Integrand size = 26, antiderivative size = 171 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=-\frac {845}{648} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}+\frac {362}{243} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {215 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1944 \sqrt {7}} \]
-1/9*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^3+115/108*(1-2*x)^(3/2)*(3+5*x)^( 3/2)/(2+3*x)^2+215/13608*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7 ^(1/2)+362/243*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+365/216*(3+5*x )^(3/2)*(1-2*x)^(1/2)/(2+3*x)-845/648*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (10304+36234 x+34341 x^2+4320 x^3\right )}{(2+3 x)^3}-20272 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+215 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{13608} \]
((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(10304 + 36234*x + 34341*x^2 + 4320*x^3)) /(2 + 3*x)^3 - 20272*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] + 215* Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/13608
Time = 0.28 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.13, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {108, 27, 166, 27, 166, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{(3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{9} \int -\frac {5 (1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{2 (3 x+2)^3}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{18} \int \frac {(1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{(3 x+2)^3}dx-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{18} \left (-\frac {1}{6} \int \frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (196 x+155)}{2 (3 x+2)^2}dx-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{18} \left (-\frac {1}{4} \int \frac {\sqrt {1-2 x} \sqrt {5 x+3} (196 x+155)}{(3 x+2)^2}dx-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{3} \int -\frac {\sqrt {5 x+3} (676 x+465)}{2 \sqrt {1-2 x} (3 x+2)}dx-\frac {73 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (-\frac {1}{6} \int \frac {\sqrt {5 x+3} (676 x+465)}{\sqrt {1-2 x} (3 x+2)}dx-\frac {73 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{6} \int -\frac {2 (5792 x+3847)}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {338}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {73 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {338}{3} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {1}{3} \int \frac {5792 x+3847}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {73 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{3} \left (\frac {43}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {5792}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )+\frac {338}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {73 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{3} \left (\frac {43}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {11584}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )+\frac {338}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {73 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{3} \left (\frac {86}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {11584}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )+\frac {338}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {73 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{3} \left (-\frac {11584}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {86 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )+\frac {338}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {73 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {5}{18} \left (\frac {1}{4} \left (\frac {1}{6} \left (\frac {1}{3} \left (-\frac {5792}{3} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {86 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )+\frac {338}{3} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {73 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}\) |
-1/9*((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^3 - (5*((-23*(1 - 2*x)^(3 /2)*(3 + 5*x)^(3/2))/(6*(2 + 3*x)^2) + ((-73*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2) )/(3*(2 + 3*x)) + ((338*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3 + ((-5792*Sqrt[2/5] *ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 - (86*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S qrt[3 + 5*x])])/(3*Sqrt[7]))/3)/6)/4))/18
3.25.6.3.1 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 = 1.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (4320 x^{3}+34341 x^{2}+36234 x +10304\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{648 \left (2+3 x \right )^{3} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {181 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{243}+\frac {215 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{27216}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(143\) |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (5805 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-547344 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+11610 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-1094688 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-181440 x^{3} \sqrt {-10 x^{2}-x +3}+7740 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -729792 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -1442322 x^{2} \sqrt {-10 x^{2}-x +3}+1720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-162176 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1521828 x \sqrt {-10 x^{2}-x +3}-432768 \sqrt {-10 x^{2}-x +3}\right )}{27216 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\) | \(270\) |
-1/648*(-1+2*x)*(3+5*x)^(1/2)*(4320*x^3+34341*x^2+36234*x+10304)/(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)-(-181/243 *10^(1/2)*arcsin(20/11*x+1/11)+215/27216*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 = 161, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=\frac {215 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20272 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (4320 \, x^{3} + 34341 \, x^{2} + 36234 \, x + 10304\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{27216 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
1/27216*(215*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37* x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 20272*sqrt(10)*(2 7*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*s qrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*(4320*x^3 + 34341*x^2 + 36234*x + 103 04)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{4}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=\frac {125}{378} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {25 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{84 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {1825}{756} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {181}{243} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {215}{27216} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {655}{4536} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {65 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{504 \, {\left (3 \, x + 2\right )}} \]
125/378*(-10*x^2 - x + 3)^(3/2) + 1/3*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54 *x^2 + 36*x + 8) + 25/84*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 1825 /756*sqrt(-10*x^2 - x + 3)*x + 181/243*sqrt(10)*arcsin(20/11*x + 1/11) - 2 15/27216*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 655/4 536*sqrt(-10*x^2 - x + 3) - 65/504*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (129) = 258\).
Time = 0.59 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.36 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=-\frac {43}{54432} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {181}{243} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {4}{81} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11 \, {\left (67 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 56000 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - 65464000 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{108 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
-43/54432*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)))) + 181/243*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)))) + 4/81*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/108*(6 7*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)))^5 + 56000*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 - 65464000*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) - sqrt(22)))^2 + 280)^3
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^4} \,d x \]