Integrand size = 26, antiderivative size = 164 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {376 (1-2 x)^{3/2} (2+3 x)^3}{75 \sqrt {3+5 x}}+\frac {69713 \sqrt {1-2 x} \sqrt {3+5 x}}{400000}+\frac {741}{250} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (3185+4392 x)}{40000}+\frac {766843 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400000 \sqrt {10}} \]
-2/15*(1-2*x)^(5/2)*(2+3*x)^3/(3+5*x)^(3/2)+766843/4000000*arcsin(1/11*22^ (1/2)*(3+5*x)^(1/2))*10^(1/2)-376/75*(1-2*x)^(3/2)*(2+3*x)^3/(3+5*x)^(1/2) +741/250*(1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(1/2)+21/40000*(1-2*x)^(3/2)*(318 5+4392*x)*(3+5*x)^(1/2)+69713/400000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.53 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.53 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx=\frac {\frac {5 \sqrt {1-2 x} \left (2322001+7876210 x+3074745 x^2-7724700 x^3+972000 x^4+6480000 x^5\right )}{(3+5 x)^{3/2}}-2300529 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{6000000} \]
((5*Sqrt[1 - 2*x]*(2322001 + 7876210*x + 3074745*x^2 - 7724700*x^3 + 97200 0*x^4 + 6480000*x^5))/(3 + 5*x)^(3/2) - 2300529*Sqrt[10]*ArcTan[Sqrt[6 + 1 0*x]/(Sqrt[11] - Sqrt[5 - 10*x])])/6000000
Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {108, 25, 167, 27, 170, 27, 164, 60, 64, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^3}{(5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{15} \int -\frac {(1-2 x)^{3/2} (3 x+2)^2 (33 x+1)}{(5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{15} \int \frac {(1-2 x)^{3/2} (3 x+2)^2 (33 x+1)}{(5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {2}{15} \left (\frac {2}{5} \int -\frac {3 (83-1482 x) \sqrt {1-2 x} (3 x+2)^2}{2 \sqrt {5 x+3}}dx+\frac {188 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{15} \left (\frac {188 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}-\frac {3}{5} \int \frac {(83-1482 x) \sqrt {1-2 x} (3 x+2)^2}{\sqrt {5 x+3}}dx\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle -\frac {2}{15} \left (\frac {188 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}-\frac {3}{5} \left (\frac {741}{20} (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}-\frac {1}{40} \int \frac {7 \sqrt {1-2 x} (3 x+2) (1647 x+110)}{\sqrt {5 x+3}}dx\right )\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{15} \left (\frac {188 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}-\frac {3}{5} \left (\frac {741}{20} (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}-\frac {7}{40} \int \frac {\sqrt {1-2 x} (3 x+2) (1647 x+110)}{\sqrt {5 x+3}}dx\right )\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {2}{15} \left (\frac {188 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}-\frac {3}{5} \left (\frac {741}{20} (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}-\frac {7}{40} \left (-\frac {9959}{160} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx-\frac {3}{80} \sqrt {5 x+3} (4392 x+3185) (1-2 x)^{3/2}\right )\right )\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {2}{15} \left (\frac {188 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}-\frac {3}{5} \left (\frac {741}{20} (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}-\frac {7}{40} \left (-\frac {9959}{160} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{80} \sqrt {5 x+3} (4392 x+3185) (1-2 x)^{3/2}\right )\right )\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle -\frac {2}{15} \left (\frac {188 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}-\frac {3}{5} \left (\frac {741}{20} (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}-\frac {7}{40} \left (-\frac {9959}{160} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{80} \sqrt {5 x+3} (4392 x+3185) (1-2 x)^{3/2}\right )\right )\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {2}{15} \left (\frac {188 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}-\frac {3}{5} \left (\frac {741}{20} (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}-\frac {7}{40} \left (-\frac {9959}{160} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{80} \sqrt {5 x+3} (4392 x+3185) (1-2 x)^{3/2}\right )\right )\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}\) |
(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (2*((188*(1 - 2*x) ^(3/2)*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) - (3*((741*(1 - 2*x)^(3/2)*(2 + 3*x) ^2*Sqrt[3 + 5*x])/20 - (7*((-3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]*(3185 + 4392* x))/80 - (9959*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sq rt[3 + 5*x]])/(5*Sqrt[10])))/160))/40))/5))/15
