Integrand size = 26, antiderivative size = 132 \[ \int \frac {(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {13246251 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac {145708761 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}} \]
-145708761/512000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+(2+3*x)^3*( 3+5*x)^(3/2)/(1-2*x)^(1/2)+27/16*(2+3*x)^2*(3+5*x)^(3/2)*(1-2*x)^(1/2)+9/1 2800*(3+5*x)^(3/2)*(62091+29320*x)*(1-2*x)^(1/2)+13246251/51200*(1-2*x)^(1 /2)*(3+5*x)^(1/2)
Time = 0.59 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {-\frac {5 \sqrt {3+5 x} \left (-22217679+15218818 x+8057880 x^2+3729600 x^3+864000 x^4\right )}{\sqrt {1-2 x}}+145708761 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{256000} \]
((-5*Sqrt[3 + 5*x]*(-22217679 + 15218818*x + 8057880*x^2 + 3729600*x^3 + 8 64000*x^4))/Sqrt[1 - 2*x] + 145708761*Sqrt[10]*ArcTan[Sqrt[6 + 10*x]/(Sqrt [11] - Sqrt[5 - 10*x])])/256000
Time = 0.23 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {108, 27, 170, 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.
| \(\displaystyle \int \frac {(3 x+2)^3 (5 x+3)^{3/2}}{(1-2 x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\int \frac {3 (3 x+2)^2 \sqrt {5 x+3} (45 x+28)}{2 \sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \int \frac {(3 x+2)^2 \sqrt {5 x+3} (45 x+28)}{\sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (-\frac {1}{40} \int -\frac {5 (3 x+2) \sqrt {5 x+3} (2199 x+1382)}{2 \sqrt {1-2 x}}dx-\frac {9}{8} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{16} \int \frac {(3 x+2) \sqrt {5 x+3} (2199 x+1382)}{\sqrt {1-2 x}}dx-\frac {9}{8} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{16} \left (\frac {4415417}{800} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {3}{400} \sqrt {1-2 x} (5 x+3)^{3/2} (29320 x+62091)\right )-\frac {9}{8} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{16} \left (\frac {4415417}{800} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{400} \sqrt {1-2 x} (5 x+3)^{3/2} (29320 x+62091)\right )-\frac {9}{8} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{16} \left (\frac {4415417}{800} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{400} \sqrt {1-2 x} (5 x+3)^{3/2} (29320 x+62091)\right )-\frac {9}{8} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {(3 x+2)^3 (5 x+3)^{3/2}}{\sqrt {1-2 x}}-\frac {3}{2} \left (\frac {1}{16} \left (\frac {4415417}{800} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{400} \sqrt {1-2 x} (5 x+3)^{3/2} (29320 x+62091)\right )-\frac {9}{8} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}\right )\) |
((2 + 3*x)^3*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] - (3*((-9*Sqrt[1 - 2*x]*(2 + 3 *x)^2*(3 + 5*x)^(3/2))/8 + ((-3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(62091 + 293 20*x))/400 + (4415417*(-1/2*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (11*ArcSin[Sqr t[2/11]*Sqrt[3 + 5*x]])/(2*Sqrt[10])))/800)/16))/2
3.26.30.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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 = 1.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(-\frac {\left (-17280000 x^{4} \sqrt {-10 x^{2}-x +3}-74592000 x^{3} \sqrt {-10 x^{2}-x +3}+291417522 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -161157600 x^{2} \sqrt {-10 x^{2}-x +3}-145708761 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-304376360 x \sqrt {-10 x^{2}-x +3}+444353580 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{1024000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(140\) |
-1/1024000*(-17280000*x^4*(-10*x^2-x+3)^(1/2)-74592000*x^3*(-10*x^2-x+3)^( 1/2)+291417522*10^(1/2)*arcsin(20/11*x+1/11)*x-161157600*x^2*(-10*x^2-x+3) ^(1/2)-145708761*10^(1/2)*arcsin(20/11*x+1/11)-304376360*x*(-10*x^2-x+3)^( 1/2)+444353580*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)/( -10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\frac {145708761 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (864000 \, x^{4} + 3729600 \, x^{3} + 8057880 \, x^{2} + 15218818 \, x - 22217679\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1024000 \, {\left (2 \, x - 1\right )}} \]
1/1024000*(145708761*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sq rt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(864000*x^4 + 3729600*x^ 3 + 8057880*x^2 + 15218818*x - 22217679)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2* x - 1)
\[ \int \frac {(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \]
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.39 \[ \int \frac {(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=-\frac {27}{32} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {155771121}{1024000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {251559}{25600} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) - \frac {2547}{640} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {2079}{64} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x - \frac {9801}{2560} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {43659}{1280} \, \sqrt {10 \, x^{2} - 21 \, x + 8} + \frac {5811399}{51200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {343 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{16 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (2 \, x - 1\right )}} - \frac {11319 \, \sqrt {-10 \, x^{2} - x + 3}}{32 \, {\left (2 \, x - 1\right )}} \]
-27/32*(-10*x^2 - x + 3)^(3/2)*x - 155771121/1024000*sqrt(5)*sqrt(2)*arcsi n(20/11*x + 1/11) - 251559/25600*I*sqrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) - 2547/640*(-10*x^2 - x + 3)^(3/2) + 2079/64*sqrt(10*x^2 - 21*x + 8)*x - 9801/2560*sqrt(-10*x^2 - x + 3)*x - 43659/1280*sqrt(10*x^2 - 21*x + 8) + 5 811399/51200*sqrt(-10*x^2 - x + 3) - 343/16*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) - 441/32*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) - 11319/32*sqrt(-10 *x^2 - x + 3)/(2*x - 1)
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.73 \[ \int \frac {(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=-\frac {145708761}{512000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (36 \, {\left (8 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 115 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 8919 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4415417 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 145708761 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1280000 \, {\left (2 \, x - 1\right )}} \]
-145708761/512000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/1280000 *(2*(36*(8*(12*sqrt(5)*(5*x + 3) + 115*sqrt(5))*(5*x + 3) + 8919*sqrt(5))* (5*x + 3) + 4415417*sqrt(5))*(5*x + 3) - 145708761*sqrt(5))*sqrt(5*x + 3)* sqrt(-10*x + 5)/(2*x - 1)
Timed out. \[ \int \frac {(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]