Integrand size = 26, antiderivative size = 143 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {1089847 \sqrt {1-2 x} \sqrt {3+5 x}}{256000}-\frac {99077 (1-2 x)^{3/2} \sqrt {3+5 x}}{25600}-\frac {9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac {153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac {11988317 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000 \sqrt {10}} \]
-9007/9600*(1-2*x)^(3/2)*(3+5*x)^(3/2)-153/800*(1-2*x)^(3/2)*(3+5*x)^(5/2) -3/50*(1-2*x)^(3/2)*(2+3*x)*(3+5*x)^(5/2)+11988317/2560000*arcsin(1/11*22^ (1/2)*(3+5*x)^(1/2))*10^(1/2)-99077/25600*(1-2*x)^(3/2)*(3+5*x)^(1/2)+1089 847/256000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.58 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-12047427-12421425 x+54502060 x^2+119936800 x^3+104688000 x^4+34560000 x^5\right )-35964951 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{7680000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(-12047427 - 12421425*x + 54502060*x^2 + 119936800*x^3 + 104688000*x^4 + 34560000*x^5) - 35964951*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(7680000*Sqrt[3 + 5*x])
Time = 0.22 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {101, 27, 90, 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.
| \(\displaystyle \int \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2} \, dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {1}{50} \int -\frac {1}{2} \sqrt {1-2 x} (5 x+3)^{3/2} (765 x+496)dx-\frac {3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{100} \int \sqrt {1-2 x} (5 x+3)^{3/2} (765 x+496)dx-\frac {3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{100} \left (\frac {9007}{16} \int \sqrt {1-2 x} (5 x+3)^{3/2}dx-\frac {153}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{100} \left (\frac {9007}{16} \left (\frac {11}{4} \int \sqrt {1-2 x} \sqrt {5 x+3}dx-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {153}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{100} \left (\frac {9007}{16} \left (\frac {11}{4} \left (\frac {11}{8} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {153}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{100} \left (\frac {9007}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {153}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{100} \left (\frac {9007}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {153}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{100} \left (\frac {9007}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {153}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}\) |
(-3*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(5/2))/50 + ((-153*(1 - 2*x)^(3/2) *(3 + 5*x)^(5/2))/8 + (9007*(-1/6*((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (11* (-1/4*((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (11*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) /5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/8))/4))/16)/100
3.23.76.3.1 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 = 3.62 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(-\frac {\left (6912000 x^{4}+16790400 x^{3}+13913120 x^{2}+2552540 x -4015809\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{768000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {11988317 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{5120000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(108\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (138240000 x^{4} \sqrt {-10 x^{2}-x +3}+335808000 x^{3} \sqrt {-10 x^{2}-x +3}+278262400 x^{2} \sqrt {-10 x^{2}-x +3}+35964951 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+51050800 x \sqrt {-10 x^{2}-x +3}-80316180 \sqrt {-10 x^{2}-x +3}\right )}{15360000 \sqrt {-10 x^{2}-x +3}}\) | \(121\) |
-1/768000*(6912000*x^4+16790400*x^3+13913120*x^2+2552540*x-4015809)*(-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)+11988317/5120000*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 = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.54 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {1}{768000} \, {\left (6912000 \, x^{4} + 16790400 \, x^{3} + 13913120 \, x^{2} + 2552540 \, x - 4015809\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {11988317}{5120000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/768000*(6912000*x^4 + 16790400*x^3 + 13913120*x^2 + 2552540*x - 4015809) *sqrt(5*x + 3)*sqrt(-2*x + 1) - 11988317/5120000*sqrt(10)*arctan(1/20*sqrt (10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.61 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx=-\frac {9}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {1677}{800} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {17971}{9600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {99077}{12800} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {11988317}{5120000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {99077}{256000} \, \sqrt {-10 \, x^{2} - x + 3} \]
-9/10*(-10*x^2 - x + 3)^(3/2)*x^2 - 1677/800*(-10*x^2 - x + 3)^(3/2)*x - 1 7971/9600*(-10*x^2 - x + 3)^(3/2) + 99077/12800*sqrt(-10*x^2 - x + 3)*x - 11988317/5120000*sqrt(10)*arcsin(-20/11*x - 1/11) + 99077/256000*sqrt(-10* x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (104) = 208\).
Time = 0.34 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.92 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\frac {3}{12800000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {19}{320000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {541}{120000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {57}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {18}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
3/12800000*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))) + 19/320000*sqrt(5)*(2*(4*(8*(6 0*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))) + 541/120000*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))) + 57/500*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))) + 18/25*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. \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx=\int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2} \,d x \]