Integrand size = 24, antiderivative size = 147 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=-\frac {256172 (2+3 x)^2}{366025 \sqrt {1-2 x}}-\frac {73 (2+3 x)^4}{3630 \sqrt {1-2 x} (3+5 x)^2}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac {3269 (2+3 x)^3}{199650 \sqrt {1-2 x} (3+5 x)}-\frac {21 \sqrt {1-2 x} (2211616+736875 x)}{3660250}-\frac {6937 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1830125 \sqrt {55}} \]
7/33*(2+3*x)^5/(1-2*x)^(3/2)/(3+5*x)^2-6937/100656875*arctanh(1/11*55^(1/2 )*(1-2*x)^(1/2))*55^(1/2)-256172/366025*(2+3*x)^2/(1-2*x)^(1/2)-73/3630*(2 +3*x)^4/(3+5*x)^2/(1-2*x)^(1/2)-3269/199650*(2+3*x)^3/(3+5*x)/(1-2*x)^(1/2 )-21/3660250*(2211616+736875*x)*(1-2*x)^(1/2)
Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.50 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {-\frac {55 \left (1463964312+253794537 x-9509366452 x^2-6510290070 x^3+5763576060 x^4+533664450 x^5\right )}{(1-2 x)^{3/2} (3+5 x)^2}-41622 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{603941250} \]
((-55*(1463964312 + 253794537*x - 9509366452*x^2 - 6510290070*x^3 + 576357 6060*x^4 + 533664450*x^5))/((1 - 2*x)^(3/2)*(3 + 5*x)^2) - 41622*Sqrt[55]* ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/603941250
Time = 0.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {109, 166, 27, 166, 27, 167, 25, 164, 73, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(3 x+2)^6}{(1-2 x)^{5/2} (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac {1}{33} \int \frac {(3 x+2)^4 (306 x+169)}{(1-2 x)^{3/2} (5 x+3)^3}dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{33} \left (-\frac {1}{110} \int \frac {7 (3 x+2)^3 (2979 x+1694)}{(1-2 x)^{3/2} (5 x+3)^2}dx-\frac {73 (3 x+2)^4}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (-\frac {7}{110} \int \frac {(3 x+2)^3 (2979 x+1694)}{(1-2 x)^{3/2} (5 x+3)^2}dx-\frac {73 (3 x+2)^4}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{33} \left (-\frac {7}{110} \left (\frac {1}{55} \int \frac {3 (3 x+2)^2 (34170 x+19511)}{(1-2 x)^{3/2} (5 x+3)}dx+\frac {467 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^4}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (-\frac {7}{110} \left (\frac {3}{55} \int \frac {(3 x+2)^2 (34170 x+19511)}{(1-2 x)^{3/2} (5 x+3)}dx+\frac {467 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^4}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{33} \left (-\frac {7}{110} \left (\frac {3}{55} \left (\frac {1}{11} \int -\frac {(3 x+2) (2210625 x+1327366)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {73192 (3 x+2)^2}{11 \sqrt {1-2 x}}\right )+\frac {467 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^4}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{33} \left (-\frac {7}{110} \left (\frac {3}{55} \left (\frac {73192 (3 x+2)^2}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(3 x+2) (2210625 x+1327366)}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {467 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^4}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{33} \left (-\frac {7}{110} \left (\frac {3}{55} \left (\frac {1}{11} \left (\frac {3}{5} \sqrt {1-2 x} (736875 x+2211616)-\frac {991}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {73192 (3 x+2)^2}{11 \sqrt {1-2 x}}\right )+\frac {467 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^4}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{33} \left (-\frac {7}{110} \left (\frac {3}{55} \left (\frac {1}{11} \left (\frac {991}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {3}{5} \sqrt {1-2 x} (736875 x+2211616)\right )+\frac {73192 (3 x+2)^2}{11 \sqrt {1-2 x}}\right )+\frac {467 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^4}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{33} \left (-\frac {7}{110} \left (\frac {3}{55} \left (\frac {1}{11} \left (\frac {1982 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}+\frac {3}{5} \sqrt {1-2 x} (736875 x+2211616)\right )+\frac {73192 (3 x+2)^2}{11 \sqrt {1-2 x}}\right )+\frac {467 (3 x+2)^3}{55 \sqrt {1-2 x} (5 x+3)}\right )-\frac {73 (3 x+2)^4}{110 \sqrt {1-2 x} (5 x+3)^2}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^2}\) |
(7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + ((-73*(2 + 3*x)^4)/(110 *Sqrt[1 - 2*x]*(3 + 5*x)^2) - (7*((467*(2 + 3*x)^3)/(55*Sqrt[1 - 2*x]*(3 + 5*x)) + (3*((73192*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]) + ((3*Sqrt[1 - 2*x]*(2 211616 + 736875*x))/5 + (1982*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[5 5]))/11))/55))/110)/33
3.22.90.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Time = 1.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(\frac {533664450 x^{5}+5763576060 x^{4}-6510290070 x^{3}-9509366452 x^{2}+253794537 x +1463964312}{10980750 \sqrt {1-2 x}\, \left (3+5 x \right )^{2} \left (-1+2 x \right )}-\frac {6937 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{100656875}\) | \(68\) |
