Integrand size = 26, antiderivative size = 209 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {2770202075 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}-\frac {391280725 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}} \]
-391280725/1229312*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2) -1/18*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^6+1/12*(1-2*x)^(3/2)*(3+5*x)^(1/ 2)/(2+3*x)^5+647/864*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+151621/36288*(1 -2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+26486645/1016064*(1-2*x)^(1/2)*(3+5*x) ^(1/2)/(2+3*x)^2+2770202075/14224896*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.39 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.43 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (3522190656+26026519504 x+76960600672 x^2+113834022672 x^3+84218501340 x^4+24931818675 x^5\right )}{(2+3 x)^6}-1173842175 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3687936} \]
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3522190656 + 26026519504*x + 76960600672* x^2 + 113834022672*x^3 + 84218501340*x^4 + 24931818675*x^5))/(2 + 3*x)^6 - 1173842175*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3687936
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} \sqrt {5 x+3}}{(3 x+2)^7} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{18} \int -\frac {5 (1-2 x)^{3/2} (12 x+5)}{2 (3 x+2)^6 \sqrt {5 x+3}}dx-\frac {(1-2 x)^{5/2} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{36} \int \frac {(1-2 x)^{3/2} (12 x+5)}{(3 x+2)^6 \sqrt {5 x+3}}dx-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{36} \left (-\frac {1}{15} \int \frac {3 (149-100 x) \sqrt {1-2 x}}{2 (3 x+2)^5 \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3} (1-2 x)^{3/2}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{36} \left (-\frac {1}{10} \int \frac {(149-100 x) \sqrt {1-2 x}}{(3 x+2)^5 \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3} (1-2 x)^{3/2}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {5}{36} \left (\frac {1}{10} \left (\frac {1}{12} \int -\frac {25727-37220 x}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {647 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{36} \left (\frac {1}{10} \left (-\frac {1}{24} \int \frac {25727-37220 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {647 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {5}{36} \left (\frac {1}{10} \left (\frac {1}{24} \left (-\frac {1}{21} \int \frac {5 (957131-1212968 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {151621 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {647 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{36} \left (\frac {1}{10} \left (\frac {1}{24} \left (-\frac {5}{42} \int \frac {957131-1212968 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {151621 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {647 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {5}{36} \left (\frac {1}{10} \left (\frac {1}{24} \left (-\frac {5}{42} \left (\frac {1}{14} \int \frac {5 (22809817-21189316 x)}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {5297329 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {151621 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {647 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{36} \left (\frac {1}{10} \left (\frac {1}{24} \left (-\frac {5}{42} \left (\frac {5}{28} \int \frac {22809817-21189316 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {5297329 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {151621 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {647 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {5}{36} \left (\frac {1}{10} \left (\frac {1}{24} \left (-\frac {5}{42} \left (\frac {5}{28} \left (\frac {1}{7} \int \frac {1267749549}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {110808083 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {5297329 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {151621 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {647 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{36} \left (\frac {1}{10} \left (\frac {1}{24} \left (-\frac {5}{42} \left (\frac {5}{28} \left (\frac {1267749549}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {110808083 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {5297329 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {151621 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {647 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {5}{36} \left (\frac {1}{10} \left (\frac {1}{24} \left (-\frac {5}{42} \left (\frac {5}{28} \left (\frac {1267749549}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {110808083 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {5297329 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {151621 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {647 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {5}{36} \left (\frac {1}{10} \left (\frac {1}{24} \left (-\frac {5}{42} \left (\frac {5}{28} \left (\frac {110808083 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {1267749549 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {5297329 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {151621 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )-\frac {647 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {3 (1-2 x)^{3/2} \sqrt {5 x+3}}{5 (3 x+2)^5}\right )-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}\) |
-1/18*((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^6 - (5*((-3*(1 - 2*x)^(3/2 )*Sqrt[3 + 5*x])/(5*(2 + 3*x)^5) + ((-647*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12 *(2 + 3*x)^4) + ((-151621*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^3) - (5*((5297329*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + (5*((11080808 3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (1267749549*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])))/28))/42)/24)/10))/36
3.24.98.3.1 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 = 1.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (24931818675 x^{5}+84218501340 x^{4}+113834022672 x^{3}+76960600672 x^{2}+26026519504 x +3522190656\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{526848 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {391280725 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2458624 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(139\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (855730945575 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+3422923782300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+5704872970500 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+349045461450 x^{5} \sqrt {-10 x^{2}-x +3}+5070998196000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+1179059018760 x^{4} \sqrt {-10 x^{2}-x +3}+2535499098000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1593676317408 x^{3} \sqrt {-10 x^{2}-x +3}+676133092800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +1077448409408 x^{2} \sqrt {-10 x^{2}-x +3}+75125899200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+364371273056 x \sqrt {-10 x^{2}-x +3}+49310669184 \sqrt {-10 x^{2}-x +3}\right )}{7375872 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\) | \(346\) |
-1/526848*(-1+2*x)*(3+5*x)^(1/2)*(24931818675*x^5+84218501340*x^4+11383402 2672*x^3+76960600672*x^2+26026519504*x+3522190656)/(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)+391280725/2458624*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 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=-\frac {1173842175 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (24931818675 \, x^{5} + 84218501340 \, x^{4} + 113834022672 \, x^{3} + 76960600672 \, x^{2} + 26026519504 \, x + 3522190656\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7375872 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
-1/7375872*(1173842175*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)*sqrt (-2*x + 1)/(10*x^2 + x - 3)) - 14*(24931818675*x^5 + 84218501340*x^4 + 113 834022672*x^3 + 76960600672*x^2 + 26026519504*x + 3522190656)*sqrt(5*x + 3 )*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 5 76*x + 64)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{7}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=\frac {391280725}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {16168625}{131712} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{18 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {19 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{12 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {4673 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{672 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {821945 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{28224 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {9701175 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{87808 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {119647825 \, \sqrt {-10 \, x^{2} - x + 3}}{526848 \, {\left (3 \, x + 2\right )}} \]
391280725/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2) ) + 16168625/131712*sqrt(-10*x^2 - x + 3) + 7/18*(-10*x^2 - x + 3)^(3/2)/( 729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 19/12* (-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 4673/672*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 821945/28224*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 9701175/87808*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 119647825/526 848*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 = 0.69 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=\frac {78256145}{4917248} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {366025 \, \sqrt {10} {\left (3207 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} - 8960840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 4031723136 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 929280844800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 111701434880000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {5519365017600000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {22077460070400000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{263424 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]
78256145/4917248*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)))) - 366025/263424*sqrt(10)*(3207*((sqrt(2)*sqrt(-1 0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^11 - 8960840*((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 - 403172 3136*((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 - 929280844800*((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 - 111701434880000*((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 - 5519365017600000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2207 7460070400000*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)*sq rt(-10*x + 5) - sqrt(22)))^2 + 280)^6
Timed out. \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^7} \,d x \]