Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=-\frac {1241596 \sqrt {1-2 x} \sqrt {3+5 x}}{750141 \sqrt {2+3 x}}-\frac {13316 \sqrt {1-2 x} (3+5 x)^{3/2}}{35721 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{27 (2+3 x)^{9/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{567 (2+3 x)^{7/2}}+\frac {2776 \sqrt {1-2 x} (3+5 x)^{5/2}}{1701 (2+3 x)^{5/2}}-\frac {100444 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{750141}-\frac {1241596 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{750141} \]
-2/27*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(9/2)+370/567*(1-2*x)^(3/2)*(3+5 *x)^(5/2)/(2+3*x)^(7/2)-100444/2250423*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2 ),1/33*1155^(1/2))*33^(1/2)-1241596/2250423*EllipticF(1/7*21^(1/2)*(1-2*x) ^(1/2),1/33*1155^(1/2))*33^(1/2)-13316/35721*(3+5*x)^(3/2)*(1-2*x)^(1/2)/( 2+3*x)^(3/2)+2776/1701*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(5/2)-1241596/7 50141*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.79 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.48 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (12903031+71920155 x+142557831 x^2+115002639 x^3+29072682 x^4\right )}{2 (2+3 x)^{9/2}}+i \sqrt {33} \left (25111 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-335510 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{2250423} \]
(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12903031 + 71920155*x + 142557831*x^2 + 115002639*x^3 + 29072682*x^4))/(2*(2 + 3*x)^(9/2)) + I*Sqrt[33]*(25111*E llipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 335510*EllipticF[I*ArcSinh[Sq rt[9 + 15*x]], -2/33])))/2250423
Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {108, 27, 167, 167, 27, 167, 27, 167, 27, 176, 123, 129}
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{27} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 (3 x+2)^{9/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{(3 x+2)^{9/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (325 x+448)}{(3 x+2)^{7/2}}dx-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \left (\frac {1388 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}-\frac {2}{15} \int -\frac {(5 x+3)^{3/2} (3690 x+5789)}{2 \sqrt {1-2 x} (3 x+2)^{5/2}}dx\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \left (\frac {1}{15} \int \frac {(5 x+3)^{3/2} (3690 x+5789)}{\sqrt {1-2 x} (3 x+2)^{5/2}}dx+\frac {1388 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \left (\frac {1}{15} \left (\frac {2}{63} \int \frac {3 \sqrt {5 x+3} (95860 x+167373)}{2 \sqrt {1-2 x} (3 x+2)^{3/2}}dx-\frac {6658 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )+\frac {1388 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \left (\frac {1}{15} \left (\frac {1}{21} \int \frac {\sqrt {5 x+3} (95860 x+167373)}{\sqrt {1-2 x} (3 x+2)^{3/2}}dx-\frac {6658 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )+\frac {1388 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \left (\frac {1}{15} \left (\frac {1}{21} \left (\frac {2}{21} \int \frac {5 (50222 x+713011)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {620798 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )-\frac {6658 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )+\frac {1388 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \left (\frac {1}{15} \left (\frac {1}{21} \left (\frac {5}{21} \int \frac {50222 x+713011}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {620798 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )-\frac {6658 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )+\frac {1388 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \left (\frac {1}{15} \left (\frac {1}{21} \left (\frac {5}{21} \left (\frac {3414389}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {50222}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {620798 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )-\frac {6658 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )+\frac {1388 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \left (\frac {1}{15} \left (\frac {1}{21} \left (\frac {5}{21} \left (\frac {3414389}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {50222}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {620798 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )-\frac {6658 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )+\frac {1388 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \left (\frac {1}{15} \left (\frac {1}{21} \left (\frac {5}{21} \left (-\frac {620798}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {50222}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {620798 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )-\frac {6658 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{3/2}}\right )+\frac {1388 \sqrt {1-2 x} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\) |
(-2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(27*(2 + 3*x)^(9/2)) - (5*((-74*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(21*(2 + 3*x)^(7/2)) - (2*((1388*Sqrt[1 - 2*x] *(3 + 5*x)^(5/2))/(15*(2 + 3*x)^(5/2)) + ((-6658*Sqrt[1 - 2*x]*(3 + 5*x)^( 3/2))/(21*(2 + 3*x)^(3/2)) + ((-620798*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*Sq rt[2 + 3*x]) + (5*((-50222*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (620798*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/21)/21)/15))/21))/27
3.28.82.3.1 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_) )^(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[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ [(b*e - a*f)/b, 0] && PosQ[-b/d] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d *e - c*f)/d, 0] && GtQ[-d/b, 0]) && !(SimplerQ[c + d*x, a + b*x] && GtQ[(( -b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ [((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f /b]))
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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Time = 1.32 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.36
| method | result | size |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{531441 \left (\frac {2}{3}+x \right )^{5}}+\frac {1406 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{177147 \left (\frac {2}{3}+x \right )^{4}}-\frac {44990 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{413343 \left (\frac {2}{3}+x \right )^{3}}+\frac {396566 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{964467 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {7178440}{750141} x^{2}-\frac {717844}{750141} x +\frac {717844}{250047}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {2852044 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{15752961 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {200888 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{15752961 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(302\) |
| default | \(-\frac {2 \left (51246756 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-4067982 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+136658016 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-10847952 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+136658016 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-10847952 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+60736896 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-4821312 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+10122816 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-803552 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-872180460 x^{6}-3537297216 x^{5}-4360088709 x^{4}-1550254392 x^{3}+680169084 x^{2}+608572302 x +116127279\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{2250423 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(504\) |
-1/(10*x^2+x-3)/(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-(-1+2*x)*(3+5* x)*(2+3*x))^(1/2)*(-98/531441*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^5+1406/ 177147*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4-44990/413343*(-30*x^3-23*x^2 +7*x+6)^(1/2)/(2/3+x)^3+396566/964467*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x) ^2+717844/2250423*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+2852044/ 15752961*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7 *x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+200888/15752961*(10+1 5*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(- 7/6*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2) ,1/35*70^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\frac {2 \, {\left (135 \, {\left (29072682 \, x^{4} + 115002639 \, x^{3} + 142557831 \, x^{2} + 71920155 \, x + 12903031\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 31507942 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2259990 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{101269035 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
2/101269035*(135*(29072682*x^4 + 115002639*x^3 + 142557831*x^2 + 71920155* x + 12903031)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 31507942*sqrt(- 30)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*weierstrassPInve rse(1159/675, 38998/91125, x + 23/90) + 2259990*sqrt(-30)*(243*x^5 + 810*x ^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*weierstrassZeta(1159/675, 38998/9112 5, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(243*x^5 + 810* x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{11/2}} \,d x \]