Integrand size = 26, antiderivative size = 108 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=-\frac {407 \sqrt {3+5 x}}{98 \sqrt {1-2 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}+\frac {25}{6} \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{147 \sqrt {7}} \]
11/21*(3+5*x)^(3/2)/(1-2*x)^(3/2)+25/12*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2) )*10^(1/2)+2/1029*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)- 407/98*(3+5*x)^(1/2)/(1-2*x)^(1/2)
Time = 1.45 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.53 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=\frac {\frac {77 \sqrt {3+5 x} (-69+292 x)}{(1-2 x)^{3/2}}-4 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )-8575 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )-4 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )}{2058} \]
((77*Sqrt[3 + 5*x]*(-69 + 292*x))/(1 - 2*x)^(3/2) - 4*Sqrt[7]*ArcTan[(Sqrt [2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] - 8575* Sqrt[10]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10*x])] - 4*Sqrt[7]*Ar cTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))]) /2058
Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{5/2} (3 x+2)} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {1}{21} \int \frac {3 \sqrt {5 x+3} (175 x+116)}{2 (1-2 x)^{3/2} (3 x+2)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {1}{14} \int \frac {\sqrt {5 x+3} (175 x+116)}{(1-2 x)^{3/2} (3 x+2)}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{14} \left (-\frac {1}{7} \int -\frac {6125 x+4082}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {407 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{14} \int \frac {6125 x+4082}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {407 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{14} \left (\frac {6125}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {4}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {407 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{14} \left (\frac {2450}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {4}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {407 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{14} \left (\frac {2450}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {8}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {407 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{14} \left (\frac {2450}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {8 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {407 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{14} \left (\frac {1225}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {8 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {407 \sqrt {5 x+3}}{7 \sqrt {1-2 x}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}\) |
(11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)) + ((-407*Sqrt[3 + 5*x])/(7*Sqrt[ 1 - 2*x]) + ((1225*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 + (8*ArcTa n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*Sqrt[7]))/14)/14
3.27.3.3.1 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. \(2(76)=152\).
Time = 1.22 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.77
| method | result | size |
| default | \(\frac {\left (34300 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-32 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-34300 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +32 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +8575 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-8 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+89936 x \sqrt {-10 x^{2}-x +3}-21252 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{8232 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(191\) |
1/8232*(34300*10^(1/2)*arcsin(20/11*x+1/11)*x^2-32*7^(1/2)*arctan(1/14*(37 *x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-34300*10^(1/2)*arcsin(20/11*x+1/11 )*x+32*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+8575*1 0^(1/2)*arcsin(20/11*x+1/11)-8*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10* x^2-x+3)^(1/2))+89936*x*(-10*x^2-x+3)^(1/2)-21252*(-10*x^2-x+3)^(1/2))*(1- 2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)
Time = 0.25 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.31 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=-\frac {8575 \, \sqrt {5} \sqrt {2} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 8 \, \sqrt {7} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 308 \, {\left (292 \, x - 69\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{8232 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/8232*(8575*sqrt(5)*sqrt(2)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2 )*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 8*sqrt(7)*(4 *x^2 - 4*x + 1)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 308*(292*x - 69)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x ^2 - 4*x + 1)
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}} \cdot \left (3 x + 2\right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (76) = 152\).
Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.51 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=-\frac {12233125 \, x^{2}}{3557763 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {625 \, x^{3}}{6 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {25}{24} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{1029} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2446625}{7115526} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {12021894385 \, x}{697321548 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {16029625 \, x^{2}}{117612 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {6953014391}{697321548 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {12465295 \, x}{205821 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2681981}{274428 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
-12233125/3557763*x^2/sqrt(-10*x^2 - x + 3) + 625/6*x^3/(-10*x^2 - x + 3)^ (3/2) + 25/24*sqrt(10)*arcsin(20/11*x + 1/11) - 1/1029*sqrt(7)*arcsin(37/1 1*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2446625/7115526*sqrt(-10*x^2 - x + 3) - 12021894385/697321548*x/sqrt(-10*x^2 - x + 3) + 16029625/117612*x^2 /(-10*x^2 - x + 3)^(3/2) - 6953014391/697321548/sqrt(-10*x^2 - x + 3) + 12 465295/205821*x/(-10*x^2 - x + 3)^(3/2) + 2681981/274428/(-10*x^2 - x + 3) ^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (76) = 152\).
Time = 0.36 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.67 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=-\frac {1}{10290} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {25}{24} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {11 \, {\left (292 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1221 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{7350 \, {\left (2 \, x - 1\right )}^{2}} \]
-1/10290*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)))) + 25/24*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqr t(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 11/7350*(292*sqrt(5)*(5*x + 3) - 1221*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )} \,d x \]