Integrand size = 26, antiderivative size = 180 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {1466281 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}-\frac {5591773 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}} \]
-5591773/307328*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1/ 15*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^5+37/840*(1-2*x)^(1/2)*(3+5*x)^(1/2 )/(2+3*x)^4+403/1680*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+14023/9408*(1-2 *x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+1466281/131712*(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 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (41933792+247045192 x+546004068 x^2+536695650 x^3+197947935 x^4\right )}{(2+3 x)^5}-27958865 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1536640} \]
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(41933792 + 247045192*x + 546004068*x^2 + 536695650*x^3 + 197947935*x^4))/(2 + 3*x)^5 - 27958865*Sqrt[7]*ArcTan[Sqrt [1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/1536640
Time = 0.27 (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, 168, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{15} \int -\frac {20 x+1}{2 \sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{30} \int \frac {20 x+1}{\sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}-\frac {1}{28} \int -\frac {3 (447-740 x)}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {3}{56} \int \frac {447-740 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {3}{56} \left (\frac {1}{21} \int \frac {35 (2525-3224 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {3}{56} \left (\frac {5}{6} \int \frac {2525-3224 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {301787-280460 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {14023 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {301787-280460 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {14023 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{30} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {16775319}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1466281 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {14023 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {16775319}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1466281 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {14023 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{30} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {16775319}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {1466281 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {14023 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{30} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1466281 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {16775319 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {14023 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}\) |
-1/15*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^5 + ((37*Sqrt[1 - 2*x]*Sqrt[ 3 + 5*x])/(28*(2 + 3*x)^4) + (3*((403*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (5*((14023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((14 66281*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (16775319*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6))/56)/30
3.23.72.3.1 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 + 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 = 134, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (197947935 x^{4}+536695650 x^{3}+546004068 x^{2}+247045192 x +41933792\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{219520 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {5591773 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{614656 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(134\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (6794004195 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+22646680650 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+30195574200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+2771271090 x^{4} \sqrt {-10 x^{2}-x +3}+20130382800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+7513739100 x^{3} \sqrt {-10 x^{2}-x +3}+6710127600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +7644056952 x^{2} \sqrt {-10 x^{2}-x +3}+894683680 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3458632688 x \sqrt {-10 x^{2}-x +3}+587073088 \sqrt {-10 x^{2}-x +3}\right )}{3073280 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) | \(298\) |
-1/219520*(-1+2*x)*(3+5*x)^(1/2)*(197947935*x^4+536695650*x^3+546004068*x^ 2+247045192*x+41933792)/(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)+5591773/614656*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 = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=-\frac {27958865 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (197947935 \, x^{4} + 536695650 \, x^{3} + 546004068 \, x^{2} + 247045192 \, x + 41933792\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3073280 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
-1/3073280*(27958865*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*(197947935*x^4 + 536695650*x^3 + 546004068*x^2 + 247045 192*x + 41933792)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080* x^3 + 720*x^2 + 240*x + 32)
\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=\int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{6}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=\frac {5591773}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {231065}{32928} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {111 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{280 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1305 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{784 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {138639 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1709881 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \]
5591773/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 231065/32928*sqrt(-10*x^2 - x + 3) + 3/35*(-10*x^2 - x + 3)^(3/2)/(243*x^ 5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 111/280*(-10*x^2 - x + 3) ^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1305/784*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 138639/21952*(-10*x^2 - x + 3)^(3/ 2)/(9*x^2 + 12*x + 4) - 1709881/131712*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 = 0.51 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=\frac {5591773}{6146560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {10} {\left (46213 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 85961680 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 30665564160 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 4732042112000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {284050977280000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1136203909120000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{21952 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
5591773/6146560*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)))) - 121/21952*sqrt(10)*(46213*((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 - 85961680*((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 - 3066556416 0*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(s qrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 4732042112000*((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 - 284050977280000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr t(5*x + 3) + 1136203909120000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr t(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)^5
Time = 20.98 (sec) , antiderivative size = 1745, normalized size of antiderivative = 9.69 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=\text {Too large to display} \]
((45503192879*((1 - 2*x)^(1/2) - 1)^7)/(478515625*(3^(1/2) - (5*x + 3)^(1/ 2))^7) - (2981176*((1 - 2*x)^(1/2) - 1)^3)/(1953125*(3^(1/2) - (5*x + 3)^( 1/2))^3) - (61599781879*((1 - 2*x)^(1/2) - 1)^5)/(3349609375*(3^(1/2) - (5 *x + 3)^(1/2))^5) - (22334164*((1 - 2*x)^(1/2) - 1))/(3349609375*(3^(1/2) - (5*x + 3)^(1/2))) - (137589446767*((1 - 2*x)^(1/2) - 1)^9)/(535937500*(3 ^(1/2) - (5*x + 3)^(1/2))^9) + (137589446767*((1 - 2*x)^(1/2) - 1)^11)/(21 4375000*(3^(1/2) - (5*x + 3)^(1/2))^11) - (45503192879*((1 - 2*x)^(1/2) - 1)^13)/(30625000*(3^(1/2) - (5*x + 3)^(1/2))^13) + (61599781879*((1 - 2*x) ^(1/2) - 1)^15)/(34300000*(3^(1/2) - (5*x + 3)^(1/2))^15) + (372647*((1 - 2*x)^(1/2) - 1)^17)/(400*(3^(1/2) - (5*x + 3)^(1/2))^17) + (5583541*((1 - 2*x)^(1/2) - 1)^19)/(219520*(3^(1/2) - (5*x + 3)^(1/2))^19) + (12070766*3^ (1/2)*((1 - 2*x)^(1/2) - 1)^2)/(133984375*(3^(1/2) - (5*x + 3)^(1/2))^2) + (2979759193*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(669921875*(3^(1/2) - (5*x + 3)^(1/2))^4) + (132301459*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(957031250*(3^ (1/2) - (5*x + 3)^(1/2))^6) - (442781811679*3^(1/2)*((1 - 2*x)^(1/2) - 1)^ 8)/(6699218750*(3^(1/2) - (5*x + 3)^(1/2))^8) + (1165566494503*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(4785156250*(3^(1/2) - (5*x + 3)^(1/2))^10) - (4427 81811679*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(1071875000*(3^(1/2) - (5*x + 3 )^(1/2))^12) + (132301459*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(24500000*(3^( 1/2) - (5*x + 3)^(1/2))^14) + (2979759193*3^(1/2)*((1 - 2*x)^(1/2) - 1)...