Integrand size = 26, antiderivative size = 193 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt {3+5 x}}+\frac {8026963 \sqrt {1-2 x} \sqrt {3+5 x}}{40000000}+\frac {23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}}{25000}+\frac {2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}+\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (64435+118392 x)}{4000000}+\frac {88296593 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40000000 \sqrt {10}} \]
-2/15*(1-2*x)^(5/2)*(2+3*x)^4/(3+5*x)^(3/2)+88296593/400000000*arcsin(1/11 *22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-508/75*(1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)^( 1/2)+23991/25000*(1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(1/2)+2514/625*(1-2*x)^(3 /2)*(2+3*x)^3*(3+5*x)^(1/2)+21/4000000*(1-2*x)^(3/2)*(64435+118392*x)*(3+5 *x)^(1/2)+8026963/40000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.59 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.48 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx=\frac {\frac {5 \sqrt {1-2 x} \left (210855251+980658710 x+865945995 x^2-1405199700 x^3-1419228000 x^4+1626480000 x^5+1555200000 x^6\right )}{(3+5 x)^{3/2}}-264889779 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{600000000} \]
((5*Sqrt[1 - 2*x]*(210855251 + 980658710*x + 865945995*x^2 - 1405199700*x^ 3 - 1419228000*x^4 + 1626480000*x^5 + 1555200000*x^6))/(3 + 5*x)^(3/2) - 2 64889779*Sqrt[10]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10*x])])/6000 00000
Time = 0.27 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {108, 167, 27, 170, 27, 170, 27, 164, 60, 64, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^4}{(5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{15} \int \frac {(2-39 x) (1-2 x)^{3/2} (3 x+2)^3}{(5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {2}{15} \left (\frac {2}{5} \int \frac {3 (184-1257 x) \sqrt {1-2 x} (3 x+2)^3}{\sqrt {5 x+3}}dx-\frac {254 (1-2 x)^{3/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {6}{5} \int \frac {(184-1257 x) \sqrt {1-2 x} (3 x+2)^3}{\sqrt {5 x+3}}dx-\frac {254 (1-2 x)^{3/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {2}{15} \left (\frac {6}{5} \left (\frac {1257}{50} (1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}-\frac {1}{50} \int -\frac {(1604-23991 x) \sqrt {1-2 x} (3 x+2)^2}{2 \sqrt {5 x+3}}dx\right )-\frac {254 (1-2 x)^{3/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {6}{5} \left (\frac {1}{100} \int \frac {(1604-23991 x) \sqrt {1-2 x} (3 x+2)^2}{\sqrt {5 x+3}}dx+\frac {1257}{50} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {254 (1-2 x)^{3/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {2}{15} \left (\frac {6}{5} \left (\frac {1}{100} \left (\frac {23991}{40} (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}-\frac {1}{40} \int -\frac {7 (2390-44397 x) \sqrt {1-2 x} (3 x+2)}{2 \sqrt {5 x+3}}dx\right )+\frac {1257}{50} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {254 (1-2 x)^{3/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {6}{5} \left (\frac {1}{100} \left (\frac {7}{80} \int \frac {(2390-44397 x) \sqrt {1-2 x} (3 x+2)}{\sqrt {5 x+3}}dx+\frac {23991}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2\right )+\frac {1257}{50} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {254 (1-2 x)^{3/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {2}{15} \left (\frac {6}{5} \left (\frac {1}{100} \left (\frac {7}{80} \left (\frac {1146709}{160} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {3}{80} \sqrt {5 x+3} (118392 x+64435) (1-2 x)^{3/2}\right )+\frac {23991}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2\right )+\frac {1257}{50} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {254 (1-2 x)^{3/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2}{15} \left (\frac {6}{5} \left (\frac {1}{100} \left (\frac {7}{80} \left (\frac {1146709}{160} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {3}{80} \sqrt {5 x+3} (118392 x+64435) (1-2 x)^{3/2}\right )+\frac {23991}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2\right )+\frac {1257}{50} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {254 (1-2 x)^{3/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {2}{15} \left (\frac {6}{5} \left (\frac {1}{100} \left (\frac {7}{80} \left (\frac {1146709}{160} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {3}{80} \sqrt {5 x+3} (118392 x+64435) (1-2 x)^{3/2}\right )+\frac {23991}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2\right )+\frac {1257}{50} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {254 (1-2 x)^{3/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {2}{15} \left (\frac {6}{5} \left (\frac {1}{100} \left (\frac {7}{80} \left (\frac {1146709}{160} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {3}{80} \sqrt {5 x+3} (118392 x+64435) (1-2 x)^{3/2}\right )+\frac {23991}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2\right )+\frac {1257}{50} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^3\right )-\frac {254 (1-2 x)^{3/2} (3 x+2)^4}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}\) |
(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^4)/(15*(3 + 5*x)^(3/2)) + (2*((-254*(1 - 2*x )^(3/2)*(2 + 3*x)^4)/(5*Sqrt[3 + 5*x]) + (6*((1257*(1 - 2*x)^(3/2)*(2 + 3* x)^3*Sqrt[3 + 5*x])/50 + ((23991*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x] )/40 + (7*((3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]*(64435 + 118392*x))/80 + (1146 709*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x] ])/(5*Sqrt[10])))/160))/80)/100))/5))/15
3.25.47.3.1 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_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_) )^(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_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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*((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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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] && IntegerQ[m]
