x + 1)^(3/2) - 671*sqrt(-2*x + 1))/(2
5*(2*x - 1)^2 + 220*x + 11)

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mupad [B]  time = 1.27, size = 71, normalized size = 0.73 \begin {gather*} \frac {18\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {2115\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}+\frac {\frac {61\,\sqrt {1-2\,x}}{55}-\frac {63\,{\left (1-2\,x\right )}^{3/2}}{121}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - 2*x)^(1/2)*(3*x + 2)*(5*x + 3)^3),x)

[Out]

(18*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/7 - (2115*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/133
1 + ((61*(1 - 2*x)^(1/2))/55 - (63*(1 - 2*x)^(3/2))/121)/((44*x)/5 + (2*x - 1)^2 + 11/25)

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sympy [C]  time = 15.83, size = 1953, normalized size = 20.13

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

24255000*sqrt(2)*I*(x - 1/2)**(11/2)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4
+ 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 79194500*sqrt(2)*I*(x - 1/2)**(9/2)/(93170000*(x - 1/2)**
6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 86182
250*sqrt(2)*I*(x - 1/2)**(7/2)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 4960
37080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 31258535*sqrt(2)*I*(x - 1/2)**(5/2)/(93170000*(x - 1/2)**6 + 40
9948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 4270000*sqr
t(55)*I*(x - 1/2)**6*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 6764
14200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) - 143780000*sqrt(55)*I*(x - 1/2)**6*atan
(sqrt(110)*sqrt(x - 1/2)/11)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037
080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 239580000*sqrt(21)*I*(x - 1/2)**6*atan(sqrt(42)*sqrt(x - 1/2)/7)/
(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*
(x - 1/2)**2) - 119790000*sqrt(21)*I*pi*(x - 1/2)**6/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 6764142
00*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 71890000*sqrt(55)*I*pi*(x - 1/2)**6/(9317
0000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x -
1/2)**2) + 18788000*sqrt(55)*I*(x - 1/2)**5*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(93170000*(x - 1/2)**6 + 409948
000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) - 632632000*sqrt(
55)*I*(x - 1/2)**5*atan(sqrt(110)*sqrt(x - 1/2)/11)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 67641420
0*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 1054152000*sqrt(21)*I*(x - 1/2)**5*atan(sq
rt(42)*sqrt(x - 1/2)/7)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(
x - 1/2)**3 + 136410197*(x - 1/2)**2) - 527076000*sqrt(21)*I*pi*(x - 1/2)**5/(93170000*(x - 1/2)**6 + 40994800
0*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 316316000*sqrt(55
)*I*pi*(x - 1/2)**5/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x -
1/2)**3 + 136410197*(x - 1/2)**2) + 31000200*sqrt(55)*I*(x - 1/2)**4*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(93170
000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1
/2)**2) - 1043842800*sqrt(55)*I*(x - 1/2)**4*atan(sqrt(110)*sqrt(x - 1/2)/11)/(93170000*(x - 1/2)**6 + 4099480
00*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 1739350800*sqrt(
21)*I*(x - 1/2)**4*atan(sqrt(42)*sqrt(x - 1/2)/7)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*
(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) - 869675400*sqrt(21)*I*pi*(x - 1/2)**4/(931700
00*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/
2)**2) + 521921400*sqrt(55)*I*pi*(x - 1/2)**4/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x -
1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 22733480*sqrt(55)*I*(x - 1/2)**3*atan(sqrt(110)/
(10*sqrt(x - 1/2)))/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x -
1/2)**3 + 136410197*(x - 1/2)**2) - 765484720*sqrt(55)*I*(x - 1/2)**3*atan(sqrt(110)*sqrt(x - 1/2)/11)/(931700
00*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/
2)**2) + 1275523920*sqrt(21)*I*(x - 1/2)**3*atan(sqrt(42)*sqrt(x - 1/2)/7)/(93170000*(x - 1/2)**6 + 409948000*
(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) - 637761960*sqrt(21)*
I*pi*(x - 1/2)**3/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/
2)**3 + 136410197*(x - 1/2)**2) + 382742360*sqrt(55)*I*pi*(x - 1/2)**3/(93170000*(x - 1/2)**6 + 409948000*(x -
1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 6251707*sqrt(55)*I*(x -
1/2)**2*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1
/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) - 210508298*sqrt(55)*I*(x - 1/2)**2*atan(sqrt(110)*s
qrt(x - 1/2)/11)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2
)**3 + 136410197*(x - 1/2)**2) + 350769078*sqrt(21)*I*(x - 1/2)**2*atan(sqrt(42)*sqrt(x - 1/2)/7)/(93170000*(x
- 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2
) - 175384539*sqrt(21)*I*pi*(x - 1/2)**2/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)
**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 105254149*sqrt(55)*I*pi*(x - 1/2)**2/(93170000*(x - 1
/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2)

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