Integrand size = 24, antiderivative size = 120 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^3} \, dx=\frac {693}{625} \sqrt {1-2 x} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {48 \sqrt {1-2 x} (2+3 x)^3}{25 (3+5 x)}+\frac {63 \sqrt {1-2 x} (92+125 x)}{6250}-\frac {5943 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \]
-1/10*(1-2*x)^(3/2)*(2+3*x)^3/(3+5*x)^2-5943/171875*arctanh(1/11*55^(1/2)* (1-2*x)^(1/2))*55^(1/2)+693/625*(2+3*x)^2*(1-2*x)^(1/2)-48/25*(2+3*x)^3*(1 -2*x)^(1/2)/(3+5*x)+63/6250*(92+125*x)*(1-2*x)^(1/2)
Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.57 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (8644+36295 x+37530 x^2-14400 x^3-27000 x^4\right )}{6250 (3+5 x)^2}-\frac {5943 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \]
(Sqrt[1 - 2*x]*(8644 + 36295*x + 37530*x^2 - 14400*x^3 - 27000*x^4))/(6250 *(3 + 5*x)^2) - (5943*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])
Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {108, 27, 166, 27, 170, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (3 x+2)^3}{(5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{10} \int \frac {3 (1-9 x) \sqrt {1-2 x} (3 x+2)^2}{(5 x+3)^2}dx-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{10} \int \frac {(1-9 x) \sqrt {1-2 x} (3 x+2)^2}{(5 x+3)^2}dx-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {3}{10} \left (\frac {1}{5} \int \frac {7 (17-66 x) (3 x+2)^2}{\sqrt {1-2 x} (5 x+3)}dx-\frac {32 \sqrt {1-2 x} (3 x+2)^3}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{5} \int \frac {(17-66 x) (3 x+2)^2}{\sqrt {1-2 x} (5 x+3)}dx-\frac {32 \sqrt {1-2 x} (3 x+2)^3}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{5} \left (\frac {66}{25} \sqrt {1-2 x} (3 x+2)^2-\frac {1}{25} \int -\frac {(58-375 x) (3 x+2)}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {32 \sqrt {1-2 x} (3 x+2)^3}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{5} \left (\frac {1}{25} \int \frac {(58-375 x) (3 x+2)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {66}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {32 \sqrt {1-2 x} (3 x+2)^3}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{5} \left (\frac {1}{25} \left (\frac {283}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {3}{5} \sqrt {1-2 x} (125 x+92)\right )+\frac {66}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {32 \sqrt {1-2 x} (3 x+2)^3}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{5} \left (\frac {1}{25} \left (\frac {3}{5} \sqrt {1-2 x} (125 x+92)-\frac {283}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {66}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {32 \sqrt {1-2 x} (3 x+2)^3}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{5} \left (\frac {1}{25} \left (\frac {3}{5} \sqrt {1-2 x} (125 x+92)-\frac {566 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}\right )+\frac {66}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {32 \sqrt {1-2 x} (3 x+2)^3}{5 (5 x+3)}\right )-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}\) |
-1/10*((1 - 2*x)^(3/2)*(2 + 3*x)^3)/(3 + 5*x)^2 + (3*((-32*Sqrt[1 - 2*x]*( 2 + 3*x)^3)/(5*(3 + 5*x)) + (7*((66*Sqrt[1 - 2*x]*(2 + 3*x)^2)/25 + ((3*Sq rt[1 - 2*x]*(92 + 125*x))/5 - (566*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*S qrt[55]))/25))/5))/10
3.20.20.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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 = 1.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.51
| method | result | size |
| risch | \(\frac {54000 x^{5}+1800 x^{4}-89460 x^{3}-35060 x^{2}+19007 x +8644}{6250 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {5943 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{171875}\) | \(61\) |
| pseudoelliptic | \(\frac {-11886 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}-55 \sqrt {1-2 x}\, \left (27000 x^{4}+14400 x^{3}-37530 x^{2}-36295 x -8644\right )}{343750 \left (3+5 x \right )^{2}}\) | \(65\) |
| derivativedivides | \(-\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{625}+\frac {18 \left (1-2 x \right )^{\frac {3}{2}}}{625}+\frac {558 \sqrt {1-2 x}}{3125}+\frac {\frac {193 \left (1-2 x \right )^{\frac {3}{2}}}{625}-\frac {429 \sqrt {1-2 x}}{625}}{\left (-6-10 x \right )^{2}}-\frac {5943 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{171875}\) | \(75\) |
