Integrand size = 26, antiderivative size = 165 \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800000 \sqrt {10}} \]
-193/2000*(1-2*x)^(7/2)*(3+5*x)^(3/2)-1/20*(1-2*x)^(7/2)*(2+3*x)*(3+5*x)^( 3/2)+105254149/128000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+8698 69/3840000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+79079/960000*(1-2*x)^(5/2)*(3+5*x)^ (1/2)-7189/32000*(1-2*x)^(7/2)*(3+5*x)^(1/2)+9568559/12800000*(1-2*x)^(1/2 )*(3+5*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.53 \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {10 \sqrt {1-2 x} \left (27911781+354090375 x+328830220 x^2-1017874400 x^3-902544000 x^4+1163520000 x^5+1152000000 x^6\right )-315762447 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{384000000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(27911781 + 354090375*x + 328830220*x^2 - 1017874400*x^3 - 902544000*x^4 + 1163520000*x^5 + 1152000000*x^6) - 315762447*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(384000000*Sqrt[3 + 5*x])
Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {101, 27, 90, 60, 60, 60, 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.
| \(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^2 \sqrt {5 x+3} \, dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {1}{60} \int -\frac {3}{2} (1-2 x)^{5/2} \sqrt {5 x+3} (193 x+124)dx-\frac {1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{40} \int (1-2 x)^{5/2} \sqrt {5 x+3} (193 x+124)dx-\frac {1}{20} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{40} \left (\frac {7189}{100} \int (1-2 x)^{5/2} \sqrt {5 x+3}dx-\frac {193}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{40} \left (\frac {7189}{100} \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {193}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{40} \left (\frac {7189}{100} \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {193}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{40} \left (\frac {7189}{100} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {193}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{40} \left (\frac {7189}{100} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {193}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{40} \left (\frac {7189}{100} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {193}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{40} \left (\frac {7189}{100} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {193}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (3 x+2) (5 x+3)^{3/2}\) |
-1/20*((1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(3/2)) + ((-193*(1 - 2*x)^(7/2) *(3 + 5*x)^(3/2))/50 + (7189*(-1/8*((1 - 2*x)^(7/2)*Sqrt[3 + 5*x]) + (11*( ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/15 + (11*(((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/ 10 + (33*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/6))/16))/100)/40
3.24.89.3.1 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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 = 113, normalized size of antiderivative = 0.68
| method | result | size |
| risch | \(-\frac {\left (230400000 x^{5}+94464000 x^{4}-237187200 x^{3}-61262560 x^{2}+102523580 x +9303927\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{38400000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {105254149 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{256000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(113\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (4608000000 x^{5} \sqrt {-10 x^{2}-x +3}+1889280000 x^{4} \sqrt {-10 x^{2}-x +3}-4743744000 x^{3} \sqrt {-10 x^{2}-x +3}-1225251200 x^{2} \sqrt {-10 x^{2}-x +3}+315762447 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2050471600 x \sqrt {-10 x^{2}-x +3}+186078540 \sqrt {-10 x^{2}-x +3}\right )}{768000000 \sqrt {-10 x^{2}-x +3}}\) | \(138\) |
-1/38400000*(230400000*x^5+94464000*x^4-237187200*x^3-61262560*x^2+1025235 80*x+9303927)*(-1+2*x)*(3+5*x)^(1/2)/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3 +5*x))^(1/2)/(1-2*x)^(1/2)+105254149/256000000*10^(1/2)*arcsin(20/11*x+1/1 1)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.50 \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {1}{38400000} \, {\left (230400000 \, x^{5} + 94464000 \, x^{4} - 237187200 \, x^{3} - 61262560 \, x^{2} + 102523580 \, x + 9303927\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {105254149}{256000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/38400000*(230400000*x^5 + 94464000*x^4 - 237187200*x^3 - 61262560*x^2 + 102523580*x + 9303927)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 105254149/256000000* sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10* x^2 + x - 3))
\[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\int \left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{2} \sqrt {5 x + 3}\, dx \]
Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.63 \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=-\frac {3}{5} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {93}{500} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {18251}{40000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {27893}{480000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {869869}{640000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {105254149}{256000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {869869}{12800000} \, \sqrt {-10 \, x^{2} - x + 3} \]
-3/5*(-10*x^2 - x + 3)^(3/2)*x^3 - 93/500*(-10*x^2 - x + 3)^(3/2)*x^2 + 18 251/40000*(-10*x^2 - x + 3)^(3/2)*x + 27893/480000*(-10*x^2 - x + 3)^(3/2) + 869869/640000*sqrt(-10*x^2 - x + 3)*x - 105254149/256000000*sqrt(10)*ar csin(-20/11*x - 1/11) + 869869/12800000*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (120) = 240\).
Time = 0.36 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.16 \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {3}{640000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {7}{40000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {79}{9600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {89}{120000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{250} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {6}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
3/640000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 15385695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*s qrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 7/40000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sq rt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 79/9600000*sqrt(5)*(2*(4*(8*( 60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 89/120000*sqr t(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4 785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/250*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqr t(5*x + 3))) + 6/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3) ) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,\sqrt {5\,x+3} \,d x \]