Integrand size = 26, antiderivative size = 106 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=-\frac {61547 \sqrt {1-2 x} \sqrt {3+5 x}}{5120}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (865+408 x)}{1280}+\frac {677017 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5120 \sqrt {10}} \]
677017/51200*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-3/40*(2+3*x)^2*( 3+5*x)^(3/2)*(1-2*x)^(1/2)-3/1280*(3+5*x)^(3/2)*(865+408*x)*(1-2*x)^(1/2)- 61547/5120*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.38 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\frac {-\sqrt {5-10 x} \sqrt {3+5 x} \left (97295+88092 x+57888 x^2+17280 x^3\right )-677017 \sqrt {2} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{5120 \sqrt {5}} \]
(-(Sqrt[5 - 10*x]*Sqrt[3 + 5*x]*(97295 + 88092*x + 57888*x^2 + 17280*x^3)) - 677017*Sqrt[2]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10*x])])/(512 0*Sqrt[5])
Time = 0.19 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {111, 27, 164, 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.
| \(\displaystyle \int \frac {(3 x+2)^3 \sqrt {5 x+3}}{\sqrt {1-2 x}} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {1}{40} \int -\frac {(3 x+2) \sqrt {5 x+3} (765 x+482)}{2 \sqrt {1-2 x}}dx-\frac {3}{40} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{80} \int \frac {(3 x+2) \sqrt {5 x+3} (765 x+482)}{\sqrt {1-2 x}}dx-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{80} \left (\frac {61547}{32} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {3}{16} \sqrt {1-2 x} (5 x+3)^{3/2} (408 x+865)\right )-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{80} \left (\frac {61547}{32} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{16} \sqrt {1-2 x} (5 x+3)^{3/2} (408 x+865)\right )-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{80} \left (\frac {61547}{32} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{16} \sqrt {1-2 x} (5 x+3)^{3/2} (408 x+865)\right )-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{80} \left (\frac {61547}{32} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{16} \sqrt {1-2 x} (5 x+3)^{3/2} (408 x+865)\right )-\frac {3}{40} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}\) |
(-3*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/40 + ((-3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(865 + 408*x))/16 + (61547*(-1/2*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x] ) + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2*Sqrt[10])))/32)/80
3.25.59.3.1 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 = 1.17 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {\left (17280 x^{3}+57888 x^{2}+88092 x +97295\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{5120 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {677017 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{102400 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-345600 x^{3} \sqrt {-10 x^{2}-x +3}-1157760 x^{2} \sqrt {-10 x^{2}-x +3}+677017 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1761840 x \sqrt {-10 x^{2}-x +3}-1945900 \sqrt {-10 x^{2}-x +3}\right )}{102400 \sqrt {-10 x^{2}-x +3}}\) | \(104\) |
1/5120*(17280*x^3+57888*x^2+88092*x+97295)*(-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)+677017/102400*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 = 72, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=-\frac {1}{5120} \, {\left (17280 \, x^{3} + 57888 \, x^{2} + 88092 \, x + 97295\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {677017}{102400} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
-1/5120*(17280*x^3 + 57888*x^2 + 88092*x + 97295)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 677017/102400*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3 )*sqrt(-2*x + 1)/(10*x^2 + x - 3))
\[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\int \frac {\left (3 x + 2\right )^{3} \sqrt {5 x + 3}}{\sqrt {1 - 2 x}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\frac {27}{80} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {677017}{102400} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {351}{320} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {4383}{256} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {114143}{5120} \, \sqrt {-10 \, x^{2} - x + 3} \]
27/80*(-10*x^2 - x + 3)^(3/2)*x + 677017/102400*sqrt(5)*sqrt(2)*arcsin(20/ 11*x + 1/11) + 351/320*(-10*x^2 - x + 3)^(3/2) - 4383/256*sqrt(-10*x^2 - x + 3)*x - 114143/5120*sqrt(-10*x^2 - x + 3)
Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.59 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=-\frac {1}{1280000} \, \sqrt {5} {\left (2 \, {\left (36 \, {\left (24 \, {\left (20 \, x + 43\right )} {\left (5 \, x + 3\right )} + 5179\right )} {\left (5 \, x + 3\right )} + 1538675\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 16925425 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
-1/1280000*sqrt(5)*(2*(36*(24*(20*x + 43)*(5*x + 3) + 5179)*(5*x + 3) + 15 38675)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 16925425*sqrt(2)*arcsin(1/11*sqrt(2 2)*sqrt(5*x + 3)))
Time = 13.17 (sec) , antiderivative size = 708, normalized size of antiderivative = 6.68 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\frac {677017\,\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 )}{25600}-\frac {\frac {431257\,\left (\sqrt {1-2\,x}-1\right )}{7812500\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {418991\,{\left (\sqrt {1-2\,x}-1\right )}^3}{625000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {284249727\,{\left (\sqrt {1-2\,x}-1\right )}^5}{31250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {157157861\,{\left (\sqrt {1-2\,x}-1\right )}^7}{12500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {157157861\,{\left (\sqrt {1-2\,x}-1\right )}^9}{5000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {284249727\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{2000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {418991\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{6400\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}-\frac {431257\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{12800\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {75776\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {2039808\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {8020992\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {5040128\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {2005248\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {127488\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {1184\,\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}} \]
(677017*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/25600 - ((431257*((1 - 2*x)^(1/2) - 1))/(7812500*(3^(1/2) - (5*x + 3)^(1/2))) - (418991*((1 - 2*x)^(1/2) - 1)^3)/(625000*(3^(1/2) - ( 5*x + 3)^(1/2))^3) - (284249727*((1 - 2*x)^(1/2) - 1)^5)/(31250000*(3^(1/2 ) - (5*x + 3)^(1/2))^5) + (157157861*((1 - 2*x)^(1/2) - 1)^7)/(12500000*(3 ^(1/2) - (5*x + 3)^(1/2))^7) - (157157861*((1 - 2*x)^(1/2) - 1)^9)/(500000 0*(3^(1/2) - (5*x + 3)^(1/2))^9) + (284249727*((1 - 2*x)^(1/2) - 1)^11)/(2 000000*(3^(1/2) - (5*x + 3)^(1/2))^11) + (418991*((1 - 2*x)^(1/2) - 1)^13) /(6400*(3^(1/2) - (5*x + 3)^(1/2))^13) - (431257*((1 - 2*x)^(1/2) - 1)^15) /(12800*(3^(1/2) - (5*x + 3)^(1/2))^15) + (75776*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (2039808*3^(1/2)*((1 - 2* x)^(1/2) - 1)^4)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (8020992*3^(1/2) *((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (504012 8*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^8) + (2005248*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(15625*(3^(1/2) - (5*x + 3)^(1 /2))^10) + (127488*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(625*(3^(1/2) - (5*x + 3)^(1/2))^12) + (1184*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 + ...