Integrand size = 26, antiderivative size = 151 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^4}-\frac {43 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {85 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {57595 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}-\frac {78045 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}} \]
-78045/153664*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+1/84 *(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4-43/504*(1-2*x)^(1/2)*(3+5*x)^(1/2)/ (2+3*x)^3+85/14112*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+57595/197568*(1-2 *x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.54 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {121 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (48240+226348 x+346760 x^2+172785 x^3\right )}{121 (2+3 x)^4}-645 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{153664} \]
(121*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(48240 + 226348*x + 346760*x^2 + 1727 85*x^3))/(121*(2 + 3*x)^4) - 645*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr t[3 + 5*x])]))/153664
Time = 0.24 (sec) , antiderivative size = 166, 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 = {109, 27, 168, 27, 168, 27, 168, 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.
| \(\displaystyle \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}-\frac {1}{84} \int -\frac {1340 x+793}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{168} \int \frac {1340 x+793}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{168} \left (\frac {1}{21} \int \frac {35 (344 x+235)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{168} \left (\frac {5}{6} \int \frac {344 x+235}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {3613-340 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {17 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {3613-340 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {17 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {140481}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {11519 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {17 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {140481}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {11519 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {17 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {140481}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {11519 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {17 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {11519 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {140481 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {17 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {43 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\) |
(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3*x)^4) + ((-43*Sqrt[1 - 2*x]*Sqrt[ 3 + 5*x])/(3*(2 + 3*x)^3) + (5*((17*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((11519*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (140481*Arc Tan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6)/168
3.25.76.3.1 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[(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 + 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_)^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])
Time = 1.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (172785 x^{3}+346760 x^{2}+226348 x +48240\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{21952 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {78045 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307328 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (6321645 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+16857720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+16857720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+2418990 x^{3} \sqrt {-10 x^{2}-x +3}+7492320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +4854640 x^{2} \sqrt {-10 x^{2}-x +3}+1248720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3168872 x \sqrt {-10 x^{2}-x +3}+675360 \sqrt {-10 x^{2}-x +3}\right )}{307328 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
-1/21952*(-1+2*x)*(3+5*x)^(1/2)*(172785*x^3+346760*x^2+226348*x+48240)/(2+ 3*x)^4/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+780 45/307328*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 = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {78045 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (172785 \, x^{3} + 346760 \, x^{2} + 226348 \, x + 48240\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{307328 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
-1/307328*(78045*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1 /14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 1 4*(172785*x^3 + 346760*x^2 + 226348*x + 48240)*sqrt(5*x + 3)*sqrt(-2*x + 1 ))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{5}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {78045}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{84 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {43 \, \sqrt {-10 \, x^{2} - x + 3}}{504 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {85 \, \sqrt {-10 \, x^{2} - x + 3}}{14112 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {57595 \, \sqrt {-10 \, x^{2} - x + 3}}{197568 \, {\left (3 \, x + 2\right )}} \]
78045/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1 /84*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) - 43/50 4*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 85/14112*sqrt(-10*x ^2 - x + 3)/(9*x^2 + 12*x + 4) + 57595/197568*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (118) = 236\).
Time = 0.54 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.44 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {15609}{614656} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {605 \, \sqrt {10} {\left (129 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 132440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 21026880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {2510681600 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {10042726400 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{10976 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
15609/614656*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)))) - 605/10976*sqrt(10)*(129*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr t(22)))^7 + 132440*((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)))^5 - 21026880*((sqrt(2 )*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr t(-10*x + 5) - sqrt(22)))^3 - 2510681600*(sqrt(2)*sqrt(-10*x + 5) - sqrt(2 2))/sqrt(5*x + 3) + 10042726400*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - s qrt(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)^4
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^5} \,d x \]