Integrand size = 26, antiderivative size = 151 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {32735 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {375265 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}} \]
-375265/153664*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1/2 8*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+1/56*(1-2*x)^(1/2)*(3+5*x)^(1/2)/( 2+3*x)^3+305/1568*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+32735/21952*(1-2*x )^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 2.53 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {5 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (278960+1230876 x+1806120 x^2+883845 x^3\right )}{5 (2+3 x)^4}+75053 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+75053 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{153664} \]
(5*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(278960 + 1230876*x + 1806120*x^2 + 883 845*x^3))/(5*(2 + 3*x)^4) + 75053*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155]) ]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 75053*Sqrt[7]*ArcTan[Sqrt [6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))]))/153664
Time = 0.23 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {110, 27, 168, 27, 168, 27, 168, 27, 104, 217}
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.
| \(\displaystyle \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {1}{28} \int \frac {60 x+47}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{56} \int \frac {60 x+47}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{56} \left (\frac {1}{21} \int \frac {105 (15-8 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{56} \left (\frac {5}{2} \int \frac {15-8 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{14} \int \frac {1369-1220 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {61 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{28} \int \frac {1369-1220 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {61 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {75053}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {6547 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {61 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {75053}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {6547 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {61 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {75053}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {6547 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {61 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {6547 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {75053 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {61 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\) |
-1/28*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^4 + ((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^3 + (5*((61*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((6547*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (75053*ArcTan[Sqrt[ 1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/2)/56
3.25.67.3.1 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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 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_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S imp[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] && ILtQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Time = 1.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (883845 x^{3}+1806120 x^{2}+1230876 x +278960\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{21952 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {375265 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307328 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (30396465 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+81057240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+81057240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+12373830 x^{3} \sqrt {-10 x^{2}-x +3}+36025440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +25285680 x^{2} \sqrt {-10 x^{2}-x +3}+6004240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+17232264 x \sqrt {-10 x^{2}-x +3}+3905440 \sqrt {-10 x^{2}-x +3}\right )}{307328 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
-1/21952*(-1+2*x)*(3+5*x)^(1/2)*(883845*x^3+1806120*x^2+1230876*x+278960)/ (2+3*x)^4/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+ 375265/307328*7^(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+ 111*x)^(1/2))*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {375265 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (883845 \, x^{3} + 1806120 \, x^{2} + 1230876 \, x + 278960\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{307328 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
-1/307328*(375265*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan( 1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(883845*x^3 + 1806120*x^2 + 1230876*x + 278960)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\int \frac {\sqrt {5 x + 3}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{5}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {375265}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {\sqrt {-10 \, x^{2} - x + 3}}{56 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {305 \, \sqrt {-10 \, x^{2} - x + 3}}{1568 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {32735 \, \sqrt {-10 \, x^{2} - x + 3}}{21952 \, {\left (3 \, x + 2\right )}} \]
375265/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/28*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1/56 *sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 305/1568*sqrt(-10*x^ 2 - x + 3)/(9*x^2 + 12*x + 4) + 32735/21952*sqrt(-10*x^2 - x + 3)/(3*x + 2 )
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (118) = 236\).
Time = 0.53 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {75053}{614656} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {55 \, \sqrt {10} {\left (6823 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 7629720 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 1915892160 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {149136243200 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {596544972800 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{10976 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
75053/614656*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3 )*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10 *x + 5) - sqrt(22)))) - 55/10976*sqrt(10)*(6823*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr t(22)))^7 - 7629720*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 1915892160*((sqr t(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)* sqrt(-10*x + 5) - sqrt(22)))^3 - 149136243200*(sqrt(2)*sqrt(-10*x + 5) - s qrt(22))/sqrt(5*x + 3) + 596544972800*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4
Time = 24.70 (sec) , antiderivative size = 1509, normalized size of antiderivative = 9.99 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\text {Too large to display} \]
((229171111*((1 - 2*x)^(1/2) - 1)^7)/(15312500*(3^(1/2) - (5*x + 3)^(1/2)) ^7) - (1983904*((1 - 2*x)^(1/2) - 1)^3)/(2734375*(3^(1/2) - (5*x + 3)^(1/2 ))^3) - (13839741*((1 - 2*x)^(1/2) - 1)^5)/(7656250*(3^(1/2) - (5*x + 3)^( 1/2))^5) - (734066*((1 - 2*x)^(1/2) - 1))/(133984375*(3^(1/2) - (5*x + 3)^ (1/2))) - (229171111*((1 - 2*x)^(1/2) - 1)^9)/(6125000*(3^(1/2) - (5*x + 3 )^(1/2))^9) + (13839741*((1 - 2*x)^(1/2) - 1)^11)/(490000*(3^(1/2) - (5*x + 3)^(1/2))^11) + (61997*((1 - 2*x)^(1/2) - 1)^13)/(875*(3^(1/2) - (5*x + 3)^(1/2))^13) + (367033*((1 - 2*x)^(1/2) - 1)^15)/(109760*(3^(1/2) - (5*x + 3)^(1/2))^15) + (1111291*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(19140625*(3^( 1/2) - (5*x + 3)^(1/2))^2) + (746274*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(546 875*(3^(1/2) - (5*x + 3)^(1/2))^4) - (7569447*3^(1/2)*((1 - 2*x)^(1/2) - 1 )^6)/(1531250*(3^(1/2) - (5*x + 3)^(1/2))^6) + (631898231*3^(1/2)*((1 - 2* x)^(1/2) - 1)^8)/(53593750*(3^(1/2) - (5*x + 3)^(1/2))^8) - (7569447*3^(1/ 2)*((1 - 2*x)^(1/2) - 1)^10)/(245000*(3^(1/2) - (5*x + 3)^(1/2))^10) + (37 3137*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(7000*(3^(1/2) - (5*x + 3)^(1/2))^1 2) + (1111291*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(78400*(3^(1/2) - (5*x + 3 )^(1/2))^14))/((45056*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3 )^(1/2))^2) + (294784*((1 - 2*x)^(1/2) - 1)^4)/(390625*(3^(1/2) - (5*x + 3 )^(1/2))^4) - (1921024*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (5828656*((1 - 2*x)^(1/2) - 1)^8)/(390625*(3^(1/2) - (5*...