Integrand size = 26, antiderivative size = 137 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=-\frac {95}{72} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {5}{6} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {155}{216} \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {59 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{27 \sqrt {7}} \]
155/432*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-59/189*arctan(1/7*(1- 2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+5/6*(3+5*x)^(3/2)*(1-2*x)^(1/2)- 1/3*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)-95/72*(1-2*x)^(1/2)*(3+5*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=\frac {\frac {42 \sqrt {1-2 x} \left (-138+175 x+1575 x^2+1500 x^3\right )}{(2+3 x) \sqrt {3+5 x}}-1085 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-944 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3024} \]
((42*Sqrt[1 - 2*x]*(-138 + 175*x + 1575*x^2 + 1500*x^3))/((2 + 3*x)*Sqrt[3 + 5*x]) - 1085*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] - 944*Sqrt[ 7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3024
Time = 0.25 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {108, 27, 171, 27, 171, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{(3 x+2)^2} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{3} \int \frac {(19-60 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {(19-60 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (5 \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {1}{12} \int -\frac {6 \sqrt {5 x+3} (95 x+24)}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {\sqrt {5 x+3} (95 x+24)}{\sqrt {1-2 x} (3 x+2)}dx+5 \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (-\frac {1}{6} \int -\frac {775 x+674}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {95}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+5 \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{12} \int \frac {775 x+674}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {95}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+5 \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{12} \left (\frac {775}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {472}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {95}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+5 \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 64 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{12} \left (\frac {472}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {310}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {95}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+5 \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{12} \left (\frac {944}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {310}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}\right )-\frac {95}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+5 \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{12} \left (\frac {310}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {944 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {95}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+5 \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {1}{12} \left (\frac {155}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {944 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 \sqrt {7}}\right )-\frac {95}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )+5 \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)}\) |
-1/3*(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x) + (5*Sqrt[1 - 2*x]*(3 + 5*x )^(3/2) + ((-95*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6 + ((155*Sqrt[10]*ArcSin[Sqr t[2/11]*Sqrt[3 + 5*x]])/3 - (944*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5* x])])/(3*Sqrt[7]))/12)/2)/6
3.23.92.3.1 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[(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_) )^(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] && 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]
Time = 1.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (300 x^{2}+135 x -46\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{72 \left (2+3 x \right ) \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {155 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{864}-\frac {59 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{378}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(138\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (3255 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +2832 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +25200 x^{2} \sqrt {-10 x^{2}-x +3}+2170 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1888 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+11340 x \sqrt {-10 x^{2}-x +3}-3864 \sqrt {-10 x^{2}-x +3}\right )}{6048 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\) | \(163\) |
-1/72*(-1+2*x)*(3+5*x)^(1/2)*(300*x^2+135*x-46)/(2+3*x)/(-(-1+2*x)*(3+5*x) )^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)-(-155/864*10^(1/2)*arcsin(20 /11*x+1/11)-59/378*7^(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^ 2+67+111*x)^(1/2)))*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=-\frac {1085 \, \sqrt {5} \sqrt {2} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 944 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 84 \, {\left (300 \, x^{2} + 135 \, x - 46\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6048 \, {\left (3 \, x + 2\right )}} \]
-1/6048*(1085*sqrt(5)*sqrt(2)*(3*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)) + 944*sqrt(7)*(3*x + 2 )*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 84*(300*x^2 + 135*x - 46)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)
\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=\int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{2}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=\frac {25}{18} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {155}{864} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {59}{378} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {65}{216} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {\sqrt {-10 \, x^{2} - x + 3}}{27 \, {\left (3 \, x + 2\right )}} \]
25/18*sqrt(-10*x^2 - x + 3)*x + 155/864*sqrt(10)*arcsin(20/11*x + 1/11) + 59/378*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 65/216* sqrt(-10*x^2 - x + 3) - 1/27*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (99) = 198\).
Time = 0.46 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.13 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=\frac {59}{3780} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{216} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 49 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {155}{864} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {22 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{27 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
59/3780*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)))) + 1/216*(12*sqrt(5)*(5*x + 3) - 49*sqrt(5))*sqrt(5*x + 3) *sqrt(-10*x + 5) + 155/864*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)))) - 22/27*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr t(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)*s qrt(-10*x + 5) - sqrt(22)))^2 + 280)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^2} \,d x \]