| pseudoelliptic | \(-\frac {243 \left (-\frac {13874 \sqrt {55}\, \left (x +\frac {3}{5}\right )^{2} \sqrt {1-2 x}\, \left (x -\frac {1}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{195676965}+x^{5}+\frac {54 x^{4}}{5}-\frac {217009669 x^{3}}{17788815}-\frac {4754683226 x^{2}}{266832225}+\frac {28199393 x}{59296050}+\frac {243994052}{88944075}\right )}{5 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{2}}\) | \(73\) |
| derivativedivides | \(\frac {243 \left (1-2 x \right )^{\frac {3}{2}}}{1000}-\frac {26973 \sqrt {1-2 x}}{5000}+\frac {\frac {37 \left (1-2 x \right )^{\frac {3}{2}}}{166375}-\frac {409 \sqrt {1-2 x}}{831875}}{\left (-6-10 x \right )^{2}}-\frac {6937 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{100656875}+\frac {117649}{31944 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {1563051}{117128 \sqrt {1-2 x}}\) | \(84\) |
| default | \(\frac {243 \left (1-2 x \right )^{\frac {3}{2}}}{1000}-\frac {26973 \sqrt {1-2 x}}{5000}+\frac {\frac {37 \left (1-2 x \right )^{\frac {3}{2}}}{166375}-\frac {409 \sqrt {1-2 x}}{831875}}{\left (-6-10 x \right )^{2}}-\frac {6937 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{100656875}+\frac {117649}{31944 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {1563051}{117128 \sqrt {1-2 x}}\) | \(84\) |
| trager | \(-\frac {\left (533664450 x^{5}+5763576060 x^{4}-6510290070 x^{3}-9509366452 x^{2}+253794537 x +1463964312\right ) \sqrt {1-2 x}}{10980750 \left (10 x^{2}+x -3\right )^{2}}+\frac {6937 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{201313750}\) | \(90\) |
1/10980750*(533664450*x^5+5763576060*x^4-6510290070*x^3-9509366452*x^2+253 794537*x+1463964312)/(1-2*x)^(1/2)/(3+5*x)^2/(-1+2*x)-6937/100656875*arcta nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {20811 \, \sqrt {55} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (533664450 \, x^{5} + 5763576060 \, x^{4} - 6510290070 \, x^{3} - 9509366452 \, x^{2} + 253794537 \, x + 1463964312\right )} \sqrt {-2 \, x + 1}}{603941250 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
1/603941250*(20811*sqrt(55)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(533664450*x^5 + 576357606 0*x^4 - 6510290070*x^3 - 9509366452*x^2 + 253794537*x + 1463964312)*sqrt(- 2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
Timed out. \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {243}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6937}{201313750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {26973}{5000} \, \sqrt {-2 \, x + 1} + \frac {73267966785 \, {\left (2 \, x - 1\right )}^{3} + 342600082649 \, {\left (2 \, x - 1\right )}^{2} + 887178503750 \, x - 345719990000}{219615000 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
243/1000*(-2*x + 1)^(3/2) + 6937/201313750*sqrt(55)*log(-(sqrt(55) - 5*sqr t(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 26973/5000*sqrt(-2*x + 1) + 1/219615000*(73267966785*(2*x - 1)^3 + 342600082649*(2*x - 1)^2 + 88717850 3750*x - 345719990000)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*( -2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.73 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {243}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {6937}{201313750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {26973}{5000} \, \sqrt {-2 \, x + 1} - \frac {16807 \, {\left (279 \, x - 101\right )}}{175692 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {185 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 409 \, \sqrt {-2 \, x + 1}}{3327500 \, {\left (5 \, x + 3\right )}^{2}} \]
243/1000*(-2*x + 1)^(3/2) + 6937/201313750*sqrt(55)*log(1/2*abs(-2*sqrt(55 ) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 26973/5000*sqrt(-2 *x + 1) - 16807/175692*(279*x - 101)/((2*x - 1)*sqrt(-2*x + 1)) + 1/332750 0*(185*(-2*x + 1)^(3/2) - 409*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 1.61 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {\frac {5865643\,x}{36300}+\frac {31145462059\,{\left (2\,x-1\right )}^2}{499125000}+\frac {4884531119\,{\left (2\,x-1\right )}^3}{366025000}-\frac {571438}{9075}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}}-\frac {26973\,\sqrt {1-2\,x}}{5000}+\frac {243\,{\left (1-2\,x\right )}^{3/2}}{1000}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,6937{}\mathrm {i}}{100656875} \]
((5865643*x)/36300 + (31145462059*(2*x - 1)^2)/499125000 + (4884531119*(2* x - 1)^3)/366025000 - 571438/9075)/((121*(1 - 2*x)^(3/2))/25 - (22*(1 - 2* x)^(5/2))/5 + (1 - 2*x)^(7/2)) + (55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)* 1i)/11)*6937i)/100656875 - (26973*(1 - 2*x)^(1/2))/5000 + (243*(1 - 2*x)^( 3/2))/1000