3.25.48.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_) )^(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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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[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 = 1.17 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(\frac {\left (129600000 x^{5} \sqrt {-10 x^{2}-x +3}+19440000 x^{4} \sqrt {-10 x^{2}-x +3}+57513225 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-154494000 x^{3} \sqrt {-10 x^{2}-x +3}+69015870 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +61494900 x^{2} \sqrt {-10 x^{2}-x +3}+20704761 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+157524200 x \sqrt {-10 x^{2}-x +3}+46440020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{24000000 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(164\) |
1/24000000*(129600000*x^5*(-10*x^2-x+3)^(1/2)+19440000*x^4*(-10*x^2-x+3)^( 1/2)+57513225*10^(1/2)*arcsin(20/11*x+1/11)*x^2-154494000*x^3*(-10*x^2-x+3 )^(1/2)+69015870*10^(1/2)*arcsin(20/11*x+1/11)*x+61494900*x^2*(-10*x^2-x+3 )^(1/2)+20704761*10^(1/2)*arcsin(20/11*x+1/11)+157524200*x*(-10*x^2-x+3)^( 1/2)+46440020*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5* x)^(3/2)
Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx=-\frac {2300529 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (6480000 \, x^{5} + 972000 \, x^{4} - 7724700 \, x^{3} + 3074745 \, x^{2} + 7876210 \, x + 2322001\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{24000000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
-1/24000000*(2300529*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20 *x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 20*(6480000*x^5 + 972000*x^4 - 7724700*x^3 + 3074745*x^2 + 7876210*x + 2322001)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{3}}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.98 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx=-\frac {395307}{8000000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {23}{11}\right ) + \frac {23221}{500000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {99}{5000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{625 \, {\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1250 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{2500 \, {\left (5 \, x + 3\right )}} + \frac {3267}{20000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} x + \frac {75141}{400000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} + \frac {3267}{25000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {11 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3750 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {99 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2500 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {99 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2500 \, {\left (5 \, x + 3\right )}} - \frac {121 \, \sqrt {-10 \, x^{2} - x + 3}}{18750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {9493 \, \sqrt {-10 \, x^{2} - x + 3}}{37500 \, {\left (5 \, x + 3\right )}} \]
-395307/8000000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 23221/500000*s qrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 99/5000*(-10*x^2 - x + 3)^(3/2) + 1/625*(-10*x^2 - x + 3)^(5/2)/(625*x^4 + 1500*x^3 + 1350*x^2 + 540*x + 81) + 9/1250*(-10*x^2 - x + 3)^(5/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 9/625 *(-10*x^2 - x + 3)^(5/2)/(25*x^2 + 30*x + 9) + 27/2500*(-10*x^2 - x + 3)^( 5/2)/(5*x + 3) + 3267/20000*sqrt(10*x^2 + 23*x + 51/5)*x + 75141/400000*sq rt(10*x^2 + 23*x + 51/5) + 3267/25000*sqrt(-10*x^2 - x + 3) - 11/3750*(-10 *x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 99/2500*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 99/2500*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 121/18750*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 9493/37500*sqrt (-10*x^2 - x + 3)/(5*x + 3)
Time = 0.45 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.21 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx=\frac {1}{10000000} \, {\left (36 \, {\left (24 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} - 57 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4915 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 338795 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{3750000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {766843}{4000000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {2079 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{312500 \, \sqrt {5 \, x + 3}} + \frac {11 \, \sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {567 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{234375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
1/10000000*(36*(24*(4*sqrt(5)*(5*x + 3) - 57*sqrt(5))*(5*x + 3) + 4915*sqr t(5))*(5*x + 3) + 338795*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/37500 00*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 76684 3/4000000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 2079/312500*sqrt( 10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 11/234375*sqrt(10 )*(5*x + 3)^(3/2)*(567*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3}{{\left (5\,x+3\right )}^{5/2}} \,d x \]