Integrand size = 26, antiderivative size = 166 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {196735 \sqrt {1-2 x}}{72 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {77 \sqrt {1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {7843 \sqrt {1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac {1784635 \sqrt {1-2 x}}{72 \sqrt {3+5 x}}-\frac {1361195 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{8 \sqrt {7}} \]
7/9*(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^(3/2)-1361195/56*arctan(1/7*(1-2*x)^(1 /2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-196735/72*(1-2*x)^(1/2)/(3+5*x)^(3/2)+7 7/4*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(3/2)+7843/24*(1-2*x)^(1/2)/(2+3*x)/(3 +5*x)^(3/2)+1784635/72*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 2.51 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\frac {1}{168} \left (\frac {7 \sqrt {1-2 x} \left (13784768+85776638 x+199977747 x^2+207031680 x^3+80308575 x^4\right )}{(2+3 x)^3 (3+5 x)^{3/2}}+4083585 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+4083585 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right ) \]
((7*Sqrt[1 - 2*x]*(13784768 + 85776638*x + 199977747*x^2 + 207031680*x^3 + 80308575*x^4))/((2 + 3*x)^3*(3 + 5*x)^(3/2)) + 4083585*Sqrt[7]*ArcTan[(Sq rt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 408 3585*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqr t[5 - 10*x]))])/168
Time = 0.26 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {109, 27, 166, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^4 (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{9} \int \frac {33 (13-12 x) \sqrt {1-2 x}}{2 (3 x+2)^3 (5 x+3)^{5/2}}dx+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{6} \int \frac {(13-12 x) \sqrt {1-2 x}}{(3 x+2)^3 (5 x+3)^{5/2}}dx+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {11}{6} \left (\frac {21 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}-\frac {1}{6} \int -\frac {3 (845-1228 x)}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{6} \left (\frac {1}{4} \int \frac {845-1228 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {21 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {11}{6} \left (\frac {1}{4} \left (\frac {1}{7} \int \frac {35 (4447-5704 x)}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {713 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {21 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{6} \left (\frac {1}{4} \left (\frac {5}{2} \int \frac {4447-5704 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {713 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {21 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {11}{6} \left (\frac {1}{4} \left (\frac {5}{2} \left (-\frac {2}{33} \int \frac {11 (45631-42924 x)}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {7154 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {713 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {21 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{6} \left (\frac {1}{4} \left (\frac {5}{2} \left (-\frac {1}{3} \int \frac {45631-42924 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {7154 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {713 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {21 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {11}{6} \left (\frac {1}{4} \left (\frac {5}{2} \left (\frac {1}{3} \left (\frac {2}{11} \int \frac {2450151}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {713854 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {7154 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {713 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {21 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{6} \left (\frac {1}{4} \left (\frac {5}{2} \left (\frac {1}{3} \left (222741 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {713854 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {7154 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {713 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {21 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {11}{6} \left (\frac {1}{4} \left (\frac {5}{2} \left (\frac {1}{3} \left (445482 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {713854 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {7154 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {713 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {21 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {11}{6} \left (\frac {1}{4} \left (\frac {5}{2} \left (\frac {1}{3} \left (\frac {713854 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {445482 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\right )-\frac {7154 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {713 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {21 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
(7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (11*((21*Sqrt[1 - 2* x])/(2*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + ((713*Sqrt[1 - 2*x])/((2 + 3*x)*(3 + 5*x)^(3/2)) + (5*((-7154*Sqrt[1 - 2*x])/(3*(3 + 5*x)^(3/2)) + ((713854*Sq rt[1 - 2*x])/(11*Sqrt[3 + 5*x]) - (445482*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sq rt[3 + 5*x])])/Sqrt[7])/3))/2)/4))/6
3.25.55.3.1 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*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_) )^(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] && ILtQ[m, -1] && GtQ[n, 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 = 1.17 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.80
| method | result | size |
| default | \(\frac {\left (2756419875 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+8820543600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+11282945355 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+1124320050 x^{4} \sqrt {-10 x^{2}-x +3}+7211611110 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+2898443520 x^{3} \sqrt {-10 x^{2}-x +3}+2303141940 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +2799688458 x^{2} \sqrt {-10 x^{2}-x +3}+294018120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1200872932 x \sqrt {-10 x^{2}-x +3}+192986752 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{336 \left (2+3 x \right )^{3} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(298\) |
1/336*(2756419875*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2 ))*x^5+8820543600*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2 ))*x^4+11282945355*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/ 2))*x^3+1124320050*x^4*(-10*x^2-x+3)^(1/2)+7211611110*7^(1/2)*arctan(1/14* (37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+2898443520*x^3*(-10*x^2-x+3)^(1 /2)+2303141940*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))* x+2799688458*x^2*(-10*x^2-x+3)^(1/2)+294018120*7^(1/2)*arctan(1/14*(37*x+2 0)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1200872932*x*(-10*x^2-x+3)^(1/2)+192986752 *(-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 = 0.24 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {4083585 \, \sqrt {7} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (80308575 \, x^{4} + 207031680 \, x^{3} + 199977747 \, x^{2} + 85776638 \, x + 13784768\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{336 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
-1/336*(4083585*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*(80308575*x^4 + 207031680*x^3 + 199977747*x^2 + 85776638*x + 13784768)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.45 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\frac {1361195}{112} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {1784635 \, x}{36 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1863329}{72 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {149501 \, x}{12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2401}{243 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {31213}{324 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {1115681}{648 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {13081615}{1944 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
1361195/112*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 17 84635/36*x/sqrt(-10*x^2 - x + 3) + 1863329/72/sqrt(-10*x^2 - x + 3) + 1495 01/12*x/(-10*x^2 - x + 3)^(3/2) + 2401/243/(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*(-10* x^2 - x + 3)^(3/2)) + 31213/324/(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)) + 1115681/648/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 13081615/1944/(-10*x^2 - x + 3)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (127) = 254\).
Time = 0.54 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.59 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {11}{48} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {272239}{224} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + 748 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {11 \, \sqrt {10} {\left (63359 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 30251200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {3730664000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {14922656000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
-11/48*sqrt(10)*((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)))^3 + 272239/224*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)))) + 748*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 11/4*sqrt(10)*(63359*(( 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 + 30251200*((sqrt(2)*sqrt(-10*x + 5) - s qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(2 2)))^3 + 3730664000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1 4922656000*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. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}} \,d x \]