Integrand size = 26, antiderivative size = 209 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\frac {164 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^5}-\frac {42863 \sqrt {1-2 x} \sqrt {3+5 x}}{41160 (2+3 x)^4}-\frac {29297 \sqrt {1-2 x} \sqrt {3+5 x}}{82320 (2+3 x)^3}-\frac {55277 \sqrt {1-2 x} \sqrt {3+5 x}}{460992 (2+3 x)^2}+\frac {426781 \sqrt {1-2 x} \sqrt {3+5 x}}{6453888 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^5}-\frac {3474273 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2151296 \sqrt {7}} \]
-3474273/15059072*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+ 11/7*(3+5*x)^(3/2)/(2+3*x)^5/(1-2*x)^(1/2)+164/735*(1-2*x)^(1/2)*(3+5*x)^( 1/2)/(2+3*x)^5-42863/41160*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4-29297/823 20*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3-55277/460992*(1-2*x)^(1/2)*(3+5*x )^(1/2)/(2+3*x)^2+426781/6453888*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.33 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.44 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\frac {121 \left (-\frac {7 \sqrt {3+5 x} \left (-16456032-95331368 x-164918884 x^2-19738914 x^3+180017865 x^4+115230870 x^5\right )}{121 \sqrt {1-2 x} (2+3 x)^5}-143565 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{75295360} \]
(121*((-7*Sqrt[3 + 5*x]*(-16456032 - 95331368*x - 164918884*x^2 - 19738914 *x^3 + 180017865*x^4 + 115230870*x^5))/(121*Sqrt[1 - 2*x]*(2 + 3*x)^5) - 1 43565*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/75295360
Time = 0.29 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {109, 27, 166, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{3/2} (3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}-\frac {1}{7} \int -\frac {\sqrt {5 x+3} (1145 x+654)}{2 \sqrt {1-2 x} (3 x+2)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {\sqrt {5 x+3} (1145 x+654)}{\sqrt {1-2 x} (3 x+2)^6}dx+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \int \frac {187255 x+110549}{\sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \left (\frac {1}{28} \int \frac {3 (857260 x+503147)}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \left (\frac {3}{56} \int \frac {857260 x+503147}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \left (\frac {3}{56} \left (\frac {1}{21} \int \frac {35 (234376 x+137825)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {29297 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \left (\frac {3}{56} \left (\frac {5}{6} \int \frac {234376 x+137825}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {29297 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {1105540 x+879287}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {55277 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {29297 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {1105540 x+879287}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {55277 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {29297 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {10422819}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {426781 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {55277 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {29297 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {10422819}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {426781 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {55277 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {29297 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {10422819}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {426781 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {55277 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {29297 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{105} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {426781 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {10422819 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )-\frac {55277 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {29297 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {42863 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {328 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^5}\) |
(11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^5) + ((328*Sqrt[1 - 2*x]*S qrt[3 + 5*x])/(105*(2 + 3*x)^5) + ((-42863*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 8*(2 + 3*x)^4) + (3*((-29297*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (5*((-55277*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((426781*Sqr t[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (10422819*ArcTan[Sqrt[1 - 2*x]/( Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6))/56)/105)/14
3.26.49.3.1 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*((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])
Leaf count of result is larger than twice the leaf count of optimal. \(352\) vs. \(2(164)=328\).
Time = 1.20 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.69
| method | result | size |
| default | \(\frac {\left (8442483390 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+23920369605 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+23451342750 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+1613232180 x^{5} \sqrt {-10 x^{2}-x +3}+6253691400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+2520250110 x^{4} \sqrt {-10 x^{2}-x +3}-4169127600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-276344796 x^{3} \sqrt {-10 x^{2}-x +3}-3057360240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -2308864376 x^{2} \sqrt {-10 x^{2}-x +3}-555883680 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1334639152 x \sqrt {-10 x^{2}-x +3}-230384448 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{150590720 \left (2+3 x \right )^{5} \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(353\) |
1/150590720*(8442483390*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3 )^(1/2))*x^6+23920369605*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+ 3)^(1/2))*x^5+23451342750*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x +3)^(1/2))*x^4+1613232180*x^5*(-10*x^2-x+3)^(1/2)+6253691400*7^(1/2)*arcta n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+2520250110*x^4*(-10*x^2- x+3)^(1/2)-4169127600*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^ (1/2))*x^2-276344796*x^3*(-10*x^2-x+3)^(1/2)-3057360240*7^(1/2)*arctan(1/1 4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-2308864376*x^2*(-10*x^2-x+3)^(1 /2)-555883680*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-1 334639152*x*(-10*x^2-x+3)^(1/2)-230384448*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/ 2)*(3+5*x)^(1/2)/(2+3*x)^5/(-1+2*x)/(-10*x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=-\frac {17371365 \, \sqrt {7} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (115230870 \, x^{5} + 180017865 \, x^{4} - 19738914 \, x^{3} - 164918884 \, x^{2} - 95331368 \, x - 16456032\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{150590720 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \]
-1/150590720*(17371365*sqrt(7)*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(- 2*x + 1)/(10*x^2 + x - 3)) - 14*(115230870*x^5 + 180017865*x^4 - 19738914* x^3 - 164918884*x^2 - 95331368*x - 16456032)*sqrt(5*x + 3)*sqrt(-2*x + 1)) /(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (164) = 328\).
Time = 0.29 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.90 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\frac {3474273}{30118144} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2133905 \, x}{9680832 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {4998019}{19361664 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{945 \, {\left (243 \, \sqrt {-10 \, x^{2} - x + 3} x^{5} + 810 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 1080 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 720 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 240 \, \sqrt {-10 \, x^{2} - x + 3} x + 32 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {331}{17640 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {83537}{740880 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {23353}{109760 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {137335}{921984 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
3474273/30118144*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2133905/9680832*x/sqrt(-10*x^2 - x + 3) + 4998019/19361664/sqrt(-10*x^2 - x + 3) + 1/945/(243*sqrt(-10*x^2 - x + 3)*x^5 + 810*sqrt(-10*x^2 - x + 3)*x^4 + 1080*sqrt(-10*x^2 - x + 3)*x^3 + 720*sqrt(-10*x^2 - x + 3)*x^2 + 240*sqrt(-10*x^2 - x + 3)*x + 32*sqrt(-10*x^2 - x + 3)) - 331/17640/(81*sq rt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3 + 216*sqrt(-10*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10*x^2 - x + 3)) + 8 3537/740880/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) - 23353/109760/(9*s qrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) - 137335/921984/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3 ))
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (164) = 328\).
Time = 0.86 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.16 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\frac {3474273}{301181440} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1936 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{588245 \, {\left (2 \, x - 1\right )}} - \frac {121 \, \sqrt {10} {\left (203039 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 265495440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 136071290880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 774949504000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {650054039040000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {2600216156160000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{7529536 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
3474273/301181440*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)*sqr t(-10*x + 5) - sqrt(22)))) - 1936/588245*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 121/7529536*sqrt(10)*(203039*((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 + 265495440*((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 + 136071290880*((s qrt(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 - 774949504000*((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)))^3 - 650054039040000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2600216156160000*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)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^6} \,d x \]