Integrand size = 26, antiderivative size = 209 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {122343637 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {52573169 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{8605184 \sqrt {7}} \]
-52573169/60236288*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2) +1/126*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^6+503/26460*(1-2*x)^(1/2)*(3+5* x)^(1/2)/(2+3*x)^5-149951/1481760*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4-71 369/2963520*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+958171/16595712*(1-2*x)^ (1/2)*(3+5*x)^(1/2)/(2+3*x)^2+122343637/232339968*(1-2*x)^(1/2)*(3+5*x)^(1 /2)/(2+3*x)
Time = 0.34 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.44 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx=\frac {1331 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (2225100096+16771747280 x+50261760608 x^2+74931979536 x^3+55658284380 x^4+16516390995 x^5\right )}{1331 (2+3 x)^6}-592485 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{903544320} \]
(1331*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2225100096 + 16771747280*x + 502617 60608*x^2 + 74931979536*x^3 + 55658284380*x^4 + 16516390995*x^5))/(1331*(2 + 3*x)^6) - 592485*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]) )/903544320
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 = {109, 27, 166, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x} (3 x+2)^7} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}-\frac {1}{126} \int -\frac {\sqrt {5 x+3} (2020 x+1179)}{2 \sqrt {1-2 x} (3 x+2)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{252} \int \frac {\sqrt {5 x+3} (2020 x+1179)}{\sqrt {1-2 x} (3 x+2)^6}dx+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{105} \int \frac {666760 x+394523}{2 \sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \int \frac {666760 x+394523}{\sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \left (\frac {1}{28} \int \frac {3 (2999020 x+1832819)}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \left (\frac {3}{56} \int \frac {2999020 x+1832819}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {1}{21} \int \frac {35 (570952 x+700025)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \int \frac {570952 x+700025}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {28005599-19163420 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {958171 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {28005599-19163420 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {958171 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {1419475563}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {122343637 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {958171 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1419475563}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {122343637 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {958171 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1419475563}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {122343637 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {958171 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{252} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {122343637 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {1419475563 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {958171 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}\) |
(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(126*(2 + 3*x)^6) + ((503*Sqrt[1 - 2*x]*Sq rt[3 + 5*x])/(105*(2 + 3*x)^5) + ((-149951*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 8*(2 + 3*x)^4) + (3*((-71369*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (5*((958171*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((122343637* Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (1419475563*ArcTan[Sqrt[1 - 2 *x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6))/56)/210)/252
3.25.88.3.1 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])
Time = 1.20 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (16516390995 x^{5}+55658284380 x^{4}+74931979536 x^{3}+50261760608 x^{2}+16771747280 x +2225100096\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{129077760 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {52573169 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{120472576 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(139\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (574887603015 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+2299550412060 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+3832584020100 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+231229473930 x^{5} \sqrt {-10 x^{2}-x +3}+3406741351200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+779215981320 x^{4} \sqrt {-10 x^{2}-x +3}+1703370675600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1049047713504 x^{3} \sqrt {-10 x^{2}-x +3}+454232180160 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +703664648512 x^{2} \sqrt {-10 x^{2}-x +3}+50470242240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+234804461920 x \sqrt {-10 x^{2}-x +3}+31151401344 \sqrt {-10 x^{2}-x +3}\right )}{1807088640 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\) | \(346\) |
-1/129077760*(-1+2*x)*(3+5*x)^(1/2)*(16516390995*x^5+55658284380*x^4+74931 979536*x^3+50261760608*x^2+16771747280*x+2225100096)/(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)+52573169/120472576*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 = 0.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx=-\frac {788597535 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (16516390995 \, x^{5} + 55658284380 \, x^{4} + 74931979536 \, x^{3} + 50261760608 \, x^{2} + 16771747280 \, x + 2225100096\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1807088640 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
-1/1807088640*(788597535*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)*sq rt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(16516390995*x^5 + 55658284380*x^4 + 7 4931979536*x^3 + 50261760608*x^2 + 16771747280*x + 2225100096)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.10 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx=\frac {52573169}{120472576} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{378 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {853 \, \sqrt {-10 \, x^{2} - x + 3}}{26460 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac {149951 \, \sqrt {-10 \, x^{2} - x + 3}}{1481760 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {71369 \, \sqrt {-10 \, x^{2} - x + 3}}{2963520 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {958171 \, \sqrt {-10 \, x^{2} - x + 3}}{16595712 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {122343637 \, \sqrt {-10 \, x^{2} - x + 3}}{232339968 \, {\left (3 \, x + 2\right )}} \]
52573169/120472576*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2 )) - 1/378*sqrt(-10*x^2 - x + 3)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 853/26460*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810 *x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) - 149951/1481760*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) - 71369/2963520*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 958171/16595712*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 122343637/232339968*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 = 0.74 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx=\frac {52573169}{1204725760} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (118497 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 188015240 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 122630175360 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 17238395059200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 3670540357120000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {197895383347200000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {791581533388800000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{12907776 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]
52573169/1204725760*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)*s qrt(-10*x + 5) - sqrt(22)))) - 1331/12907776*sqrt(10)*(118497*((sqrt(2)*sq rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1 0*x + 5) - sqrt(22)))^11 + 188015240*((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 + 122630175360*((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 - 17238395059200*((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)))^5 - 3670540357120000*((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 - 197895383347200000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5 *x + 3) + 791581533388800000*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)^6
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^7} \,d x \]