Integrand size = 26, antiderivative size = 179 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=-\frac {155777 \sqrt {1-2 x} \sqrt {3+5 x}}{31104}+\frac {1453}{288} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{324} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {660959 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{93312 \sqrt {10}}-\frac {1295}{729} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
-5/18*(1-2*x)^(3/2)*(3+5*x)^(5/2)-1/3*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)- 1295/729*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-660959/93 3120*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+1453/288*(3+5*x)^(3/2)*( 1-2*x)^(1/2)-247/324*(3+5*x)^(5/2)*(1-2*x)^(1/2)-155777/31104*(1-2*x)^(1/2 )*(3+5*x)^(1/2)
Time = 0.34 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=\frac {-\frac {30 \sqrt {1-2 x} \left (136974+9743 x-183201 x^2+945420 x^3+295200 x^4-1296000 x^5\right )}{(2+3 x) \sqrt {3+5 x}}+660959 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-1657600 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{933120} \]
((-30*Sqrt[1 - 2*x]*(136974 + 9743*x - 183201*x^2 + 945420*x^3 + 295200*x^ 4 - 1296000*x^5))/((2 + 3*x)*Sqrt[3 + 5*x]) + 660959*Sqrt[10]*ArcTan[Sqrt[ 5/2 - 5*x]/Sqrt[3 + 5*x]] - 1657600*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]* Sqrt[3 + 5*x])])/933120
Time = 0.28 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.15, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {108, 27, 171, 27, 171, 27, 171, 27, 171, 27, 175, 64, 104, 217, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^2} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{3} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 (3 x+2)}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{6} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{3 x+2}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{60} \int -\frac {10 (8-247 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{3 x+2}dx+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {1}{6} \int \frac {(8-247 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{3 x+2}dx\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {1}{45} \int \frac {(10106-39231 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)}dx\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {1}{90} \int \frac {(10106-39231 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{90} \left (\frac {1}{12} \int -\frac {3 \sqrt {5 x+3} (155777 x+7158)}{2 \sqrt {1-2 x} (3 x+2)}dx-\frac {13077}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{90} \left (-\frac {1}{8} \int \frac {\sqrt {5 x+3} (155777 x+7158)}{\sqrt {1-2 x} (3 x+2)}dx-\frac {13077}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {660959 x+53866}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {155777}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {13077}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \int \frac {660959 x+53866}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {155777}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {13077}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {660959}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1160320}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {155777}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {13077}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {1321918}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1160320}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {155777}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {13077}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {1321918}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {2320640}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {155777}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {13077}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {1321918}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {331520}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )+\frac {155777}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {13077}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {5}{6} \left (\frac {1}{6} \left (\frac {1}{90} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {660959}{3} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {331520}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )+\frac {155777}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {13077}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {247}{45} \sqrt {1-2 x} (5 x+3)^{5/2}\right )+\frac {1}{3} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
-1/3*((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x) - (5*(((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/3 + ((247*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/45 + ((-13077*Sqrt [1 - 2*x]*(3 + 5*x)^(3/2))/4 + ((155777*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6 + ( (660959*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 + (331520*Sqrt[7]*Ar cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3)/12)/8)/90)/6))/6
3.25.16.3.1 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[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 _)), 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[(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_) )^(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] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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])
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 = 148, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (259200 x^{4}-214560 x^{3}-60348 x^{2}+72849 x -45658\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{31104 \left (2+3 x \right ) \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {660959 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{1866240}-\frac {1295 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{1458}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(148\) |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-15552000 x^{4} \sqrt {-10 x^{2}-x +3}+12873600 x^{3} \sqrt {-10 x^{2}-x +3}+1982877 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -4972800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +3620880 x^{2} \sqrt {-10 x^{2}-x +3}+1321918 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3315200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-4370940 x \sqrt {-10 x^{2}-x +3}+2739480 \sqrt {-10 x^{2}-x +3}\right )}{1866240 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\) | \(197\) |
-1/31104*(-1+2*x)*(3+5*x)^(1/2)*(259200*x^4-214560*x^3-60348*x^2+72849*x-4 5658)/(2+3*x)/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1 /2)-(660959/1866240*10^(1/2)*arcsin(20/11*x+1/11)-1295/1458*7^(1/2)*arctan (9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2)))*((1-2*x)*(3+5 *x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=-\frac {1657600 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 660959 \, \sqrt {10} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 60 \, {\left (259200 \, x^{4} - 214560 \, x^{3} - 60348 \, x^{2} + 72849 \, x - 45658\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1866240 \, {\left (3 \, x + 2\right )}} \]
-1/1866240*(1657600*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt (5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 660959*sqrt(10)*(3*x + 2)*arc tan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3) ) - 60*(259200*x^4 - 214560*x^3 - 60348*x^2 + 72849*x - 45658)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{2}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=-\frac {25}{18} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {695}{648} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3 \, {\left (3 \, x + 2\right )}} + \frac {11045}{2592} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {660959}{1866240} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {1295}{1458} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {76253}{31104} \, \sqrt {-10 \, x^{2} - x + 3} \]
-25/18*(-10*x^2 - x + 3)^(3/2)*x + 695/648*(-10*x^2 - x + 3)^(3/2) - 1/3*( -10*x^2 - x + 3)^(5/2)/(3*x + 2) + 11045/2592*sqrt(-10*x^2 - x + 3)*x - 66 0959/1866240*sqrt(10)*arcsin(20/11*x + 1/11) + 1295/1458*sqrt(7)*arcsin(37 /11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 76253/31104*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (131) = 262\).
Time = 0.50 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.78 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=\frac {259}{2916} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{777600} \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} - 593 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 26185 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 622085 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {660959}{1866240} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1078 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{243 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
259/2916*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)))) + 1/777600*(12*(8*(36*sqrt(5)*(5*x + 3) - 593*sqrt(5))*( 5*x + 3) + 26185*sqrt(5))*(5*x + 3) - 622085*sqrt(5))*sqrt(5*x + 3)*sqrt(- 10*x + 5) - 660959/1866240*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sq rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5 ) - sqrt(22)))) - 1078/243*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)))/(((s qrt(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)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^2} \,d x \]