Integrand size = 26, antiderivative size = 144 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {415 \sqrt {3+5 x}}{22638 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {765 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \]
-765/9604*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+2/21*(3+ 5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^2+415/22638*(3+5*x)^(1/2)/(1-2*x)^(1/2)-1 /14*(3+5*x)^(1/2)/(2+3*x)^2/(1-2*x)^(1/2)+5/196*(3+5*x)^(1/2)/(2+3*x)/(1-2 *x)^(1/2)
Time = 0.18 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {7 \sqrt {3+5 x} \left (-6708-8633 x+19380 x^2+14940 x^3\right )-25245 \sqrt {7-14 x} (-1+2 x) (2+3 x)^2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{316932 (1-2 x)^{3/2} (2+3 x)^2} \]
-1/316932*(7*Sqrt[3 + 5*x]*(-6708 - 8633*x + 19380*x^2 + 14940*x^3) - 2524 5*Sqrt[7 - 14*x]*(-1 + 2*x)*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt [3 + 5*x])])/((1 - 2*x)^(3/2)*(2 + 3*x)^2)
Time = 0.23 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {110, 27, 168, 27, 168, 27, 169, 27, 104, 217}
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.
| \(\displaystyle \int \frac {\sqrt {5 x+3}}{(1-2 x)^{5/2} (3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {2}{21} \int -\frac {90 x+53}{2 (1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \int \frac {90 x+53}{(1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{21} \left (\frac {1}{14} \int \frac {35 (24 x+17)}{2 (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {5}{4} \int \frac {24 x+17}{(1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{21} \left (\frac {5}{4} \left (\frac {1}{7} \int \frac {113-60 x}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx+\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {5}{4} \left (\frac {1}{14} \int \frac {113-60 x}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx+\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{21} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {332 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {2}{77} \int -\frac {5049}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{21} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {459}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {332 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{21} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {918}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {332 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{21} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {332 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {918 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {3 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + ((-3*Sqrt[3 + 5*x])/( 2*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*((3*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + ((332*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) - (918*ArcTan[Sqrt[1 - 2* x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/14))/4)/21
3.26.87.3.1 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[(((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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs. \(2(111)=222\).
Time = 4.06 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.78
| method | result | size |
| default | \(\frac {\left (908820 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+302940 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-580635 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-209160 x^{3} \sqrt {-10 x^{2}-x +3}-100980 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -271320 x^{2} \sqrt {-10 x^{2}-x +3}+100980 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+120862 x \sqrt {-10 x^{2}-x +3}+93912 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{633864 \left (2+3 x \right )^{2} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(257\) |
1/633864*(908820*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2) )*x^4+302940*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^ 3-580635*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-20 9160*x^3*(-10*x^2-x+3)^(1/2)-100980*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/ (-10*x^2-x+3)^(1/2))*x-271320*x^2*(-10*x^2-x+3)^(1/2)+100980*7^(1/2)*arcta n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+120862*x*(-10*x^2-x+3)^(1/2) +93912*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2/(-1+2*x) ^2/(-10*x^2-x+3)^(1/2)
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {25245 \, \sqrt {7} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (14940 \, x^{3} + 19380 \, x^{2} - 8633 \, x - 6708\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{633864 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
-1/633864*(25245*sqrt(7)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*arctan(1/14* sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(1 4940*x^3 + 19380*x^2 - 8633*x - 6708)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(36*x^ 4 + 12*x^3 - 23*x^2 - 4*x + 4)
\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{3}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {765}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2075 \, x}{22638 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {4415}{45276 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {125 \, x}{294 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{126 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {23}{252 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {5}{1764 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
765/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2075 /22638*x/sqrt(-10*x^2 - x + 3) + 4415/45276/sqrt(-10*x^2 - x + 3) + 125/29 4*x/(-10*x^2 - x + 3)^(3/2) - 1/126/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(- 10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 23/252/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 5/1764/(-10*x^2 - x + 3)^( 3/2)
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (111) = 222\).
Time = 0.53 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {153}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {8 \, {\left (524 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3267 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1980825 \, {\left (2 \, x - 1\right )}^{2}} - \frac {297 \, \sqrt {10} {\left (19 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {840 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {3360 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4802 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
153/38416*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)))) - 8/1980825*(524*sqrt(5)*(5*x + 3) - 3267*sqrt(5))*sqrt (5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 297/4802*sqrt(10)*(19*((sqrt(2)*sq rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1 0*x + 5) - sqrt(22)))^3 - 840*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5* x + 3) + 3360*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)*sq rt(-10*x + 5) - sqrt(22)))^2 + 280)^2
Timed out. \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3} \,d x \]