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 = 1.18 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(\frac {\left (31104000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+32529600000 x^{5} \sqrt {-10 x^{2}-x +3}-28384560000 x^{4} \sqrt {-10 x^{2}-x +3}+6622244475 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-28103994000 x^{3} \sqrt {-10 x^{2}-x +3}+7946693370 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +17318919900 x^{2} \sqrt {-10 x^{2}-x +3}+2384008011 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+19613174200 x \sqrt {-10 x^{2}-x +3}+4217105020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{2400000000 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(181\) |
1/2400000000*(31104000000*(-10*x^2-x+3)^(1/2)*x^6+32529600000*x^5*(-10*x^2 -x+3)^(1/2)-28384560000*x^4*(-10*x^2-x+3)^(1/2)+6622244475*10^(1/2)*arcsin (20/11*x+1/11)*x^2-28103994000*x^3*(-10*x^2-x+3)^(1/2)+7946693370*10^(1/2) *arcsin(20/11*x+1/11)*x+17318919900*x^2*(-10*x^2-x+3)^(1/2)+2384008011*10^ (1/2)*arcsin(20/11*x+1/11)+19613174200*x*(-10*x^2-x+3)^(1/2)+4217105020*(- 10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)
Time = 0.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx=-\frac {264889779 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (1555200000 \, x^{6} + 1626480000 \, x^{5} - 1419228000 \, x^{4} - 1405199700 \, x^{3} + 865945995 \, x^{2} + 980658710 \, x + 210855251\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2400000000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
-1/2400000000*(264889779*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10) *(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 20*(155520000 0*x^6 + 1626480000*x^5 - 1419228000*x^4 - 1405199700*x^3 + 865945995*x^2 + 980658710*x + 210855251)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9 )
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{4}}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.83 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx=\frac {81}{15625} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {891}{25000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {70759953}{800000000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {23}{11}\right ) + \frac {27401}{1250000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {8811}{500000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3125 \, {\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac {6 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3125 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {18 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3125 \, {\left (5 \, x + 3\right )}} + \frac {584793}{2000000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} x + \frac {13450239}{40000000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} + \frac {3267}{62500} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {11 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{18750 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {99 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{6250 \, {\left (5 \, x + 3\right )}} - \frac {121 \, \sqrt {-10 \, x^{2} - x + 3}}{93750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {638 \, \sqrt {-10 \, x^{2} - x + 3}}{9375 \, {\left (5 \, x + 3\right )}} \]
81/15625*(-10*x^2 - x + 3)^(5/2) + 891/25000*(-10*x^2 - x + 3)^(3/2)*x - 7 0759953/800000000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 27401/125000 0*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 8811/500000*(-10*x^2 - x + 3)^( 3/2) + 1/3125*(-10*x^2 - x + 3)^(5/2)/(625*x^4 + 1500*x^3 + 1350*x^2 + 540 *x + 81) + 6/3125*(-10*x^2 - x + 3)^(5/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 18/3125*(-10*x^2 - x + 3)^(5/2)/(25*x^2 + 30*x + 9) + 27/3125*(-10*x^2 - x + 3)^(5/2)/(5*x + 3) + 584793/2000000*sqrt(10*x^2 + 23*x + 51/5)*x + 1 3450239/40000000*sqrt(10*x^2 + 23*x + 51/5) + 3267/62500*sqrt(-10*x^2 - x + 3) - 11/18750*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 33/3125*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 99/6250*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 121/93750*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 638/9375*sqrt(-10*x^2 - x + 3)/(5*x + 3)
Time = 0.47 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.09 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx=\frac {1}{1000000000} \, {\left (12 \, {\left (24 \, {\left (12 \, {\left (48 \, \sqrt {5} {\left (5 \, x + 3\right )} - 613 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 19439 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1264235 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 10674335 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{18750000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {88296593}{400000000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {561 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{312500 \, \sqrt {5 \, x + 3}} + \frac {11 \, \sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {765 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{1171875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
1/1000000000*(12*(24*(12*(48*sqrt(5)*(5*x + 3) - 613*sqrt(5))*(5*x + 3) + 19439*sqrt(5))*(5*x + 3) + 1264235*sqrt(5))*(5*x + 3) - 10674335*sqrt(5))* sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/18750000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 88296593/400000000*sqrt(10)*arcsin(1/1 1*sqrt(22)*sqrt(5*x + 3)) - 561/312500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 11/1171875*sqrt(10)*(5*x + 3)^(3/2)*(765*(sqrt( 2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4}{{\left (5\,x+3\right )}^{5/2}} \,d x \]