Integrand size = 26, antiderivative size = 173 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt {3+5 x}} \, dx=-\frac {7986105 \sqrt {3+5 x}}{845152 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^4}+\frac {263 \sqrt {3+5 x}}{392 \sqrt {1-2 x} (2+3 x)^3}+\frac {6621 \sqrt {3+5 x}}{1568 \sqrt {1-2 x} (2+3 x)^2}+\frac {698295 \sqrt {3+5 x}}{21952 \sqrt {1-2 x} (2+3 x)}-\frac {24922335 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{153664 \sqrt {7}} \]
-24922335/1075648*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)- 7986105/845152*(3+5*x)^(1/2)/(1-2*x)^(1/2)+3/28*(3+5*x)^(1/2)/(2+3*x)^4/(1 -2*x)^(1/2)+263/392*(3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2)+6621/1568*(3+5*x )^(1/2)/(2+3*x)^2/(1-2*x)^(1/2)+698295/21952*(3+5*x)^(1/2)/(2+3*x)/(1-2*x) ^(1/2)
Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt {3+5 x}} \, dx=\frac {-7 \sqrt {3+5 x} \left (-205593328-491393004 x+482249808 x^2+1998242055 x^3+1293749010 x^4\right )-274145685 \sqrt {7-14 x} (2+3 x)^4 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{11832128 \sqrt {1-2 x} (2+3 x)^4} \]
(-7*Sqrt[3 + 5*x]*(-205593328 - 491393004*x + 482249808*x^2 + 1998242055*x ^3 + 1293749010*x^4) - 274145685*Sqrt[7 - 14*x]*(2 + 3*x)^4*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(11832128*Sqrt[1 - 2*x]*(2 + 3*x)^4)
Time = 0.26 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {114, 27, 168, 27, 168, 27, 168, 27, 169, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^5 \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{28} \int \frac {103-240 x}{2 (1-2 x)^{3/2} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{56} \int \frac {103-240 x}{(1-2 x)^{3/2} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{56} \left (\frac {1}{21} \int \frac {9 (1643-5260 x)}{2 (1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {263 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{56} \left (\frac {3}{14} \int \frac {1643-5260 x}{(1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {263 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{56} \left (\frac {3}{14} \left (\frac {1}{14} \int \frac {35 (3747-17656 x)}{2 (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {2207 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {263 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{56} \left (\frac {3}{14} \left (\frac {5}{4} \int \frac {3747-17656 x}{(1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {2207 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {263 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{56} \left (\frac {3}{14} \left (\frac {5}{4} \left (\frac {1}{7} \int -\frac {931060 x+66877}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx+\frac {46553 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2207 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {263 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{56} \left (\frac {3}{14} \left (\frac {5}{4} \left (\frac {46553 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {1}{14} \int \frac {931060 x+66877}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {2207 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {263 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{56} \left (\frac {3}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {2}{77} \int \frac {18276379}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {2129628 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {46553 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2207 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {263 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{56} \left (\frac {3}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {1661489}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {2129628 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {46553 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2207 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {263 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{56} \left (\frac {3}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {3322978}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {2129628 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {46553 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2207 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {263 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{56} \left (\frac {3}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (-\frac {3322978 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {2129628 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {46553 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {2207 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {263 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {3 \sqrt {5 x+3}}{28 \sqrt {1-2 x} (3 x+2)^4}\) |
(3*Sqrt[3 + 5*x])/(28*Sqrt[1 - 2*x]*(2 + 3*x)^4) + ((263*Sqrt[3 + 5*x])/(7 *Sqrt[1 - 2*x]*(2 + 3*x)^3) + (3*((2207*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*((46553*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + ((-21 29628*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) - (3322978*ArcTan[Sqrt[1 - 2*x]/(S qrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/14))/4))/14)/56
3.26.60.3.1 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[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] + Simp[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] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 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 + 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_))^(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] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(134)=268\).
Time = 1.20 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.76
| method | result | size |
| default | \(\frac {\left (44411600970 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+96225135435 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+59215467960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+18112486140 x^{4} \sqrt {-10 x^{2}-x +3}-6579496440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+27975388770 x^{3} \sqrt {-10 x^{2}-x +3}-17545323840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +6751497312 x^{2} \sqrt {-10 x^{2}-x +3}-4386330960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-6879502056 x \sqrt {-10 x^{2}-x +3}-2878306592 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{23664256 \left (2+3 x \right )^{4} \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(305\) |
1/23664256*(44411600970*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3 )^(1/2))*x^5+96225135435*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+ 3)^(1/2))*x^4+59215467960*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x +3)^(1/2))*x^3+18112486140*x^4*(-10*x^2-x+3)^(1/2)-6579496440*7^(1/2)*arct an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+27975388770*x^3*(-10*x^ 2-x+3)^(1/2)-17545323840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+ 3)^(1/2))*x+6751497312*x^2*(-10*x^2-x+3)^(1/2)-4386330960*7^(1/2)*arctan(1 /14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-6879502056*x*(-10*x^2-x+3)^(1/2 )-2878306592*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(2+3*x)^4/(- 1+2*x)/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt {3+5 x}} \, dx=-\frac {274145685 \, \sqrt {7} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1293749010 \, x^{4} + 1998242055 \, x^{3} + 482249808 \, x^{2} - 491393004 \, x - 205593328\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{23664256 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
-1/23664256*(274145685*sqrt(7)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64* x - 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x ^2 + x - 3)) - 14*(1293749010*x^4 + 1998242055*x^3 + 482249808*x^2 - 49139 3004*x - 205593328)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(162*x^5 + 351*x^4 + 216 *x^3 - 24*x^2 - 64*x - 16)
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt {3+5 x}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{5} \sqrt {5 x + 3}}\, dx \]
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{5} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (134) = 268\).
Time = 0.60 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.28 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt {3+5 x}} \, dx=\frac {4984467}{4302592} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {64 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{924385 \, {\left (2 \, x - 1\right )}} + \frac {99 \, \sqrt {10} {\left (4411181 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 2388710520 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 506212728000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {38676680000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {154706720000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{537824 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
4984467/4302592*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)))) - 64/924385*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) /(2*x - 1) + 99/537824*sqrt(10)*(4411181*((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))) ^7 + 2388710520*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sq rt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 506212728000*((sqrt( 2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq rt(-10*x + 5) - sqrt(22)))^3 + 38676680000000*(sqrt(2)*sqrt(-10*x + 5) - s qrt(22))/sqrt(5*x + 3) - 154706720000000*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*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt {3+5 x}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^5\,\sqrt {5\,x+3}} \,d x \]