Integrand size = 26, antiderivative size = 166 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {21891025 \sqrt {1-2 x}}{90552 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {325 \sqrt {1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {79335 \sqrt {1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac {2184369575 \sqrt {1-2 x}}{996072 \sqrt {3+5 x}}-\frac {41307885 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]
-41307885/19208*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-21 891025/90552*(1-2*x)^(1/2)/(3+5*x)^(3/2)+1/7*(1-2*x)^(1/2)/(2+3*x)^3/(3+5* x)^(3/2)+325/196*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(3/2)+79335/2744*(1-2*x)^ (1/2)/(2+3*x)/(3+5*x)^(3/2)+2184369575/996072*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=\frac {\sqrt {1-2 x} \left (50617099616+314968389410 x+734310313245 x^2+760212086400 x^3+294889892625 x^4\right )}{996072 (2+3 x)^3 (3+5 x)^{3/2}}-\frac {41307885 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]
(Sqrt[1 - 2*x]*(50617099616 + 314968389410*x + 734310313245*x^2 + 76021208 6400*x^3 + 294889892625*x^4))/(996072*(2 + 3*x)^3*(3 + 5*x)^(3/2)) - (4130 7885*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])
Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.11, 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, 169, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{21} \int \frac {15 (11-16 x)}{2 \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{5/2}}dx+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \int \frac {11-16 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{5/2}}dx+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{14} \int \frac {2689-3900 x}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {65 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \int \frac {2689-3900 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {65 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {494833-634680 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {15867 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {65 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {1}{14} \int \frac {494833-634680 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {15867 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {65 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {1}{14} \left (-\frac {2}{33} \int \frac {55851707-52538460 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {8756410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {15867 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {65 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {1}{14} \left (-\frac {1}{33} \int \frac {55851707-52538460 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {8756410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {15867 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {65 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {1}{14} \left (\frac {1}{33} \left (\frac {2}{11} \int \frac {2998952451}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {873747830 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {8756410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {15867 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {65 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {1}{14} \left (\frac {1}{33} \left (272632041 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {873747830 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {8756410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {15867 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {65 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {1}{14} \left (\frac {1}{33} \left (545264082 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {873747830 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {8756410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {15867 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {65 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{28} \left (\frac {1}{14} \left (\frac {1}{33} \left (\frac {873747830 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {545264082 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\right )-\frac {8756410 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {15867 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {65 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}\) |
Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (5*((65*Sqrt[1 - 2*x])/(14 *(2 + 3*x)^2*(3 + 5*x)^(3/2)) + ((15867*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5 *x)^(3/2)) + ((-8756410*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) + ((873747830* Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) - (545264082*ArcTan[Sqrt[1 - 2*x]/(Sqrt[ 7]*Sqrt[3 + 5*x])])/Sqrt[7])/33)/14)/28))/14
3.26.16.3.1 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. \(297\) vs. \(2(127)=254\).
Time = 3.96 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.80
| method | result | size |
| default | \(\frac {\left (10121464522125 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+32388686470800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+41430528110565 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+4128458496750 x^{4} \sqrt {-10 x^{2}-x +3}+26480750142330 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+10642969209600 x^{3} \sqrt {-10 x^{2}-x +3}+8457045911820 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +10280344385430 x^{2} \sqrt {-10 x^{2}-x +3}+1079622882360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4409557451740 x \sqrt {-10 x^{2}-x +3}+708639394624 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{13945008 \left (2+3 x \right )^{3} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(298\) |
1/13945008*(10121464522125*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2- x+3)^(1/2))*x^5+32388686470800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10* x^2-x+3)^(1/2))*x^4+41430528110565*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/( -10*x^2-x+3)^(1/2))*x^3+4128458496750*x^4*(-10*x^2-x+3)^(1/2)+264807501423 30*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+10642969 209600*x^3*(-10*x^2-x+3)^(1/2)+8457045911820*7^(1/2)*arctan(1/14*(37*x+20) *7^(1/2)/(-10*x^2-x+3)^(1/2))*x+10280344385430*x^2*(-10*x^2-x+3)^(1/2)+107 9622882360*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4409 557451740*x*(-10*x^2-x+3)^(1/2)+708639394624*(-10*x^2-x+3)^(1/2))*(1-2*x)^ (1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)
Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {14994762255 \, \sqrt {7} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (294889892625 \, x^{4} + 760212086400 \, x^{3} + 734310313245 \, x^{2} + 314968389410 \, x + 50617099616\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{13945008 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
-1/13945008*(14994762255*sqrt(7)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1 )/(10*x^2 + x - 3)) - 14*(294889892625*x^4 + 760212086400*x^3 + 7343103132 45*x^2 + 314968389410*x + 50617099616)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(675* x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{4} \sqrt {-2 \, x + 1}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (127) = 254\).
Time = 0.49 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.59 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {125}{5808} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {8261577}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {8125}{121} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {1485 \, \sqrt {10} {\left (13759 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 6614720 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {818950720 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {3275802880 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1372 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
-125/5808*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)))^3 + 8261577/76832*sqr t(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(2 2)))) + 8125/121*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))) + 1485/1372*sq rt(10)*(13759*((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)))^5 + 6614720*((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 + 818950720*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr t(5*x + 3) - 3275802880*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. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}} \,d x \]