Integrand size = 24, antiderivative size = 155 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {11763 \sqrt {1-2 x}}{78125}+\frac {1903 (1-2 x)^{3/2} (2+3 x)^2}{4375}+\frac {1117}{750} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{5/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {127 (1-2 x)^{3/2} (2+3 x)^4}{50 (3+5 x)}+\frac {(1-2 x)^{3/2} (734+24939 x)}{93750}-\frac {11763 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
1903/4375*(1-2*x)^(3/2)*(2+3*x)^2+1117/750*(1-2*x)^(3/2)*(2+3*x)^3-1/10*(1 -2*x)^(5/2)*(2+3*x)^4/(3+5*x)^2-127/50*(1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)+1/9 3750*(1-2*x)^(3/2)*(734+24939*x)-11763/390625*arctanh(1/11*55^(1/2)*(1-2*x )^(1/2))*55^(1/2)+11763/78125*(1-2*x)^(1/2)
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.50 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {\frac {5 \sqrt {1-2 x} \left (871208+6891315 x+9372960 x^2-11139550 x^3-16051500 x^4+15075000 x^5+15750000 x^6\right )}{(3+5 x)^2}-164682 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5468750} \]
((5*Sqrt[1 - 2*x]*(871208 + 6891315*x + 9372960*x^2 - 11139550*x^3 - 16051 500*x^4 + 15075000*x^5 + 15750000*x^6))/(3 + 5*x)^2 - 164682*Sqrt[55]*ArcT anh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5468750
Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {108, 166, 27, 170, 27, 170, 27, 164, 60, 73, 219}
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^4}{(5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{10} \int \frac {(2-39 x) (1-2 x)^{3/2} (3 x+2)^3}{(5 x+3)^2}dx-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{5} \int \frac {3 (114-1117 x) \sqrt {1-2 x} (3 x+2)^3}{5 x+3}dx-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {3}{5} \int \frac {(114-1117 x) \sqrt {1-2 x} (3 x+2)^3}{5 x+3}dx-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{10} \left (\frac {3}{5} \left (\frac {1117}{45} (1-2 x)^{3/2} (3 x+2)^3-\frac {1}{45} \int -\frac {3 (69-3806 x) \sqrt {1-2 x} (3 x+2)^2}{5 x+3}dx\right )-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {3}{5} \left (\frac {1}{15} \int \frac {(69-3806 x) \sqrt {1-2 x} (3 x+2)^2}{5 x+3}dx+\frac {1117}{45} (1-2 x)^{3/2} (3 x+2)^3\right )-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{10} \left (\frac {3}{5} \left (\frac {1}{15} \left (\frac {3806}{35} (1-2 x)^{3/2} (3 x+2)^2-\frac {1}{35} \int -\frac {7 (690-2771 x) \sqrt {1-2 x} (3 x+2)}{5 x+3}dx\right )+\frac {1117}{45} (1-2 x)^{3/2} (3 x+2)^3\right )-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {3}{5} \left (\frac {1}{15} \left (\frac {1}{5} \int \frac {(690-2771 x) \sqrt {1-2 x} (3 x+2)}{5 x+3}dx+\frac {3806}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {1117}{45} (1-2 x)^{3/2} (3 x+2)^3\right )-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{10} \left (\frac {3}{5} \left (\frac {1}{15} \left (\frac {1}{5} \left (\frac {11763}{25} \int \frac {\sqrt {1-2 x}}{5 x+3}dx+\frac {1}{75} (24939 x+734) (1-2 x)^{3/2}\right )+\frac {3806}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {1117}{45} (1-2 x)^{3/2} (3 x+2)^3\right )-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{10} \left (\frac {3}{5} \left (\frac {1}{15} \left (\frac {1}{5} \left (\frac {11763}{25} \left (\frac {11}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {2}{5} \sqrt {1-2 x}\right )+\frac {1}{75} (24939 x+734) (1-2 x)^{3/2}\right )+\frac {3806}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {1117}{45} (1-2 x)^{3/2} (3 x+2)^3\right )-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{10} \left (\frac {3}{5} \left (\frac {1}{15} \left (\frac {1}{5} \left (\frac {11763}{25} \left (\frac {2}{5} \sqrt {1-2 x}-\frac {11}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {1}{75} (24939 x+734) (1-2 x)^{3/2}\right )+\frac {3806}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {1117}{45} (1-2 x)^{3/2} (3 x+2)^3\right )-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{10} \left (\frac {3}{5} \left (\frac {1}{15} \left (\frac {1}{5} \left (\frac {11763}{25} \left (\frac {2}{5} \sqrt {1-2 x}-\frac {2}{5} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {1}{75} (24939 x+734) (1-2 x)^{3/2}\right )+\frac {3806}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {1117}{45} (1-2 x)^{3/2} (3 x+2)^3\right )-\frac {127 (1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}\right )-\frac {(1-2 x)^{5/2} (3 x+2)^4}{10 (5 x+3)^2}\) |
