Integrand size = 26, antiderivative size = 121 \[ \int \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {14443 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1313 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {158873 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800 \sqrt {10}} \]
-37/160*(1-2*x)^(3/2)*(3+5*x)^(3/2)-3/40*(1-2*x)^(3/2)*(2+3*x)*(3+5*x)^(3/ 2)+158873/128000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-1313/1280*(1 -2*x)^(3/2)*(3+5*x)^(1/2)+14443/12800*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {10 \sqrt {1-2 x} \left (-39981+865 x+267540 x^2+344800 x^3+144000 x^4\right )-158873 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{128000 \sqrt {3+5 x}} \]
(10*Sqrt[1 - 2*x]*(-39981 + 865*x + 267540*x^2 + 344800*x^3 + 144000*x^4) - 158873*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(128000*Sq rt[3 + 5*x])
Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {101, 27, 90, 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.
| \(\displaystyle \int \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3} \, dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle -\frac {1}{40} \int -\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3} (555 x+356)dx-\frac {3}{40} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{80} \int \sqrt {1-2 x} \sqrt {5 x+3} (555 x+356)dx-\frac {3}{40} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{80} \left (\frac {1313}{4} \int \sqrt {1-2 x} \sqrt {5 x+3}dx-\frac {37}{2} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {3}{40} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{80} \left (\frac {1313}{4} \left (\frac {11}{8} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {37}{2} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {3}{40} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{80} \left (\frac {1313}{4} \left (\frac {11}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {37}{2} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {3}{40} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{80} \left (\frac {1313}{4} \left (\frac {11}{8} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {37}{2} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {3}{40} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{80} \left (\frac {1313}{4} \left (\frac {11}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {37}{2} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {3}{40} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}\) |
(-3*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/40 + ((-37*(1 - 2*x)^(3/2)* (3 + 5*x)^(3/2))/2 + (1313*(-1/4*((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (11*((S qrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*S qrt[10])))/8))/4)/80
3.23.64.3.1 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 = 1.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(-\frac {\left (28800 x^{3}+51680 x^{2}+22500 x -13327\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{12800 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {158873 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{256000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (576000 x^{3} \sqrt {-10 x^{2}-x +3}+1033600 x^{2} \sqrt {-10 x^{2}-x +3}+158873 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+450000 x \sqrt {-10 x^{2}-x +3}-266540 \sqrt {-10 x^{2}-x +3}\right )}{256000 \sqrt {-10 x^{2}-x +3}}\) | \(104\) |
-1/12800*(28800*x^3+51680*x^2+22500*x-13327)*(-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)+158873/256000*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 = 72, normalized size of antiderivative = 0.60 \[ \int \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {1}{12800} \, {\left (28800 \, x^{3} + 51680 \, x^{2} + 22500 \, x - 13327\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {158873}{256000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
1/12800*(28800*x^3 + 51680*x^2 + 22500*x - 13327)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 158873/256000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3 )*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right )^{2} \sqrt {5 x + 3}\, dx \]
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58 \[ \int \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx=-\frac {9}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {61}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1313}{640} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {158873}{256000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1313}{12800} \, \sqrt {-10 \, x^{2} - x + 3} \]
-9/40*(-10*x^2 - x + 3)^(3/2)*x - 61/160*(-10*x^2 - x + 3)^(3/2) + 1313/64 0*sqrt(-10*x^2 - x + 3)*x - 158873/256000*sqrt(10)*arcsin(-20/11*x - 1/11) + 1313/12800*sqrt(-10*x^2 - x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (88) = 176\).
Time = 0.32 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.68 \[ \int \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {3}{640000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {29}{40000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {7}{250} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {6}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
3/640000*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))) + 29/40000*sqrt(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))) + 7/250*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2 )*arcsin(1/11*sqrt(22)*sqrt(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))
Time = 11.84 (sec) , antiderivative size = 708, normalized size of antiderivative = 5.85 \[ \int \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {158873\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{64000}-\frac {\frac {38070349\,{\left (\sqrt {1-2\,x}-1\right )}^5}{15625000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {3947\,{\left (\sqrt {1-2\,x}-1\right )}^3}{7812500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {148327\,\left (\sqrt {1-2\,x}-1\right )}{19531250\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {88735647\,{\left (\sqrt {1-2\,x}-1\right )}^7}{6250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {88735647\,{\left (\sqrt {1-2\,x}-1\right )}^9}{2500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}-\frac {38070349\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{1000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {3947\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{80000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}+\frac {148327\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{32000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {16384\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {137728\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1014272\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {364288\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {253568\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}-\frac {8608\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {256\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}}{\frac {1024\,{\left (\sqrt {1-2\,x}-1\right )}^2}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^6}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {224\,{\left (\sqrt {1-2\,x}-1\right )}^8}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {448\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {112\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {16\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{16}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}+\frac {256}{390625}} \]
(158873*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/64000 - ((38070349*((1 - 2*x)^(1/2) - 1)^5)/(15625000*(3^(1 /2) - (5*x + 3)^(1/2))^5) - (3947*((1 - 2*x)^(1/2) - 1)^3)/(7812500*(3^(1/ 2) - (5*x + 3)^(1/2))^3) - (148327*((1 - 2*x)^(1/2) - 1))/(19531250*(3^(1/ 2) - (5*x + 3)^(1/2))) - (88735647*((1 - 2*x)^(1/2) - 1)^7)/(6250000*(3^(1 /2) - (5*x + 3)^(1/2))^7) + (88735647*((1 - 2*x)^(1/2) - 1)^9)/(2500000*(3 ^(1/2) - (5*x + 3)^(1/2))^9) - (38070349*((1 - 2*x)^(1/2) - 1)^11)/(100000 0*(3^(1/2) - (5*x + 3)^(1/2))^11) + (3947*((1 - 2*x)^(1/2) - 1)^13)/(80000 *(3^(1/2) - (5*x + 3)^(1/2))^13) + (148327*((1 - 2*x)^(1/2) - 1)^15)/(3200 0*(3^(1/2) - (5*x + 3)^(1/2))^15) + (16384*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2 )/(390625*(3^(1/2) - (5*x + 3)^(1/2))^2) - (137728*3^(1/2)*((1 - 2*x)^(1/2 ) - 1)^4)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (1014272*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (364288*3^(1/2 )*((1 - 2*x)^(1/2) - 1)^8)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^8) + (253568 *3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^10) - (8608*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(625*(3^(1/2) - (5*x + 3)^(1/2)) ^12) + (256*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(25*(3^(1/2) - (5*x + 3)^(1/ 2))^14))/((1024*((1 - 2*x)^(1/2) - 1)^2)/(78125*(3^(1/2) - (5*x + 3)^(1/2) )^2) + (1792*((1 - 2*x)^(1/2) - 1)^4)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^4 ) + (1792*((1 - 2*x)^(1/2) - 1)^6)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^6)...