Integrand size = 26, antiderivative size = 180 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=-\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{2520 (2+3 x)^4}+\frac {641 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {17981 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {1852307 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}-\frac {783959 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}} \]
-783959/307328*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1/1 5*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^5-107/2520*(1-2*x)^(1/2)*(3+5*x)^(1/ 2)/(2+3*x)^4+641/15120*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+17981/84672*( 1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+1852307/1185408*(1-2*x)^(1/2)*(3+5*x) ^(1/2)/(2+3*x)
Time = 0.33 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (17507808+103856008 x+230080132 x^2+226052850 x^3+83353815 x^4\right )}{(2+3 x)^5}-11759385 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4609920} \]
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(17507808 + 103856008*x + 230080132*x^2 + 226052850*x^3 + 83353815*x^4))/(2 + 3*x)^5 - 11759385*Sqrt[7]*ArcTan[Sqrt[ 1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/4609920
Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {108, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{15} \int \frac {(9-40 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^5}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \int \frac {(9-40 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^5}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{84} \int -\frac {4780 x+1691}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (-\frac {1}{168} \int \frac {4780 x+1691}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{168} \left (\frac {641 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}-\frac {1}{21} \int -\frac {35 (2575-5128 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx\right )-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{168} \left (\frac {5}{6} \int \frac {2575-5128 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {641 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {377689-359620 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {17981 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {641 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {377689-359620 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {17981 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {641 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {21166893}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1852307 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {17981 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {641 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {21166893}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1852307 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {17981 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {641 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {21166893}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {1852307 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {17981 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {641 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1852307 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {21166893 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {17981 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {641 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}\) |
-1/15*(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^5 + ((-107*Sqrt[1 - 2*x]*S qrt[3 + 5*x])/(84*(2 + 3*x)^4) + ((641*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (5*((17981*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((1 852307*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (21166893*ArcTan[Sqrt[ 1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6)/168)/30
3.23.84.3.1 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 = 3.64 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (83353815 x^{4}+226052850 x^{3}+230080132 x^{2}+103856008 x +17507808\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{658560 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {783959 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{614656 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(134\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (2857530555 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+9525101850 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+12700135800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+1166953410 x^{4} \sqrt {-10 x^{2}-x +3}+8466757200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+3164739900 x^{3} \sqrt {-10 x^{2}-x +3}+2822252400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +3221121848 x^{2} \sqrt {-10 x^{2}-x +3}+376300320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1453984112 x \sqrt {-10 x^{2}-x +3}+245109312 \sqrt {-10 x^{2}-x +3}\right )}{9219840 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) | \(298\) |
-1/658560*(-1+2*x)*(3+5*x)^(1/2)*(83353815*x^4+226052850*x^3+230080132*x^2 +103856008*x+17507808)/(2+3*x)^5/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x ))^(1/2)/(1-2*x)^(1/2)+783959/614656*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 = 131, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=-\frac {11759385 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (83353815 \, x^{4} + 226052850 \, x^{3} + 230080132 \, x^{2} + 103856008 \, x + 17507808\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{9219840 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
-1/9219840*(11759385*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240 *x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10* x^2 + x - 3)) - 14*(83353815*x^4 + 226052850*x^3 + 230080132*x^2 + 1038560 08*x + 17507808)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x ^3 + 720*x^2 + 240*x + 32)
\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{6}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\frac {783959}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {32395}{32928} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {13 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{280 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {545 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2352 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {19437 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {239723 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \]
783959/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 32395/32928*sqrt(-10*x^2 - x + 3) - 1/35*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 13/280*(-10*x^2 - x + 3)^(3 /2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 545/2352*(-10*x^2 - x + 3)^ (3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 19437/21952*(-10*x^2 - x + 3)^(3/2)/( 9*x^2 + 12*x + 4) - 239723/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (141) = 282\).
Time = 0.53 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\frac {783959}{6146560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (1767 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 2308880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 925245440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 177804928000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {10860971520000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {43443886080000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{65856 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
783959/6146560*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)))) - 1331/65856*sqrt(10)*(1767*((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 + 2308880*((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 - 925245440*(( 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 - 177804928000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq rt(22)))^3 - 10860971520000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 43443886080000*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)/(sq rt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^6} \,d x \]