-1/10*((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(3 + 5*x)^2 + ((-127*(1 - 2*x)^(3/2)*( 2 + 3*x)^4)/(5*(3 + 5*x)) + (3*((1117*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/45 + (( 3806*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/35 + (((1 - 2*x)^(3/2)*(734 + 24939*x))/ 75 + (11763*((2*Sqrt[1 - 2*x])/5 - (2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5))/25)/5)/15))/5)/10
3.20.91.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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_ ))*((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] && 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] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Time = 1.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(-\frac {31500000 x^{7}+14400000 x^{6}-47178000 x^{5}-6227600 x^{4}+29885470 x^{3}+4409670 x^{2}-5148899 x -871208}{1093750 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {11763 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(71\) |
| pseudoelliptic | \(\frac {-164682 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}+5 \sqrt {1-2 x}\, \left (15750000 x^{6}+15075000 x^{5}-16051500 x^{4}-11139550 x^{3}+9372960 x^{2}+6891315 x +871208\right )}{5468750 \left (3+5 x \right )^{2}}\) | \(75\) |
| trager | \(\frac {\left (15750000 x^{6}+15075000 x^{5}-16051500 x^{4}-11139550 x^{3}+9372960 x^{2}+6891315 x +871208\right ) \sqrt {1-2 x}}{1093750 \left (3+5 x \right )^{2}}+\frac {11763 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{781250}\) | \(92\) |
| derivativedivides | \(\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{250}-\frac {1107 \left (1-2 x \right )^{\frac {7}{2}}}{8750}+\frac {108 \left (1-2 x \right )^{\frac {5}{2}}}{15625}+\frac {76 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {2404 \sqrt {1-2 x}}{15625}+\frac {\frac {561 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {31097 \sqrt {1-2 x}}{78125}}{\left (-6-10 x \right )^{2}}-\frac {11763 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(93\) |
| default | \(\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{250}-\frac {1107 \left (1-2 x \right )^{\frac {7}{2}}}{8750}+\frac {108 \left (1-2 x \right )^{\frac {5}{2}}}{15625}+\frac {76 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {2404 \sqrt {1-2 x}}{15625}+\frac {\frac {561 \left (1-2 x \right )^{\frac {3}{2}}}{3125}-\frac {31097 \sqrt {1-2 x}}{78125}}{\left (-6-10 x \right )^{2}}-\frac {11763 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(93\) |
-1/1093750*(31500000*x^7+14400000*x^6-47178000*x^5-6227600*x^4+29885470*x^ 3+4409670*x^2-5148899*x-871208)/(3+5*x)^2/(1-2*x)^(1/2)-11763/390625*arcta nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {82341 \, \sqrt {11} \sqrt {5} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (15750000 \, x^{6} + 15075000 \, x^{5} - 16051500 \, x^{4} - 11139550 \, x^{3} + 9372960 \, x^{2} + 6891315 \, x + 871208\right )} \sqrt {-2 \, x + 1}}{5468750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
1/5468750*(82341*sqrt(11)*sqrt(5)*(25*x^2 + 30*x + 9)*log((sqrt(11)*sqrt(5 )*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 5*(15750000*x^6 + 15075000*x^5 - 16051500*x^4 - 11139550*x^3 + 9372960*x^2 + 6891315*x + 871208)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)
