Integrand size = 26, antiderivative size = 178 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=-\frac {97235 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac {2675 \sqrt {1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}-\frac {40}{243} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {3244595 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{108864 \sqrt {7}} \]
-1/12*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^4+115/216*(1-2*x)^(3/2)*(3+5*x)^ (3/2)/(2+3*x)^3-3244595/762048*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1 /2))*7^(1/2)-40/243*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+2675/864* (3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-97235/36288*(1-2*x)^(1/2)*(3+5*x)^(1 /2)/(2+3*x)
Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (677168+2947548 x+4103592 x^2+1790325 x^3\right )}{(2+3 x)^4}+125440 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-3244595 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{762048} \]
((21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(677168 + 2947548*x + 4103592*x^2 + 17903 25*x^3))/(2 + 3*x)^4 + 125440*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x ]] - 3244595*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/762048
Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.11, 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, 166, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{(3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{12} \int -\frac {5 (1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{2 (3 x+2)^4}dx-\frac {(1-2 x)^{5/2} (5 x+3)^{3/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{24} \int \frac {(1-2 x)^{3/2} \sqrt {5 x+3} (16 x+3)}{(3 x+2)^4}dx-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{24} \left (-\frac {1}{9} \int \frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (64 x+221)}{2 (3 x+2)^3}dx-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{24} \left (-\frac {1}{6} \int \frac {\sqrt {1-2 x} \sqrt {5 x+3} (64 x+221)}{(3 x+2)^3}dx-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{24} \left (\frac {1}{6} \left (\frac {1}{6} \int -\frac {(6141-512 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}dx-\frac {535 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{24} \left (\frac {1}{6} \left (-\frac {1}{12} \int \frac {(6141-512 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {535 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{24} \left (\frac {1}{6} \left (\frac {1}{12} \left (\frac {19447 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}-\frac {1}{21} \int \frac {192413-35840 x}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {535 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{24} \left (\frac {1}{6} \left (\frac {1}{12} \left (\frac {19447 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}-\frac {1}{42} \int \frac {192413-35840 x}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {535 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {5}{24} \left (\frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \left (\frac {35840}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {648919}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {19447 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {535 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle -\frac {5}{24} \left (\frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \left (\frac {14336}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {648919}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {19447 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {535 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {5}{24} \left (\frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \left (\frac {14336}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1297838}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {19447 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {535 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {5}{24} \left (\frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \left (\frac {14336}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1297838 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )+\frac {19447 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {535 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {5}{24} \left (\frac {1}{6} \left (\frac {1}{12} \left (\frac {1}{42} \left (\frac {7168}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {1297838 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )+\frac {19447 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )-\frac {535 \sqrt {1-2 x} (5 x+3)^{3/2}}{6 (3 x+2)^2}\right )-\frac {23 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}\) |
-1/12*((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4 - (5*((-23*(1 - 2*x)^( 3/2)*(3 + 5*x)^(3/2))/(9*(2 + 3*x)^3) + ((-535*Sqrt[1 - 2*x]*(3 + 5*x)^(3/ 2))/(6*(2 + 3*x)^2) + ((19447*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)) + ((7168*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 + (1297838*ArcTan[Sq rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*Sqrt[7]))/42)/12)/6))/24
3.25.7.3.1 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[(((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 = 1.15 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.80
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (1790325 x^{3}+4103592 x^{2}+2947548 x +677168\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{36288 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {20 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{243}-\frac {3244595 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{1524096}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(143\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (262812195 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-10160640 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}+700832520 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-27095040 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+700832520 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-27095040 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+75193650 x^{3} \sqrt {-10 x^{2}-x +3}+311481120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -12042240 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +172350864 x^{2} \sqrt {-10 x^{2}-x +3}+51913520 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2007040 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+123797016 x \sqrt {-10 x^{2}-x +3}+28441056 \sqrt {-10 x^{2}-x +3}\right )}{1524096 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(315\) |
-1/36288*(-1+2*x)*(3+5*x)^(1/2)*(1790325*x^3+4103592*x^2+2947548*x+677168) /(2+3*x)^4/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2) -(20/243*10^(1/2)*arcsin(20/11*x+1/11)-3244595/1524096*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 = 0.25 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=-\frac {3244595 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 125440 \, \sqrt {10} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \, {\left (1790325 \, x^{3} + 4103592 \, x^{2} + 2947548 \, x + 677168\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1524096 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
-1/1524096*(3244595*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arcta n(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 125440*sqrt(10)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/20*sqr t(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(1790 325*x^3 + 4103592*x^2 + 2947548*x + 677168)*sqrt(5*x + 3)*sqrt(-2*x + 1))/ (81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{5}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.11 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\frac {21775}{21168} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{4 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {95 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{168 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {4355 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{4704 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {539675}{42336} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {20}{243} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {3244595}{1524096} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {1460395}{254016} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {18245 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{28224 \, {\left (3 \, x + 2\right )}} \]
21775/21168*(-10*x^2 - x + 3)^(3/2) + 1/4*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 95/168*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 4355/4704*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4 ) + 539675/42336*sqrt(-10*x^2 - x + 3)*x - 20/243*sqrt(10)*arcsin(20/11*x + 1/11) + 3244595/1524096*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs( 3*x + 2)) - 1460395/254016*sqrt(-10*x^2 - x + 3) + 18245/28224*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (136) = 272\).
Time = 0.66 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.44 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\frac {648919}{3048192} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {20}{243} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {55 \, \sqrt {10} {\left (19447 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 19946472 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 6199166400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {348224576000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1392898304000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{18144 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
648919/3048192*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)))) - 20/243*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)))) - 55/18144*sqrt(10)*(19447*((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)))^7 + 19946472*((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 - 6199166400*((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)))^3 - 348224576000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1392898304000*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)^4
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^5} \,d x \]