Integrand size = 26, antiderivative size = 91 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {2}{9} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {37 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{9 \sqrt {7}} \]
-37/63*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-2/9*arcsin( 1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-1/3*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3 *x)
Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{6+9 x}+\frac {2}{9} \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-\frac {37 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{9 \sqrt {7}} \]
-((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6 + 9*x)) + (2*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/9 - (37*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]) ])/(9*Sqrt[7])
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {108, 27, 175, 64, 104, 217, 223}
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.
\(\displaystyle \int \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {1}{3} \int -\frac {20 x+1}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{6} \int \frac {20 x+1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\) |
\(\Big \downarrow \) 175 |
\(\displaystyle \frac {1}{6} \left (\frac {37}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {20}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {1}{6} \left (\frac {37}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {8}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {1}{6} \left (\frac {74}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {8}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{6} \left (-\frac {8}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {74 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{6} \left (-\frac {4}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {74 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}\) |
-1/3*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x) + ((-4*Sqrt[10]*ArcSin[Sqrt[2 /11]*Sqrt[3 + 5*x]])/3 - (74*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])] )/(3*Sqrt[7]))/6
3.23.68.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [2/b Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] || PosQ[b])
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/(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[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Time = 1.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.41
method | result | size |
risch | \(\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3 \left (2+3 x \right ) \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {\sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{9}-\frac {37 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{126}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(128\) |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (42 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -111 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +28 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-74 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+42 \sqrt {-10 x^{2}-x +3}\right )}{126 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\) | \(131\) |
1/3*(-1+2*x)/(2+3*x)*(3+5*x)^(1/2)/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5 *x))^(1/2)/(1-2*x)^(1/2)-(1/9*10^(1/2)*arcsin(20/11*x+1/11)-37/126*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 = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {37 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, \sqrt {10} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{126 \, {\left (3 \, x + 2\right )}} \]
-1/126*(37*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3) *sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*sqrt(10)*(3*x + 2)*arctan(1/20*sqrt (10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*sqrt(5 *x + 3)*sqrt(-2*x + 1))/(3*x + 2)
\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=\int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{2}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=-\frac {1}{9} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {37}{126} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{3 \, {\left (3 \, x + 2\right )}} \]
-1/9*sqrt(10)*arcsin(20/11*x + 1/11) + 37/126*sqrt(7)*arcsin(37/11*x/abs(3 *x + 2) + 20/11/abs(3*x + 2)) - 1/3*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (67) = 134\).
Time = 0.38 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.86 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=\frac {37}{1260} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{9} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {22 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
37/1260*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 1/9*sqrt(10)*(pi + 2*arctan(-1/4*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)))) - 22/3*sqrt(10)*((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)))/(((sqrt(2)*s qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(- 10*x + 5) - sqrt(22)))^2 + 280)
Time = 7.22 (sec) , antiderivative size = 752, normalized size of antiderivative = 8.26 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx=\frac {37\,\sqrt {7}\,\mathrm {atan}\left (\frac {31139252096\,\sqrt {7}\,\left (\sqrt {1-2\,x}-1\right )}{38759765625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\frac {1076274944\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1550390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {14849685376\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{12919921875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2152549888}{7751953125}\right )}-\frac {901743872\,\sqrt {3}\,\sqrt {7}}{4306640625\,\left (\frac {1076274944\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1550390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {14849685376\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{12919921875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2152549888}{7751953125}\right )}+\frac {450871936\,\sqrt {3}\,\sqrt {7}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{861328125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\frac {1076274944\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1550390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {14849685376\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{12919921875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2152549888}{7751953125}\right )}\right )}{63}-\frac {4\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{9}-\frac {2\,{\left (\sqrt {1-2\,x}-1\right )}^3}{3\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )}+\frac {4\,\left (\sqrt {1-2\,x}-1\right )}{15\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )}-\frac {37\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{75\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )} \]
(37*7^(1/2)*atan((31139252096*7^(1/2)*((1 - 2*x)^(1/2) - 1))/(38759765625* (3^(1/2) - (5*x + 3)^(1/2))*((1076274944*((1 - 2*x)^(1/2) - 1)^2)/(1550390 625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (14849685376*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(12919921875*(3^(1/2) - (5*x + 3)^(1/2))) - 2152549888/7751953125)) - (901743872*3^(1/2)*7^(1/2))/(4306640625*((1076274944*((1 - 2*x)^(1/2) - 1)^2)/(1550390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (14849685376*3^(1/2)*( (1 - 2*x)^(1/2) - 1))/(12919921875*(3^(1/2) - (5*x + 3)^(1/2))) - 21525498 88/7751953125)) + (450871936*3^(1/2)*7^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(861 328125*(3^(1/2) - (5*x + 3)^(1/2))^2*((1076274944*((1 - 2*x)^(1/2) - 1)^2) /(1550390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (14849685376*3^(1/2)*((1 - 2 *x)^(1/2) - 1))/(12919921875*(3^(1/2) - (5*x + 3)^(1/2))) - 2152549888/775 1953125))))/63 - (4*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^( 1/2) - (5*x + 3)^(1/2)))))/9 - (2*((1 - 2*x)^(1/2) - 1)^3)/(3*(3^(1/2) - ( 5*x + 3)^(1/2))^3*((14*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^( 1/2))^2) + ((1 - 2*x)^(1/2) - 1)^4/(3^(1/2) - (5*x + 3)^(1/2))^4 + (6*3^(1 /2)*((1 - 2*x)^(1/2) - 1)^3)/(5*(3^(1/2) - (5*x + 3)^(1/2))^3) - (12*3^(1/ 2)*((1 - 2*x)^(1/2) - 1))/(25*(3^(1/2) - (5*x + 3)^(1/2))) + 4/25)) + (4*( (1 - 2*x)^(1/2) - 1))/(15*(3^(1/2) - (5*x + 3)^(1/2))*((14*((1 - 2*x)^(1/2 ) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + ((1 - 2*x)^(1/2) - 1)^4/(3^ (1/2) - (5*x + 3)^(1/2))^4 + (6*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(5*(3^...