Integrand size = 26, antiderivative size = 115 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {1111 \sqrt {1-2 x}}{15 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}-\frac {8}{45} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {665}{9} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
-8/225*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+665/9*arctan(1/7*(1-2* x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+7/3*(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^ (1/2)-1111/15*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=\frac {1}{225} \left (-\frac {15 \sqrt {1-2 x} (2187+3403 x)}{(2+3 x) \sqrt {3+5 x}}+8 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+16625 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \]
((-15*Sqrt[1 - 2*x]*(2187 + 3403*x))/((2 + 3*x)*Sqrt[3 + 5*x]) + 8*Sqrt[10 ]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] + 16625*Sqrt[7]*ArcTan[Sqrt[1 - 2* x]/(Sqrt[7]*Sqrt[3 + 5*x])])/225
Time = 0.21 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {109, 27, 167, 27, 175, 64, 104, 217, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^2 (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{3} \int \frac {\sqrt {1-2 x} (8 x+227)}{2 (3 x+2) (5 x+3)^{3/2}}dx+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {\sqrt {1-2 x} (8 x+227)}{(3 x+2) (5 x+3)^{3/2}}dx+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{6} \left (\frac {2}{5} \int -\frac {16 x+7769}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {2222 \sqrt {1-2 x}}{5 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (-\frac {1}{5} \int \frac {16 x+7769}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {2222 \sqrt {1-2 x}}{5 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (-\frac {16}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {23275}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {2222 \sqrt {1-2 x}}{5 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (-\frac {23275}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {32}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {2222 \sqrt {1-2 x}}{5 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (-\frac {46550}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {32}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {2222 \sqrt {1-2 x}}{5 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {6650}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {32}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {2222 \sqrt {1-2 x}}{5 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {6650}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {16}{3} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )\right )-\frac {2222 \sqrt {1-2 x}}{5 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}\) |
(7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*Sqrt[3 + 5*x]) + ((-2222*Sqrt[1 - 2*x])/( 5*Sqrt[3 + 5*x]) + ((-16*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 + ( 6650*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3)/5)/6
3.25.42.3.1 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 _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(190\) vs. \(2(83)=166\).
Time = 3.84 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.66
| method | result | size |
| default | \(-\frac {\left (120 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+249375 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+152 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +315875 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +48 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+99750 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+102090 x \sqrt {-10 x^{2}-x +3}+65610 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{450 \left (2+3 x \right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(191\) |
-1/450*(120*10^(1/2)*arcsin(20/11*x+1/11)*x^2+249375*7^(1/2)*arctan(1/14*( 37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+152*10^(1/2)*arcsin(20/11*x+1/11 )*x+315875*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+48 *10^(1/2)*arcsin(20/11*x+1/11)+99750*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2) /(-10*x^2-x+3)^(1/2))+102090*x*(-10*x^2-x+3)^(1/2)+65610*(-10*x^2-x+3)^(1/ 2))*(1-2*x)^(1/2)/(2+3*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.23 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=\frac {8 \, \sqrt {5} \sqrt {2} {\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 16625 \, \sqrt {7} {\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 30 \, {\left (3403 \, x + 2187\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{450 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \]
1/450*(8*sqrt(5)*sqrt(2)*(15*x^2 + 19*x + 6)*arctan(1/20*sqrt(5)*sqrt(2)*( 20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 16625*sqrt(7)*( 15*x^2 + 19*x + 6)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 30*(3403*x + 2187)*sqrt(5*x + 3)*sqrt(-2*x + 1)) /(15*x^2 + 19*x + 6)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {4}{225} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {665}{18} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {6806 \, x}{45 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {10699}{135 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343}{27 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
-4/225*sqrt(10)*arcsin(20/11*x + 1/11) - 665/18*sqrt(7)*arcsin(37/11*x/abs (3*x + 2) + 20/11/abs(3*x + 2)) + 6806/45*x/sqrt(-10*x^2 - x + 3) - 10699/ 135/sqrt(-10*x^2 - x + 3) + 343/27/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10 *x^2 - x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (83) = 166\).
Time = 0.39 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.77 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {133}{36} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {4}{225} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121}{50} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {1078 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
-133/36*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 4/225*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt (2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 121/50*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt (5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 1078/3 *sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - s qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(2 2)))^2 + 280)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}} \,d x \]