Integrand size = 26, antiderivative size = 195 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=-\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}-\frac {21935 \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1458}-\frac {408665 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5832 \sqrt {7}} \]
-1/9*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^3+185/108*(1-2*x)^(3/2)*(3+5*x)^( 5/2)/(2+3*x)^2-21935/2916*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-408 665/40824*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+2075/72* (3+5*x)^(3/2)*(1-2*x)^(1/2)-10385/648*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)- 48625/1944*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.35 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (-107984-391014 x-420531 x^2-93420 x^3+32400 x^4\right )}{(2+3 x)^3}+307090 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-408665 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{40824} \]
((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-107984 - 391014*x - 420531*x^2 - 93420* x^3 + 32400*x^4))/(2 + 3*x)^3 + 307090*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqr t[3 + 5*x]] - 408665*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])] )/40824
Time = 0.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {108, 27, 166, 27, 166, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{9} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 (3 x+2)^3}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{18} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{(3 x+2)^3}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{18} \left (-\frac {1}{6} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (1640 x+401)}{2 (3 x+2)^2}dx-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{18} \left (-\frac {1}{12} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (1640 x+401)}{(3 x+2)^2}dx-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {1}{3} \int -\frac {(21973-89640 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)}dx+\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}-\frac {1}{6} \int \frac {(21973-89640 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {1}{6} \left (\frac {1}{12} \int \frac {12 (1311-38900 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}dx-7470 \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {1}{6} \left (\int \frac {(1311-38900 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}dx-7470 \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {1}{6} \left (-\frac {1}{6} \int -\frac {2 (87740 x+31249)}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-7470 \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {19450}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {1}{6} \left (\frac {1}{3} \int \frac {87740 x+31249}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-7470 \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {19450}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {1}{6} \left (\frac {1}{3} \left (\frac {87740}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {81733}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-7470 \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {19450}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {1}{6} \left (\frac {1}{3} \left (\frac {35096}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {81733}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-7470 \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {19450}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {1}{6} \left (\frac {1}{3} \left (\frac {35096}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {163466}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-7470 \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {19450}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {1}{6} \left (\frac {1}{3} \left (\frac {35096}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {163466 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-7470 \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {19450}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{12} \left (\frac {1}{6} \left (\frac {1}{3} \left (\frac {17548}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {163466 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-7470 \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {19450}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {2077 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\right )-\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}\) |
-1/9*((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^3 - (5*((-37*(1 - 2*x)^(3 /2)*(3 + 5*x)^(5/2))/(6*(2 + 3*x)^2) + ((2077*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2 ))/(3*(2 + 3*x)) + ((19450*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3 - 7470*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2) + ((17548*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/ 3 + (163466*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*Sqrt[7]))/3) /6)/12))/18
3.25.18.3.1 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[(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[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 = 3.88 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (32400 x^{4}-93420 x^{3}-420531 x^{2}-391014 x -107984\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1944 \left (2+3 x \right )^{3} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {21935 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{5832}-\frac {408665 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{81648}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(148\) |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (8291430 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-11033955 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-1360800 x^{4} \sqrt {-10 x^{2}-x +3}+16582860 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-22067910 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+3923640 x^{3} \sqrt {-10 x^{2}-x +3}+11055240 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -14711940 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +17662302 x^{2} \sqrt {-10 x^{2}-x +3}+2456720 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3269320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+16422588 x \sqrt {-10 x^{2}-x +3}+4535328 \sqrt {-10 x^{2}-x +3}\right )}{81648 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\) | \(287\) |
-1/1944*(-1+2*x)*(3+5*x)^(1/2)*(32400*x^4-93420*x^3-420531*x^2-391014*x-10 7984)/(2+3*x)^3/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^ (1/2)-(21935/5832*10^(1/2)*arcsin(20/11*x+1/11)-408665/81648*7^(1/2)*arcta n(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 = 172, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {307090 \, \sqrt {5} \sqrt {2} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 408665 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (32400 \, x^{4} - 93420 \, x^{3} - 420531 \, x^{2} - 391014 \, x - 107984\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{81648 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
1/81648*(307090*sqrt(5)*sqrt(2)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*s qrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 408665*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 2 0)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*(32400*x^4 - 93420* x^3 - 420531*x^2 - 391014*x - 107984)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^ 3 + 54*x^2 + 36*x + 8)
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{4}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=-\frac {185}{882} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {16075}{1764} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {189865}{31752} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {6347 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3528 \, {\left (3 \, x + 2\right )}} + \frac {41225}{2268} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {21935}{5832} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {408665}{81648} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {191965}{13608} \, \sqrt {-10 \, x^{2} - x + 3} \]
-185/882*(-10*x^2 - x + 3)^(5/2) + 1/7*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 5 4*x^2 + 36*x + 8) - 37/196*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 16 075/1764*(-10*x^2 - x + 3)^(3/2)*x + 189865/31752*(-10*x^2 - x + 3)^(3/2) - 6347/3528*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 41225/2268*sqrt(-10*x^2 - x + 3)*x - 21935/5832*sqrt(10)*arcsin(20/11*x + 1/11) + 408665/81648*sqrt( 7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 191965/13608*sqrt(- 10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (145) = 290\).
Time = 0.59 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.14 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\frac {81733}{163296} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{486} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 329 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {21935}{5832} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, {\left (2803 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1982400 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 411208000 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{324 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
81733/163296*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)))) + 1/486*(12*sqrt(5)*(5*x + 3) - 329*sqrt(5))*sqrt(5* x + 3)*sqrt(-10*x + 5) - 21935/5832*sqrt(10)*(pi + 2*arctan(-1/4*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)))) - 11/324*(2803*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) - sq rt(22)))^5 + 1982400*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 + 411208 000*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))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq rt(22)))^2 + 280)^3
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^4} \,d x \]