Integrand size = 26, antiderivative size = 150 \[ \int (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=\frac {41137943 \sqrt {1-2 x} \sqrt {3+5 x}}{25600000}+\frac {3739813 (1-2 x)^{3/2} \sqrt {3+5 x}}{7680000}-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {452517373 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{25600000 \sqrt {10}} \]
-1/20*(1-2*x)^(5/2)*(2+3*x)^2*(3+5*x)^(3/2)-1/160000*(1-2*x)^(5/2)*(3+5*x) ^(3/2)*(88987+63120*x)+452517373/256000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1 /2))*10^(1/2)+3739813/7680000*(1-2*x)^(3/2)*(3+5*x)^(1/2)-339983/384000*(1 -2*x)^(5/2)*(3+5*x)^(1/2)+41137943/25600000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.59 \[ \int (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=\frac {-10 \sqrt {1-2 x} \left (244217763-716691375 x-3743568940 x^2-2192231200 x^3+5295888000 x^4+8328960000 x^5+3456000000 x^6\right )-1357552119 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{768000000 \sqrt {3+5 x}} \]
(-10*Sqrt[1 - 2*x]*(244217763 - 716691375*x - 3743568940*x^2 - 2192231200* x^3 + 5295888000*x^4 + 8328960000*x^5 + 3456000000*x^6) - 1357552119*Sqrt[ 30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(768000000*Sqrt[3 + 5*x] )
Time = 0.22 (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.
| \(\displaystyle \int (1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {1}{60} \int -\frac {3}{2} (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3} (263 x+166)dx-\frac {1}{20} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{40} \int (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3} (263 x+166)dx-\frac {1}{20} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{40} \left (\frac {339983 \int (1-2 x)^{3/2} \sqrt {5 x+3}dx}{1600}-\frac {(1-2 x)^{5/2} (5 x+3)^{3/2} (63120 x+88987)}{4000}\right )-\frac {1}{20} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{40} \left (\frac {339983 \left (\frac {11}{12} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )}{1600}-\frac {(1-2 x)^{5/2} (5 x+3)^{3/2} (63120 x+88987)}{4000}\right )-\frac {1}{20} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{40} \left (\frac {339983 \left (\frac {11}{12} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )}{1600}-\frac {(1-2 x)^{5/2} (5 x+3)^{3/2} (63120 x+88987)}{4000}\right )-\frac {1}{20} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{40} \left (\frac {339983 \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )}{1600}-\frac {(1-2 x)^{5/2} (5 x+3)^{3/2} (63120 x+88987)}{4000}\right )-\frac {1}{20} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{40} \left (\frac {339983 \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )}{1600}-\frac {(1-2 x)^{5/2} (5 x+3)^{3/2} (63120 x+88987)}{4000}\right )-\frac {1}{20} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{40} \left (\frac {339983 \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )}{1600}-\frac {(1-2 x)^{5/2} (5 x+3)^{3/2} (63120 x+88987)}{4000}\right )-\frac {1}{20} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{3/2}\) |
-1/20*((1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (-1/4000*((1 - 2*x)^ (5/2)*(3 + 5*x)^(3/2)*(88987 + 63120*x)) + (339983*(-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*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10]) ))/20))/12))/1600)/40
3.24.27.3.1 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 = 1.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(\frac {\left (691200000 x^{5}+1251072000 x^{4}+308534400 x^{3}-623566880 x^{2}-374573660 x +81405921\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{76800000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {452517373 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{512000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(113\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-13824000000 x^{5} \sqrt {-10 x^{2}-x +3}-25021440000 x^{4} \sqrt {-10 x^{2}-x +3}-6170688000 x^{3} \sqrt {-10 x^{2}-x +3}+12471337600 x^{2} \sqrt {-10 x^{2}-x +3}+1357552119 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+7491473200 x \sqrt {-10 x^{2}-x +3}-1628118420 \sqrt {-10 x^{2}-x +3}\right )}{1536000000 \sqrt {-10 x^{2}-x +3}}\) | \(138\) |
1/76800000*(691200000*x^5+1251072000*x^4+308534400*x^3-623566880*x^2-37457 3660*x+81405921)*(-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)+452517373/512000000*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 = 82, normalized size of antiderivative = 0.55 \[ \int (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=-\frac {1}{76800000} \, {\left (691200000 \, x^{5} + 1251072000 \, x^{4} + 308534400 \, x^{3} - 623566880 \, x^{2} - 374573660 \, x + 81405921\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {452517373}{512000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
-1/76800000*(691200000*x^5 + 1251072000*x^4 + 308534400*x^3 - 623566880*x^ 2 - 374573660*x + 81405921)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 452517373/51200 0000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1) /(10*x^2 + x - 3))
\[ \int (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=\int \left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{3} \sqrt {5 x + 3}\, dx \]
Time = 0.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.69 \[ \int (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=\frac {9}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {1539}{1000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {41427}{80000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {385939}{960000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3739813}{1280000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {452517373}{512000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3739813}{25600000} \, \sqrt {-10 \, x^{2} - x + 3} \]
9/10*(-10*x^2 - x + 3)^(3/2)*x^3 + 1539/1000*(-10*x^2 - x + 3)^(3/2)*x^2 + 41427/80000*(-10*x^2 - x + 3)^(3/2)*x - 385939/960000*(-10*x^2 - x + 3)^( 3/2) + 3739813/1280000*sqrt(-10*x^2 - x + 3)*x - 452517373/512000000*sqrt( 10)*arcsin(-20/11*x - 1/11) + 3739813/25600000*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 = 0.38 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.37 \[ \int (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=-\frac {9}{1280000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {189}{320000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {111}{3200000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {23}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{20} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {12}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
-9/1280000000*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) *sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 189/320000000*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) + 103921 95*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 111/3200000*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))) + 23/6000 0*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))) + 1/20*sqrt(5)*(2*(2 0*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22) *sqrt(5*x + 3))) + 12/25*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. \[ \int (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x} \, dx=\int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3\,\sqrt {5\,x+3} \,d x \]