Integrand size = 26, antiderivative size = 173 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=-\frac {16985 \sqrt {3+5 x}}{316932 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac {3 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)^2}+\frac {605 \sqrt {3+5 x}}{2744 \sqrt {1-2 x} (2+3 x)}-\frac {25365 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208 \sqrt {7}} \]
-25365/134456*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+2/21 *(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^3-16985/316932*(3+5*x)^(1/2)/(1-2*x)^ (1/2)-3/49*(3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2)-1/196*(3+5*x)^(1/2)/(2+3* x)^2/(1-2*x)^(1/2)+605/2744*(3+5*x)^(1/2)/(2+3*x)/(1-2*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {\frac {7 \sqrt {3+5 x} \left (302352-39530 x-1465461 x^2+235980 x^3+1834380 x^4\right )}{(1-2 x)^{3/2} (2+3 x)^3}-837045 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4437048} \]
((7*Sqrt[3 + 5*x]*(302352 - 39530*x - 1465461*x^2 + 235980*x^3 + 1834380*x ^4))/((1 - 2*x)^(3/2)*(2 + 3*x)^3) - 837045*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/( Sqrt[7]*Sqrt[3 + 5*x])])/4437048
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 = {110, 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.
| \(\displaystyle \int \frac {\sqrt {5 x+3}}{(1-2 x)^{5/2} (3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac {2}{21} \int -\frac {120 x+71}{2 (1-2 x)^{3/2} (3 x+2)^4 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \int \frac {120 x+71}{(1-2 x)^{3/2} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{21} \int \frac {3 (540 x+353)}{2 (1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {9 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{14} \int \frac {540 x+353}{(1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {9 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{14} \left (\frac {1}{14} \int \frac {35 (24 x+137)}{2 (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {9 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{14} \left (\frac {5}{4} \int \frac {24 x+137}{(1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {9 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{7} \int \frac {233-7260 x}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx+\frac {363 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {9 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \int \frac {233-7260 x}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx+\frac {363 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {9 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (-\frac {2}{77} \int -\frac {167409}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {13588 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {363 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {9 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {15219}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {13588 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {363 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {9 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {30438}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {13588 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {363 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {9 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (-\frac {30438 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {13588 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {363 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {9 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}\) |
(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + ((-9*Sqrt[3 + 5*x])/( 7*Sqrt[1 - 2*x]*(2 + 3*x)^3) + ((-3*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]*(2 + 3 *x)^2) + (5*((363*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + ((-13588*Sq rt[3 + 5*x])/(77*Sqrt[1 - 2*x]) - (30438*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr t[3 + 5*x])])/(7*Sqrt[7]))/14))/4)/14)/21
3.26.88.3.1 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_))^(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 = 4.07 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.76
| method | result | size |
| default | \(\frac {\left (90400860 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+90400860 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-37667025 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+25681320 x^{4} \sqrt {-10 x^{2}-x +3}-48548610 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+3303720 x^{3} \sqrt {-10 x^{2}-x +3}+3348180 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -20516454 x^{2} \sqrt {-10 x^{2}-x +3}+6696360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-553420 x \sqrt {-10 x^{2}-x +3}+4232928 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{8874096 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(305\) |
1/8874096*(90400860*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1 /2))*x^5+90400860*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2 ))*x^4-37667025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) *x^3+25681320*x^4*(-10*x^2-x+3)^(1/2)-48548610*7^(1/2)*arctan(1/14*(37*x+2 0)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+3303720*x^3*(-10*x^2-x+3)^(1/2)+334818 0*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-20516454*x^ 2*(-10*x^2-x+3)^(1/2)+6696360*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x ^2-x+3)^(1/2))-553420*x*(-10*x^2-x+3)^(1/2)+4232928*(-10*x^2-x+3)^(1/2))*( 1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)
Time = 0.24 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=-\frac {837045 \, \sqrt {7} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1834380 \, x^{4} + 235980 \, x^{3} - 1465461 \, x^{2} - 39530 \, x + 302352\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{8874096 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
-1/8874096*(837045*sqrt(7)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8) *arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1834380*x^4 + 235980*x^3 - 1465461*x^2 - 39530*x + 302352)*sqr t(5*x + 3)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)
\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{4}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {25365}{268912} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {84925 \, x}{316932 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {131015}{633864 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {375 \, x}{1372 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{189 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {11}{196 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {377}{3528 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {3215}{74088 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
25365/268912*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 8 4925/316932*x/sqrt(-10*x^2 - x + 3) + 131015/633864/sqrt(-10*x^2 - x + 3) + 375/1372*x/(-10*x^2 - x + 3)^(3/2) - 1/189/(27*(-10*x^2 - x + 3)^(3/2)*x ^3 + 54*(-10*x^2 - x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-1 0*x^2 - x + 3)^(3/2)) + 11/196/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^ 2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) - 377/3528/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 3215/74088/(-10*x^2 - x + 3)^ (3/2)
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (134) = 268\).
Time = 0.62 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {5073}{537824} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {32 \, {\left (361 \, \sqrt {5} {\left (5 \, x + 3\right )} - 2178 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{13865775 \, {\left (2 \, x - 1\right )}^{2}} - \frac {297 \, \sqrt {10} {\left (603 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 235200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {37240000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {148960000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{67228 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
5073/537824*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)))) - 32/13865775*(361*sqrt(5)*(5*x + 3) - 2178*sqrt(5))* sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 297/67228*sqrt(10)*(603*((sqrt (2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s qrt(-10*x + 5) - sqrt(22)))^5 - 235200*((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)))^3 - 37240000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 148960000 *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)^3
Timed out. \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4} \,d x \]