Integrand size = 26, antiderivative size = 171 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {7591 \sqrt {1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac {511 (2+3 x)^4}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {261331 \sqrt {1-2 x} (2+3 x)^2}{2196150 \sqrt {3+5 x}}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (190406711+78981180 x)}{117128000}+\frac {753543 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8000 \sqrt {10}} \]
7/33*(2+3*x)^5/(1-2*x)^(3/2)/(3+5*x)^(3/2)+753543/80000*arcsin(1/11*22^(1/ 2)*(3+5*x)^(1/2))*10^(1/2)-511/242*(2+3*x)^4/(3+5*x)^(3/2)/(1-2*x)^(1/2)+7 591/39930*(2+3*x)^3*(1-2*x)^(1/2)/(3+5*x)^(3/2)+261331/2196150*(2+3*x)^2*( 1-2*x)^(1/2)/(3+5*x)^(1/2)-7/117128000*(190406711+78981180*x)*(1-2*x)^(1/2 )*(3+5*x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.46 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {44437106459+19932058554 x-274128335769 x^2-252342435560 x^3+97980793020 x^4+12807946800 x^5}{351384000 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {753543 \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{8000 \sqrt {10}} \]
-1/351384000*(44437106459 + 19932058554*x - 274128335769*x^2 - 25234243556 0*x^3 + 97980793020*x^4 + 12807946800*x^5)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2 )) - (753543*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(8000*Sqrt[10])
Time = 0.25 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {109, 27, 167, 27, 167, 27, 167, 27, 164, 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.
| \(\displaystyle \int \frac {(3 x+2)^6}{(1-2 x)^{5/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{33} \int \frac {3 (3 x+2)^4 (239 x+136)}{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{22} \int \frac {(3 x+2)^4 (239 x+136)}{(1-2 x)^{3/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (-\frac {1}{11} \int -\frac {(3 x+2)^3 (38547 x+21610)}{2 \sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \int \frac {(3 x+2)^3 (38547 x+21610)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {2}{165} \int \frac {7 (3 x+2)^2 (571191 x+335248)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \int \frac {(3 x+2)^2 (571191 x+335248)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \left (\frac {2}{55} \int \frac {3 (3 x+2) (6581765 x+4039402)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {74666 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \left (\frac {3}{55} \int \frac {(3 x+2) (6581765 x+4039402)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {74666 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \left (\frac {3}{55} \left (\frac {1576089009}{160} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (78981180 x+190406711)\right )+\frac {74666 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \left (\frac {3}{55} \left (\frac {1576089009}{400} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (78981180 x+190406711)\right )+\frac {74666 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \left (\frac {3}{55} \left (\frac {1576089009 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80 \sqrt {10}}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (78981180 x+190406711)\right )+\frac {74666 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
(7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + ((-511*(2 + 3*x)^4) /(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((15182*Sqrt[1 - 2*x]*(2 + 3*x)^3)/( 165*(3 + 5*x)^(3/2)) + (7*((74666*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(55*Sqrt[3 + 5*x]) + (3*(-1/80*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(190406711 + 78981180*x)) + (1576089009*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80*Sqrt[10])))/55))/165)/2 2)/22
3.27.27.3.1 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_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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[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 = 4.17 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.16
| method | result | size |
| default | \(\frac {\sqrt {1-2 x}\, \left (3309786918900 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}-256158936000 x^{5} \sqrt {-10 x^{2}-x +3}+661957383780 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-1959615860400 x^{4} \sqrt {-10 x^{2}-x +3}-1952774282151 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+5046848711200 x^{3} \sqrt {-10 x^{2}-x +3}-198587215134 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +5482566715380 x^{2} \sqrt {-10 x^{2}-x +3}+297880822701 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-398641171080 x \sqrt {-10 x^{2}-x +3}-888742129180 \sqrt {-10 x^{2}-x +3}\right )}{7027680000 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(199\) |
1/7027680000*(1-2*x)^(1/2)*(3309786918900*10^(1/2)*arcsin(20/11*x+1/11)*x^ 4-256158936000*x^5*(-10*x^2-x+3)^(1/2)+661957383780*10^(1/2)*arcsin(20/11* x+1/11)*x^3-1959615860400*x^4*(-10*x^2-x+3)^(1/2)-1952774282151*10^(1/2)*a rcsin(20/11*x+1/11)*x^2+5046848711200*x^3*(-10*x^2-x+3)^(1/2)-198587215134 *10^(1/2)*arcsin(20/11*x+1/11)*x+5482566715380*x^2*(-10*x^2-x+3)^(1/2)+297 880822701*10^(1/2)*arcsin(20/11*x+1/11)-398641171080*x*(-10*x^2-x+3)^(1/2) -888742129180*(-10*x^2-x+3)^(1/2))/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^ (3/2)
Time = 0.23 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.74 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {33097869189 \, \sqrt {10} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (12807946800 \, x^{5} + 97980793020 \, x^{4} - 252342435560 \, x^{3} - 274128335769 \, x^{2} + 19932058554 \, x + 44437106459\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7027680000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
-1/7027680000*(33097869189*sqrt(10)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)* arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(12807946800*x^5 + 97980793020*x^4 - 252342435560*x^3 - 27412833 5769*x^2 + 19932058554*x + 44437106459)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(100 *x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
\[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{6}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.25 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {729 \, x^{5}}{20 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {111537 \, x^{4}}{400 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {251181}{234256000} \, x {\left (\frac {7220 \, x}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {361}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} - \frac {753543}{160000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {90676341}{117128000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {170985889 \, x}{7027680 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {766611 \, x^{2}}{1000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1005653687}{878460000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {416356591 \, x}{3630000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {496819753}{3630000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
-729/20*x^5/(-10*x^2 - x + 3)^(3/2) - 111537/400*x^4/(-10*x^2 - x + 3)^(3/ 2) + 251181/234256000*x*(7220*x/sqrt(-10*x^2 - x + 3) + 439230*x^2/(-10*x^ 2 - x + 3)^(3/2) + 361/sqrt(-10*x^2 - x + 3) + 21901*x/(-10*x^2 - x + 3)^( 3/2) - 87483/(-10*x^2 - x + 3)^(3/2)) - 753543/160000*sqrt(10)*arcsin(-20/ 11*x - 1/11) + 90676341/117128000*sqrt(-10*x^2 - x + 3) - 170985889/702768 0*x/sqrt(-10*x^2 - x + 3) + 766611/1000*x^2/(-10*x^2 - x + 3)^(3/2) + 1005 653687/878460000/sqrt(-10*x^2 - x + 3) + 416356591/3630000*x/(-10*x^2 - x + 3)^(3/2) - 496819753/3630000/(-10*x^2 - x + 3)^(3/2)
Time = 0.35 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.20 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{2196150000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {753543}{80000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {37 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{16637500 \, \sqrt {5 \, x + 3}} - \frac {{\left (4 \, {\left (32019867 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 93 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 110347010662 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1820310410259 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{219615000000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1221 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{137259375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
-1/2196150000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3 /2) + 753543/80000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 37/16637 500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/219615 000000*(4*(32019867*(4*sqrt(5)*(5*x + 3) + 93*sqrt(5))*(5*x + 3) - 1103470 10662*sqrt(5))*(5*x + 3) + 1820310410259*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 1/137259375*sqrt(10)*(5*x + 3)^(3/2)*(1221*(sqrt(2)*sq rt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt (22))^3
Timed out. \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^6}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]