Integrand size = 26, antiderivative size = 209 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=-\frac {6533 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {47279 \sqrt {1-2 x} \sqrt {3+5 x}}{1270080 (2+3 x)^3}+\frac {1057139 \sqrt {1-2 x} \sqrt {3+5 x}}{7112448 (2+3 x)^2}+\frac {106751933 \sqrt {1-2 x} \sqrt {3+5 x}}{99574272 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}-\frac {15036307 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1229312 \sqrt {7}} \]
-15036307/8605184*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)- 59/1260*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^5-1/18*(3+5*x)^(5/2)*(1-2*x)^( 1/2)/(2+3*x)^6-6533/211680*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+47279/127 0080*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+1057139/7112448*(1-2*x)^(1/2)*( 3+5*x)^(1/2)/(2+3*x)^2+106751933/99574272*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3 *x)
Time = 0.37 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (665270208+4978384240 x+14818971424 x^2+21960917808 x^3+16234789140 x^4+4803836985 x^5\right )}{(2+3 x)^6}-225544605 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{129077760} \]
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(665270208 + 4978384240*x + 14818971424*x^ 2 + 21960917808*x^3 + 16234789140*x^4 + 4803836985*x^5))/(2 + 3*x)^6 - 225 544605*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/129077760
Time = 0.29 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {108, 27, 166, 27, 166, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{(3 x+2)^7} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{18} \int \frac {(19-60 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)^6}dx-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \int \frac {(19-60 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^6}dx-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{105} \int -\frac {3 \sqrt {5 x+3} (3460 x+129)}{2 \sqrt {1-2 x} (3 x+2)^5}dx-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (-\frac {1}{70} \int \frac {\sqrt {5 x+3} (3460 x+129)}{\sqrt {1-2 x} (3 x+2)^5}dx-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (-\frac {1}{84} \int \frac {576820 x+274229}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (-\frac {1}{168} \int \frac {576820 x+274229}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (\frac {47279 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}-\frac {1}{21} \int -\frac {35 (100225-378232 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx\right )-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (\frac {5}{6} \int \frac {100225-378232 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {47279 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {21488791-21142780 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {1057139 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {47279 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {21488791-21142780 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {1057139 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {47279 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {1217940867}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {106751933 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {1057139 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {47279 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1217940867}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {106751933 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {1057139 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {47279 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1217940867}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {106751933 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {1057139 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {47279 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {106751933 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {1217940867 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {1057139 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {47279 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}\) |
-1/18*(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^6 + ((-59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(35*(2 + 3*x)^5) + ((-6533*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84 *(2 + 3*x)^4) + ((47279*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (5* ((1057139*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((106751933*Sqrt [1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (1217940867*ArcTan[Sqrt[1 - 2*x]/ (Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6)/168)/70)/36
3.23.97.3.1 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/(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[(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[((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 = 1.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (4803836985 x^{5}+16234789140 x^{4}+21960917808 x^{3}+14818971424 x^{2}+4978384240 x +665270208\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{18439680 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {15036307 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{17210368 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(139\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (164422017045 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+657688068180 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+1096146780300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+67253717790 x^{5} \sqrt {-10 x^{2}-x +3}+974352693600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+227287047960 x^{4} \sqrt {-10 x^{2}-x +3}+487176346800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+307452849312 x^{3} \sqrt {-10 x^{2}-x +3}+129913692480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +207465599936 x^{2} \sqrt {-10 x^{2}-x +3}+14434854720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+69697379360 x \sqrt {-10 x^{2}-x +3}+9313782912 \sqrt {-10 x^{2}-x +3}\right )}{258155520 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\) | \(346\) |
-1/18439680*(-1+2*x)*(3+5*x)^(1/2)*(4803836985*x^5+16234789140*x^4+2196091 7808*x^3+14818971424*x^2+4978384240*x+665270208)/(2+3*x)^6/(-(-1+2*x)*(3+5 *x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+15036307/17210368*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 = 146, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=-\frac {225544605 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (4803836985 \, x^{5} + 16234789140 \, x^{4} + 21960917808 \, x^{3} + 14818971424 \, x^{2} + 4978384240 \, x + 665270208\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{258155520 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
-1/258155520*(225544605*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqr t(-2*x + 1)/(10*x^2 + x - 3)) - 14*(4803836985*x^5 + 16234789140*x^4 + 219 60917808*x^3 + 14818971424*x^2 + 4978384240*x + 665270208)*sqrt(5*x + 3)*s qrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576* x + 64)
\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{7}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {15036307}{17210368} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {621335}{921984} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{126 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} - \frac {169 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2940 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {547 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{23520 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {31055 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{197568 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {372801 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {4597879 \, \sqrt {-10 \, x^{2} - x + 3}}{3687936 \, {\left (3 \, x + 2\right )}} \]
15036307/17210368*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2) ) + 621335/921984*sqrt(-10*x^2 - x + 3) + 1/126*(-10*x^2 - x + 3)^(3/2)/(7 29*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) - 169/294 0*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 547/23520*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96 *x + 16) + 31055/197568*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 372801/614656*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 4597879/36 87936*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (164) = 328\).
Time = 0.70 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {15036307}{172103680} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {14641 \, \sqrt {10} {\left (3081 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 4888520 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 3188465280 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 599903001600 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 103716175360000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {5302514380800000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {21210057523200000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1843968 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]
15036307/172103680*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)*sq rt(-10*x + 5) - sqrt(22)))) - 14641/1843968*sqrt(10)*(3081*((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)))^11 + 4888520*((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)))^9 + 3188 465280*((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)))^7 - 599903001600*((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 - 103716175360000*((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)))^3 - 5302514380800000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 21 210057523200000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqr t(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)^6
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^7} \,d x \]