| default | \(-\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{625}+\frac {18 \left (1-2 x \right )^{\frac {3}{2}}}{625}+\frac {558 \sqrt {1-2 x}}{3125}+\frac {\frac {193 \left (1-2 x \right )^{\frac {3}{2}}}{625}-\frac {429 \sqrt {1-2 x}}{625}}{\left (-6-10 x \right )^{2}}-\frac {5943 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{171875}\) | \(75\) |
| trager | \(-\frac {\left (27000 x^{4}+14400 x^{3}-37530 x^{2}-36295 x -8644\right ) \sqrt {1-2 x}}{6250 \left (3+5 x \right )^{2}}-\frac {5943 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{343750}\) | \(82\) |
1/6250*(54000*x^5+1800*x^4-89460*x^3-35060*x^2+19007*x+8644)/(3+5*x)^2/(1- 2*x)^(1/2)-5943/171875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^3} \, dx=\frac {5943 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (27000 \, x^{4} + 14400 \, x^{3} - 37530 \, x^{2} - 36295 \, x - 8644\right )} \sqrt {-2 \, x + 1}}{343750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
1/343750*(5943*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(27000*x^4 + 14400*x^3 - 37530*x^2 - 36295*x - 8 644)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)
Time = 135.39 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.05 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^3} \, dx=- \frac {27 \left (1 - 2 x\right )^{\frac {5}{2}}}{625} + \frac {18 \left (1 - 2 x\right )^{\frac {3}{2}}}{625} + \frac {558 \sqrt {1 - 2 x}}{3125} + \frac {23 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{1375} - \frac {836 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{625} + \frac {968 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{3125} \]
-27*(1 - 2*x)**(5/2)/625 + 18*(1 - 2*x)**(3/2)/625 + 558*sqrt(1 - 2*x)/312 5 + 23*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqr t(55)/5))/1375 - 836*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 2*x )/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/625 + 968*Piecewise((sqrt(55 )*(3*log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(1 - 2*x)/ 11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) + 1/(16*(sqrt(55)*sqrt (1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)) - 1/(16*(sq rt(55)*sqrt(1 - 2*x)/11 - 1)**2))/6655, (sqrt(1 - 2*x) > -sqrt(55)/5) & (s qrt(1 - 2*x) < sqrt(55)/5)))/3125
Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^3} \, dx=-\frac {27}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {18}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {5943}{343750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {558}{3125} \, \sqrt {-2 \, x + 1} + \frac {193 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 429 \, \sqrt {-2 \, x + 1}}{625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
-27/625*(-2*x + 1)^(5/2) + 18/625*(-2*x + 1)^(3/2) + 5943/343750*sqrt(55)* log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 558/31 25*sqrt(-2*x + 1) + 1/625*(193*(-2*x + 1)^(3/2) - 429*sqrt(-2*x + 1))/(25* (2*x - 1)^2 + 220*x + 11)
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^3} \, dx=-\frac {27}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {18}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {5943}{343750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {558}{3125} \, \sqrt {-2 \, x + 1} + \frac {193 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 429 \, \sqrt {-2 \, x + 1}}{2500 \, {\left (5 \, x + 3\right )}^{2}} \]
-27/625*(2*x - 1)^2*sqrt(-2*x + 1) + 18/625*(-2*x + 1)^(3/2) + 5943/343750 *sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt( -2*x + 1))) + 558/3125*sqrt(-2*x + 1) + 1/2500*(193*(-2*x + 1)^(3/2) - 429 *sqrt(-2*x + 1))/(5*x + 3)^2
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^3} \, dx=\frac {558\,\sqrt {1-2\,x}}{3125}+\frac {18\,{\left (1-2\,x\right )}^{3/2}}{625}-\frac {27\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {\frac {429\,\sqrt {1-2\,x}}{15625}-\frac {193\,{\left (1-2\,x\right )}^{3/2}}{15625}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,5943{}\mathrm {i}}{171875} \]