Integrand size = 26, antiderivative size = 122 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {32 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {25}{9} \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {169 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{441 \sqrt {7}} \]
-25/18*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-169/3087*arctan(1/7*(1 -2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+11/7*(3+5*x)^(3/2)/(2+3*x)/(1-2 *x)^(1/2)+32/147*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {42 \sqrt {3+5 x} (725+1091 x)+8575 \sqrt {10-20 x} (2+3 x) \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-338 \sqrt {7-14 x} (2+3 x) \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6174 \sqrt {1-2 x} (2+3 x)} \]
(42*Sqrt[3 + 5*x]*(725 + 1091*x) + 8575*Sqrt[10 - 20*x]*(2 + 3*x)*ArcTan[S qrt[5/2 - 5*x]/Sqrt[3 + 5*x]] - 338*Sqrt[7 - 14*x]*(2 + 3*x)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(6174*Sqrt[1 - 2*x]*(2 + 3*x))
Time = 0.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {109, 27, 166, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{3/2} (3 x+2)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {1}{7} \int \frac {\sqrt {5 x+3} (175 x+138)}{2 \sqrt {1-2 x} (3 x+2)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {1}{14} \int \frac {\sqrt {5 x+3} (175 x+138)}{\sqrt {1-2 x} (3 x+2)^2}dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{14} \left (\frac {64 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}-\frac {1}{21} \int \frac {6125 x+4027}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{21} \left (\frac {169}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {6125}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )+\frac {64 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{21} \left (\frac {169}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {2450}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )+\frac {64 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{21} \left (\frac {338}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {2450}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )+\frac {64 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{21} \left (-\frac {2450}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {338 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )+\frac {64 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{21} \left (-\frac {1225}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {338 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )+\frac {64 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}\) |
(11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + ((64*Sqrt[1 - 2*x]*Sqrt [3 + 5*x])/(21*(2 + 3*x)) + ((-1225*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5* x]])/3 - (338*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*Sqrt[7]))/ 21)/14
3.26.45.3.1 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] && ILtQ[m, -1] && GtQ[n, 0]
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. \(197\) vs. \(2(90)=180\).
Time = 1.19 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.62
| method | result | size |
| default | \(-\frac {\left (51450 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-2028 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+8575 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -338 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -17150 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+676 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+91644 x \sqrt {-10 x^{2}-x +3}+60900 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{12348 \left (2+3 x \right ) \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(198\) |
-1/12348*(51450*10^(1/2)*arcsin(20/11*x+1/11)*x^2-2028*7^(1/2)*arctan(1/14 *(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+8575*10^(1/2)*arcsin(20/11*x+1 /11)*x-338*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-17 150*10^(1/2)*arcsin(20/11*x+1/11)+676*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2 )/(-10*x^2-x+3)^(1/2))+91644*x*(-10*x^2-x+3)^(1/2)+60900*(-10*x^2-x+3)^(1/ 2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)/(-1+2*x)/(-10*x^2-x+3)^(1/2)
Time = 0.25 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {8575 \, \sqrt {5} \sqrt {2} {\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 338 \, \sqrt {7} {\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 84 \, {\left (1091 \, x + 725\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12348 \, {\left (6 \, x^{2} + x - 2\right )}} \]
1/12348*(8575*sqrt(5)*sqrt(2)*(6*x^2 + x - 2)*arctan(1/20*sqrt(5)*sqrt(2)* (20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 338*sqrt(7)*(6 *x^2 + x - 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1) /(10*x^2 + x - 3)) - 84*(1091*x + 725)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(6*x^ 2 + x - 2)
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=-\frac {25}{36} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {169}{6174} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {5455 \, x}{441 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {9784}{1323 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{189 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
-25/36*sqrt(10)*arcsin(20/11*x + 1/11) + 169/6174*sqrt(7)*arcsin(37/11*x/a bs(3*x + 2) + 20/11/abs(3*x + 2)) + 5455/441*x/sqrt(-10*x^2 - x + 3) + 978 4/1323/sqrt(-10*x^2 - x + 3) + 1/189/(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. 286 vs. \(2 (90) = 180\).
Time = 0.43 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.34 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\frac {169}{61740} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {25}{36} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{245 \, {\left (2 \, x - 1\right )}} - \frac {22 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{147 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
169/61740*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*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)))) - 25/36*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sq rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5 ) - sqrt(22)))) - 121/245*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 22/147*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) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2} \,d x \]