Integrand size = 26, antiderivative size = 166 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {73435}{15092 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {36657025 \sqrt {1-2 x}}{332024 \sqrt {3+5 x}}+\frac {1}{7 \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}+\frac {37}{28 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {6525}{392 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {2079585 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]
2079585/19208*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-7343 5/15092/(1-2*x)^(1/2)/(3+5*x)^(1/2)+1/7/(2+3*x)^3/(1-2*x)^(1/2)/(3+5*x)^(1 /2)+37/28/(2+3*x)^2/(1-2*x)^(1/2)/(3+5*x)^(1/2)+6525/392/(2+3*x)/(1-2*x)^( 1/2)/(3+5*x)^(1/2)-36657025/332024*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=\frac {\frac {7 \left (-283149136-723664682 x+622325745 x^2+2925598635 x^3+1979479350 x^4\right )}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}}+251629785 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2324168} \]
((7*(-283149136 - 723664682*x + 622325745*x^2 + 2925598635*x^3 + 197947935 0*x^4))/(Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x]) + 251629785*Sqrt[7]*ArcT an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/2324168
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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^4 (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{21} \int \frac {3 (33-80 x)}{2 (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{3/2}}dx+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {33-80 x}{(1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{3/2}}dx+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{14} \int \frac {35 (139-444 x)}{2 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {37}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \int \frac {139-444 x}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {37}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{7} \int \frac {11413-52200 x}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}dx+\frac {1305}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\right )+\frac {37}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \int \frac {11413-52200 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}dx+\frac {1305}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\right )+\frac {37}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (-\frac {2}{77} \int -\frac {937549-881220 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {58748}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {1305}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\right )+\frac {37}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {1}{77} \int \frac {937549-881220 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {58748}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {1305}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\right )+\frac {37}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {1}{77} \left (-\frac {2}{11} \int \frac {50325957}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {14662810 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {58748}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {1305}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\right )+\frac {37}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {1}{77} \left (-4575087 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {14662810 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {58748}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {1305}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\right )+\frac {37}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {1}{77} \left (-9150174 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {14662810 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {58748}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {1305}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\right )+\frac {37}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {1}{77} \left (\frac {9150174 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}-\frac {14662810 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {58748}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {1305}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\right )+\frac {37}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {1}{7 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}\) |
1/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (37/(2*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (5*(1305/(7*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x]) + (-58748/(77*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + ((-14662810*Sqrt[1 - 2*x])/( 11*Sqrt[3 + 5*x]) + (9150174*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])] )/Sqrt[7])/77)/14))/4)/14
3.26.70.3.1 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. \(2(127)=254\).
Time = 1.22 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.84
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \left (67940041950 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+142674088095 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+83792718405 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+27712710900 x^{4} \sqrt {-10 x^{2}-x +3}-11574970110 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+40958380890 x^{3} \sqrt {-10 x^{2}-x +3}-25162978500 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +8712560430 x^{2} \sqrt {-10 x^{2}-x +3}-6039114840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-10131305548 x \sqrt {-10 x^{2}-x +3}-3964087904 \sqrt {-10 x^{2}-x +3}\right )}{4648336 \left (2+3 x \right )^{3} \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(305\) |
-1/4648336*(1-2*x)^(1/2)*(67940041950*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2 )/(-10*x^2-x+3)^(1/2))*x^5+142674088095*7^(1/2)*arctan(1/14*(37*x+20)*7^(1 /2)/(-10*x^2-x+3)^(1/2))*x^4+83792718405*7^(1/2)*arctan(1/14*(37*x+20)*7^( 1/2)/(-10*x^2-x+3)^(1/2))*x^3+27712710900*x^4*(-10*x^2-x+3)^(1/2)-11574970 110*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+4095838 0890*x^3*(-10*x^2-x+3)^(1/2)-25162978500*7^(1/2)*arctan(1/14*(37*x+20)*7^( 1/2)/(-10*x^2-x+3)^(1/2))*x+8712560430*x^2*(-10*x^2-x+3)^(1/2)-6039114840* 7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-10131305548*x*( -10*x^2-x+3)^(1/2)-3964087904*(-10*x^2-x+3)^(1/2))/(2+3*x)^3/(-1+2*x)/(-10 *x^2-x+3)^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=\frac {251629785 \, \sqrt {7} {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1979479350 \, x^{4} + 2925598635 \, x^{3} + 622325745 \, x^{2} - 723664682 \, x - 283149136\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4648336 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} \]
1/4648336*(251629785*sqrt(7)*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^ 2 + x - 3)) - 14*(1979479350*x^4 + 2925598635*x^3 + 622325745*x^2 - 723664 682*x - 283149136)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(270*x^5 + 567*x^4 + 333* x^3 - 46*x^2 - 100*x - 24)
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {2079585}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {36657025 \, x}{166012 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {38272595}{332024 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{7 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {37}{28 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {6525}{392 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
-2079585/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 36657025/166012*x/sqrt(-10*x^2 - x + 3) - 38272595/332024/sqrt(-10*x^2 - x + 3) + 1/7/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 37/28/(9*sqrt(-1 0*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 6525/392/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (127) = 254\).
Time = 0.57 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.38 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {415917}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {625}{242} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {64 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1452605 \, {\left (2 \, x - 1\right )}} - \frac {297 \, \sqrt {10} {\left (37841 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 16959040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {2009470400 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {8037881600 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9604 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
-415917/76832*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(-1 0*x + 5) - sqrt(22)))) - 625/242*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)) ) - 64/1452605*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 297/9604* sqrt(10)*(37841*((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 + 16959040*((sqrt(2)*s qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(- 10*x + 5) - sqrt(22)))^3 + 2009470400*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)) /sqrt(5*x + 3) - 8037881600*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. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^{3/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}} \,d x \]