Integrand size = 26, antiderivative size = 150 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {115431701 \sqrt {1-2 x} \sqrt {3+5 x}}{10240000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {3+5 x}}{1024000}-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {1269748711 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10240000 \sqrt {10}} \]
-953981/384000*(1-2*x)^(3/2)*(3+5*x)^(3/2)-1/20*(1-2*x)^(3/2)*(2+3*x)^2*(3 +5*x)^(5/2)-7/32000*(1-2*x)^(3/2)*(3+5*x)^(5/2)*(3821+2256*x)+1269748711/1 02400000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-10493791/1024000*(1- 2*x)^(3/2)*(3+5*x)^(1/2)+115431701/10240000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.59 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-1451592441-2354035875 x+4157008580 x^2+14549698400 x^3+19494864000 x^4+12890880000 x^5+3456000000 x^6\right )-3809246133 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{307200000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(-1451592441 - 2354035875*x + 4157008580*x^2 + 145496984 00*x^3 + 19494864000*x^4 + 12890880000*x^5 + 3456000000*x^6) - 3809246133* Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(307200000*Sqrt[3 + 5*x])
Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {111, 27, 164, 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.
| \(\displaystyle \int \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{3/2} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {1}{60} \int -\frac {21}{2} \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2} (47 x+30)dx-\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{40} \int \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2} (47 x+30)dx-\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {7}{40} \left (\frac {136283 \int \sqrt {1-2 x} (5 x+3)^{3/2}dx}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (2256 x+3821)\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {7}{40} \left (\frac {136283 \left (\frac {11}{4} \int \sqrt {1-2 x} \sqrt {5 x+3}dx-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (2256 x+3821)\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {7}{40} \left (\frac {136283 \left (\frac {11}{4} \left (\frac {11}{8} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (2256 x+3821)\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {7}{40} \left (\frac {136283 \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (2256 x+3821)\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {7}{40} \left (\frac {136283 \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (2256 x+3821)\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {7}{40} \left (\frac {136283 \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (2256 x+3821)\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\) |
-1/20*((1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2)) + (7*(-1/800*((1 - 2*x )^(3/2)*(3 + 5*x)^(5/2)*(3821 + 2256*x)) + (136283*(-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))/1600))/40
3.23.75.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
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[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 = 3.60 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(-\frac {\left (691200000 x^{5}+2163456000 x^{4}+2600899200 x^{3}+1349400160 x^{2}+21761620 x -483864147\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{30720000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1269748711 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{204800000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(113\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (13824000000 x^{5} \sqrt {-10 x^{2}-x +3}+43269120000 x^{4} \sqrt {-10 x^{2}-x +3}+52017984000 x^{3} \sqrt {-10 x^{2}-x +3}+26988003200 x^{2} \sqrt {-10 x^{2}-x +3}+3809246133 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+435232400 x \sqrt {-10 x^{2}-x +3}-9677282940 \sqrt {-10 x^{2}-x +3}\right )}{614400000 \sqrt {-10 x^{2}-x +3}}\) | \(138\) |
-1/30720000*(691200000*x^5+2163456000*x^4+2600899200*x^3+1349400160*x^2+21 761620*x-483864147)*(-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)+1269748711/204800000*10^(1/2)*arcsin(20/1 1*x+1/11)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.55 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {1}{30720000} \, {\left (691200000 \, x^{5} + 2163456000 \, x^{4} + 2600899200 \, x^{3} + 1349400160 \, x^{2} + 21761620 \, x - 483864147\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1269748711}{204800000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/30720000*(691200000*x^5 + 2163456000*x^4 + 2600899200*x^3 + 1349400160*x ^2 + 21761620*x - 483864147)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1269748711/204 800000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.69 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx=-\frac {9}{4} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {2727}{400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {270711}{32000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {2147273}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {10493791}{512000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1269748711}{204800000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {10493791}{10240000} \, \sqrt {-10 \, x^{2} - x + 3} \]
-9/4*(-10*x^2 - x + 3)^(3/2)*x^3 - 2727/400*(-10*x^2 - x + 3)^(3/2)*x^2 - 270711/32000*(-10*x^2 - x + 3)^(3/2)*x - 2147273/384000*(-10*x^2 - x + 3)^ (3/2) + 10493791/512000*sqrt(-10*x^2 - x + 3)*x - 1269748711/204800000*sqr t(10)*arcsin(-20/11*x - 1/11) + 10493791/10240000*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (111) = 222\).
Time = 0.35 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.37 \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\frac {9}{512000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{4000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {921}{3200000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {883}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {141}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {36}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
9/512000000*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))) + 9/4000000*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*sqr t(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 921/3200000*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))) + 883/60000*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))) + 141/500*sqrt(5)*(2*(20* x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*s qrt(5*x + 3))) + 36/25*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. \[ \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx=\int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2} \,d x \]