Time = 164.62 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.51 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {9 \left (1 - 2 x\right )^{\frac {9}{2}}}{250} - \frac {1107 \left (1 - 2 x\right )^{\frac {7}{2}}}{8750} + \frac {108 \left (1 - 2 x\right )^{\frac {5}{2}}}{15625} + \frac {76 \left (1 - 2 x\right )^{\frac {3}{2}}}{3125} + \frac {2404 \sqrt {1 - 2 x}}{15625} + \frac {5754 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{390625} - \frac {60984 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{78125} + \frac {10648 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{78125} \]
9*(1 - 2*x)**(9/2)/250 - 1107*(1 - 2*x)**(7/2)/8750 + 108*(1 - 2*x)**(5/2) /15625 + 76*(1 - 2*x)**(3/2)/3125 + 2404*sqrt(1 - 2*x)/15625 + 5754*sqrt(5 5)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/390 625 - 60984*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + l og(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1) ) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55) /5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/78125 + 10648*Piecewise((sqrt(55)*(3* log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) + 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)) - 1/(16*(sqrt(55 )*sqrt(1 - 2*x)/11 - 1)**2))/6655, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/78125
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {9}{250} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {1107}{8750} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {108}{15625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {76}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11763}{781250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2404}{15625} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (1275 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2827 \, \sqrt {-2 \, x + 1}\right )}}{78125 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
9/250*(-2*x + 1)^(9/2) - 1107/8750*(-2*x + 1)^(7/2) + 108/15625*(-2*x + 1) ^(5/2) + 76/3125*(-2*x + 1)^(3/2) + 11763/781250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2404/15625*sqrt(-2*x + 1) + 11/78125*(1275*(-2*x + 1)^(3/2) - 2827*sqrt(-2*x + 1))/(25*(2*x - 1) ^2 + 220*x + 11)
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {9}{250} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {1107}{8750} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {108}{15625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {76}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11763}{781250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2404}{15625} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (1275 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2827 \, \sqrt {-2 \, x + 1}\right )}}{312500 \, {\left (5 \, x + 3\right )}^{2}} \]
9/250*(2*x - 1)^4*sqrt(-2*x + 1) + 1107/8750*(2*x - 1)^3*sqrt(-2*x + 1) + 108/15625*(2*x - 1)^2*sqrt(-2*x + 1) + 76/3125*(-2*x + 1)^(3/2) + 11763/78 1250*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s qrt(-2*x + 1))) + 2404/15625*sqrt(-2*x + 1) + 11/312500*(1275*(-2*x + 1)^( 3/2) - 2827*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^3} \, dx=\frac {2404\,\sqrt {1-2\,x}}{15625}+\frac {76\,{\left (1-2\,x\right )}^{3/2}}{3125}+\frac {108\,{\left (1-2\,x\right )}^{5/2}}{15625}-\frac {1107\,{\left (1-2\,x\right )}^{7/2}}{8750}+\frac {9\,{\left (1-2\,x\right )}^{9/2}}{250}-\frac {\frac {31097\,\sqrt {1-2\,x}}{1953125}-\frac {561\,{\left (1-2\,x\right )}^{3/2}}{78125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,11763{}\mathrm {i}}{390625} \]
(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*11763i)/390625 + (2404*(1 - 2*x)^(1/2))/15625 + (76*(1 - 2*x)^(3/2))/3125 + (108*(1 - 2*x)^(5/2))/1 5625 - (1107*(1 - 2*x)^(7/2))/8750 + (9*(1 - 2*x)^(9/2))/250 - ((31097*(1 - 2*x)^(1/2))/1953125 - (561*(1 - 2*x)^(3/2))/78125)/((44*x)/5 + (2*x - 1) ^2 + 11/25)