Integrand size = 26, antiderivative size = 110 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3200 \sqrt {10}} \]
-1463447/32000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+(2+3*x)^3*(3+5 *x)^(1/2)/(1-2*x)^(1/2)+7/4*(2+3*x)^2*(1-2*x)^(1/2)*(3+5*x)^(1/2)+1/3200*( 176833+73380*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {-10 \sqrt {3+5 x} \left (-224833+142686 x+57960 x^2+14400 x^3\right )+1463447 \sqrt {10-20 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{32000 \sqrt {1-2 x}} \]
(-10*Sqrt[3 + 5*x]*(-224833 + 142686*x + 57960*x^2 + 14400*x^3) + 1463447* Sqrt[10 - 20*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(32000*Sqrt[1 - 2*x ])
Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {108, 27, 170, 27, 164, 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.
\(\displaystyle \int \frac {(3 x+2)^3 \sqrt {5 x+3}}{(1-2 x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {(3 x+2)^3 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\int \frac {(3 x+2)^2 (105 x+64)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(3 x+2)^3 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {1}{2} \int \frac {(3 x+2)^2 (105 x+64)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\) |
\(\Big \downarrow \) 170 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{30} \int -\frac {15 (3 x+2) (1223 x+750)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {7}{2} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )+\frac {\sqrt {5 x+3} (3 x+2)^3}{\sqrt {1-2 x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {7}{2} \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}-\frac {1}{4} \int \frac {(3 x+2) (1223 x+750)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )+\frac {\sqrt {5 x+3} (3 x+2)^3}{\sqrt {1-2 x}}\) |
\(\Big \downarrow \) 164 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{400} \sqrt {1-2 x} \sqrt {5 x+3} (73380 x+176833)-\frac {1463447}{800} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )+\frac {7}{2} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )+\frac {\sqrt {5 x+3} (3 x+2)^3}{\sqrt {1-2 x}}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{400} \sqrt {1-2 x} \sqrt {5 x+3} (73380 x+176833)-\frac {1463447 \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}}{2000}\right )+\frac {7}{2} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )+\frac {\sqrt {5 x+3} (3 x+2)^3}{\sqrt {1-2 x}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{400} \sqrt {1-2 x} \sqrt {5 x+3} (73380 x+176833)-\frac {1463447 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}}\right )+\frac {7}{2} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2\right )+\frac {\sqrt {5 x+3} (3 x+2)^3}{\sqrt {1-2 x}}\) |
((2 + 3*x)^3*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + ((7*Sqrt[1 - 2*x]*(2 + 3*x)^2* Sqrt[3 + 5*x])/2 + ((Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(176833 + 73380*x))/400 - (1463447*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*Sqrt[10]))/4)/2
3.26.20.3.1 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[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 = 3.98 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.12
method | result | size |
default | \(-\frac {\left (-288000 x^{3} \sqrt {-10 x^{2}-x +3}+2926894 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -1159200 x^{2} \sqrt {-10 x^{2}-x +3}-1463447 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2853720 x \sqrt {-10 x^{2}-x +3}+4496660 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{64000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(123\) |
-1/64000*(-288000*x^3*(-10*x^2-x+3)^(1/2)+2926894*10^(1/2)*arcsin(20/11*x+ 1/11)*x-1159200*x^2*(-10*x^2-x+3)^(1/2)-1463447*10^(1/2)*arcsin(20/11*x+1/ 11)-2853720*x*(-10*x^2-x+3)^(1/2)+4496660*(-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 = 86, normalized size of antiderivative = 0.78 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {1463447 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (14400 \, x^{3} + 57960 \, x^{2} + 142686 \, x - 224833\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{64000 \, {\left (2 \, x - 1\right )}} \]
1/64000*(1463447*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5 *x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(14400*x^3 + 57960*x^2 + 142 686*x - 224833)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)
\[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int \frac {\left (3 x + 2\right )^{3} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=-\frac {1463447}{64000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {9}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1593}{160} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {89793}{3200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {343 \, \sqrt {-10 \, x^{2} - x + 3}}{8 \, {\left (2 \, x - 1\right )}} \]
-1463447/64000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9/40*(-10*x^2 - x + 3)^(3/2) + 1593/160*sqrt(-10*x^2 - x + 3)*x + 89793/3200*sqrt(-10*x^2 - x + 3) - 343/8*sqrt(-10*x^2 - x + 3)/(2*x - 1)
Time = 0.49 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=-\frac {1463447}{32000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (18 \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 89 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4927 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 1463447 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{80000 \, {\left (2 \, x - 1\right )}} \]
-1463447/32000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/80000*(18* (4*(8*sqrt(5)*(5*x + 3) + 89*sqrt(5))*(5*x + 3) + 4927*sqrt(5))*(5*x + 3) - 1463447*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)
Timed out. \[ \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^3\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]