Integrand size = 24, antiderivative size = 160 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx=\frac {2767149 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}+\frac {83853 (1-2 x)^{3/2} \sqrt {3+5 x}}{256000}+\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {30438639 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2560000 \sqrt {10}} \]
-63/400*(1-2*x)^(7/2)*(3+5*x)^(3/2)-1/20*(1-2*x)^(7/2)*(3+5*x)^(5/2)+30438 639/25600000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+83853/256000*(1- 2*x)^(3/2)*(3+5*x)^(1/2)+7623/64000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-2079/6400* (1-2*x)^(7/2)*(3+5*x)^(1/2)+2767149/2560000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.55 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx=\frac {10 \sqrt {1-2 x} \left (2152197+34806375 x+36544140 x^2-102392800 x^3-102288000 x^4+119040000 x^5+128000000 x^6\right )-30438639 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{25600000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(2152197 + 34806375*x + 36544140*x^2 - 102392800*x^3 - 1 02288000*x^4 + 119040000*x^5 + 128000000*x^6) - 30438639*Sqrt[30 + 50*x]*A rcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(25600000*Sqrt[3 + 5*x])
Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {90, 60, 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.
| \(\displaystyle \int (1-2 x)^{5/2} (3 x+2) (5 x+3)^{3/2} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {63}{40} \int (1-2 x)^{5/2} (5 x+3)^{3/2}dx-\frac {1}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {63}{40} \left (\frac {33}{20} \int (1-2 x)^{5/2} \sqrt {5 x+3}dx-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {63}{40} \left (\frac {33}{20} \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {63}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {63}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {63}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {63}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {63}{40} \left (\frac {33}{20} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {1}{10} (1-2 x)^{7/2} (5 x+3)^{3/2}\right )-\frac {1}{20} (1-2 x)^{7/2} (5 x+3)^{5/2}\) |
-1/20*((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2)) + (63*(-1/10*((1 - 2*x)^(7/2)*(3 + 5*x)^(3/2)) + (33*(-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))/20))/40
3.25.1.3.1 Defintions of rubi rules used
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[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 = 1.14 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {\left (25600000 x^{5}+8448000 x^{4}-25526400 x^{3}-5162720 x^{2}+10406460 x +717399\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2560000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {30438639 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{51200000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(113\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (512000000 x^{5} \sqrt {-10 x^{2}-x +3}+168960000 x^{4} \sqrt {-10 x^{2}-x +3}-510528000 x^{3} \sqrt {-10 x^{2}-x +3}-103254400 x^{2} \sqrt {-10 x^{2}-x +3}+30438639 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+208129200 x \sqrt {-10 x^{2}-x +3}+14347980 \sqrt {-10 x^{2}-x +3}\right )}{51200000 \sqrt {-10 x^{2}-x +3}}\) | \(138\) |
-1/2560000*(25600000*x^5+8448000*x^4-25526400*x^3-5162720*x^2+10406460*x+7 17399)*(-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)+30438639/51200000*10^(1/2)*arcsin(20/11*x+1/11)*((1-2* x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx=\frac {1}{2560000} \, {\left (25600000 \, x^{5} + 8448000 \, x^{4} - 25526400 \, x^{3} - 5162720 \, x^{2} + 10406460 \, x + 717399\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {30438639}{51200000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/2560000*(25600000*x^5 + 8448000*x^4 - 25526400*x^3 - 5162720*x^2 + 10406 460*x + 717399)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 30438639/51200000*sqrt(10)* arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx=\int \left (1 - 2 x\right )^{\frac {5}{2}} \cdot \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.62 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx=\frac {1}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {13}{1000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {693}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {693}{32000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {251559}{128000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {30438639}{51200000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {251559}{2560000} \, \sqrt {-10 \, x^{2} - x + 3} \]
1/10*(-10*x^2 - x + 3)^(5/2)*x + 13/1000*(-10*x^2 - x + 3)^(5/2) + 693/160 0*(-10*x^2 - x + 3)^(3/2)*x + 693/32000*(-10*x^2 - x + 3)^(3/2) + 251559/1 28000*sqrt(-10*x^2 - x + 3)*x - 30438639/51200000*sqrt(10)*arcsin(-20/11*x - 1/11) + 251559/2560000*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (115) = 230\).
Time = 0.40 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.22 \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx=\frac {1}{128000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {13}{48000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {137}{9600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {17}{15000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {3}{400} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
1/128000000*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))) + 13/48000000*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*s qrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 137/9600000*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))) - 17/15000*sq rt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/400*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sq rt(5*x + 3))) + 9/25*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. \[ \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{3/2} \,d x \]