Integrand size = 24, antiderivative size = 138 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx=\frac {147741 \sqrt {1-2 x} \sqrt {3+5 x}}{128000}+\frac {4477 (1-2 x)^{3/2} \sqrt {3+5 x}}{12800}-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {1625151 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{128000 \sqrt {10}} \]
-37/160*(1-2*x)^(5/2)*(3+5*x)^(3/2)-3/50*(1-2*x)^(5/2)*(3+5*x)^(5/2)+16251 51/1280000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+4477/12800*(1-2*x) ^(3/2)*(3+5*x)^(1/2)-407/640*(1-2*x)^(5/2)*(3+5*x)^(1/2)+147741/128000*(1- 2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.60 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx=\frac {-10 \sqrt {1-2 x} \left (140427-1233975 x-3539660 x^2+415200 x^3+6032000 x^4+3840000 x^5\right )-1625151 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1280000 \sqrt {3+5 x}} \]
(-10*Sqrt[1 - 2*x]*(140427 - 1233975*x - 3539660*x^2 + 415200*x^3 + 603200 0*x^4 + 3840000*x^5) - 1625151*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt [3 + 5*x]])/(1280000*Sqrt[3 + 5*x])
Time = 0.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {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.
| \(\displaystyle \int (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {37}{20} \int (1-2 x)^{3/2} (5 x+3)^{3/2}dx-\frac {3}{50} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {37}{20} \left (\frac {33}{16} \int (1-2 x)^{3/2} \sqrt {5 x+3}dx-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {37}{20} \left (\frac {33}{16} \left (\frac {11}{12} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {37}{20} \left (\frac {33}{16} \left (\frac {11}{12} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {37}{20} \left (\frac {33}{16} \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {37}{20} \left (\frac {33}{16} \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {37}{20} \left (\frac {33}{16} \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (5 x+3)^{5/2}\) |
(-3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/50 + (37*(-1/8*((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)) + (33*(-1/6*((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]) + (11*(((1 - 2*x)^ (3/2)*Sqrt[3 + 5*x])/10 + (33*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*ArcSi n[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/12))/16))/20
3.24.39.3.1 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 = 3.71 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(\frac {\left (768000 x^{4}+745600 x^{3}-364320 x^{2}-489340 x +46809\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{128000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1625151 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2560000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(108\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-15360000 x^{4} \sqrt {-10 x^{2}-x +3}-14912000 x^{3} \sqrt {-10 x^{2}-x +3}+7286400 x^{2} \sqrt {-10 x^{2}-x +3}+1625151 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+9786800 x \sqrt {-10 x^{2}-x +3}-936180 \sqrt {-10 x^{2}-x +3}\right )}{2560000 \sqrt {-10 x^{2}-x +3}}\) | \(121\) |
1/128000*(768000*x^4+745600*x^3-364320*x^2-489340*x+46809)*(-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)+16 25151/2560000*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 = 77, normalized size of antiderivative = 0.56 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx=-\frac {1}{128000} \, {\left (768000 \, x^{4} + 745600 \, x^{3} - 364320 \, x^{2} - 489340 \, x + 46809\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1625151}{2560000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
-1/128000*(768000*x^4 + 745600*x^3 - 364320*x^2 - 489340*x + 46809)*sqrt(5 *x + 3)*sqrt(-2*x + 1) - 1625151/2560000*sqrt(10)*arctan(1/20*sqrt(10)*(20 *x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx=\int \left (1 - 2 x\right )^{\frac {3}{2}} \cdot \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.61 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx=-\frac {3}{50} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {37}{80} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {37}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {13431}{6400} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1625151}{2560000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {13431}{128000} \, \sqrt {-10 \, x^{2} - x + 3} \]
-3/50*(-10*x^2 - x + 3)^(5/2) + 37/80*(-10*x^2 - x + 3)^(3/2)*x + 37/1600* (-10*x^2 - x + 3)^(3/2) + 13431/6400*sqrt(-10*x^2 - x + 3)*x - 1625151/256 0000*sqrt(10)*arcsin(-20/11*x - 1/11) + 13431/128000*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (99) = 198\).
Time = 0.34 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx=-\frac {1}{6400000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {41}{1920000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {17}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {51}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
-1/6400000*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))) - 41/1920000*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))) - 17/60000*sqrt (5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 47 85*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 51/2000*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. \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{3/2}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{3/2} \,d x \]