Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{7/2}}+\frac {13136 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {613276 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{3/2}}+\frac {42623864 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 \sqrt {2+3 x}}-\frac {42623864 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{972405}-\frac {1282376 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{972405} \]
-42623864/2917215*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33 ^(1/2)-1282376/2917215*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2 ))*33^(1/2)-2/27*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)+82/567*(1-2*x)^ (1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+13136/19845*(1-2*x)^(1/2)*(3+5*x)^(1/2)/ (2+3*x)^(5/2)+613276/138915*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+4262 3864/972405*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.48 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (359554583+2127363207 x+4722182964 x^2+4661331894 x^3+1726266492 x^4\right )}{4 (2+3 x)^{9/2}}+i \sqrt {33} \left (5327983 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5488280 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{2917215} \]
(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(359554583 + 2127363207*x + 4722182964* x^2 + 4661331894*x^3 + 1726266492*x^4))/(4*(2 + 3*x)^(9/2)) + I*Sqrt[33]*( 5327983*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 5488280*EllipticF[I* ArcSinh[Sqrt[9 + 15*x]], -2/33])))/2917215
Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {108, 27, 167, 169, 27, 169, 169, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{(3 x+2)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{27} \int -\frac {\sqrt {1-2 x} (40 x+13)}{2 (3 x+2)^{9/2} \sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{27} \int \frac {\sqrt {1-2 x} (40 x+13)}{(3 x+2)^{9/2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{27} \left (\frac {2}{21} \int \frac {598-745 x}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {82 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{27} \left (\frac {2}{21} \left (\frac {2}{35} \int \frac {3 (29213-32840 x)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {6568 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {82 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{27} \left (\frac {2}{21} \left (\frac {3}{35} \int \frac {29213-32840 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {6568 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {82 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{27} \left (\frac {2}{21} \left (\frac {3}{35} \left (\frac {2}{21} \int \frac {1264931-766595 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {306638 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {6568 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {82 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{27} \left (\frac {2}{21} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {5 (10655966 x+6746233)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {10655966 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {306638 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {6568 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {82 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{27} \left (\frac {2}{21} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \int \frac {10655966 x+6746233}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {10655966 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {306638 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {6568 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {82 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{27} \left (\frac {2}{21} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {1763267}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {10655966}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {10655966 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {306638 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {6568 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {82 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{27} \left (\frac {2}{21} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {1763267}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {10655966}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {10655966 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {306638 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {6568 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {82 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{27} \left (\frac {2}{21} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (-\frac {320594}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {10655966}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {10655966 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {306638 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {6568 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {82 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
(-2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) + ((82*Sqrt[1 - 2* x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(7/2)) + (2*((6568*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + (3*((306638*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21* (2 + 3*x)^(3/2)) + (2*((10655966*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (5*((-10655966*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] ], 35/33])/5 - (320594*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] ], 35/33])/5))/7))/21))/35))/21)/27
3.28.7.3.1 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[((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] && LtQ[m, -1] && 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 = 1.28 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.31
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {110 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45927 \left (\frac {2}{3}+x \right )^{4}}+\frac {13136 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{535815 \left (\frac {2}{3}+x \right )^{3}}+\frac {613276 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1250235 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {85247728}{194481} x^{2}-\frac {42623864}{972405} x +\frac {42623864}{324135}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {53969864 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{20420505 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {85247728 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{20420505 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19683 \left (\frac {2}{3}+x \right )^{5}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(291\) |
| default | \(-\frac {2 \left (1676591136 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1726266492 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+4470909696 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-4603377312 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+4470909696 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-4603377312 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1987070976 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2045945472 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+331178496 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-340990912 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-51787994760 x^{6}-145018756296 x^{5}-140113086174 x^{4}-36035458056 x^{3}+25330919565 x^{2}+18067605114 x +3235991247\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{2917215 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(504\) |
(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*(110/45927*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+13136/535815*(-30*x^3- 23*x^2+7*x+6)^(1/2)/(2/3+x)^3+613276/1250235*(-30*x^3-23*x^2+7*x+6)^(1/2)/ (2/3+x)^2+42623864/2917215*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2) +53969864/20420505*(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))+85247728/20 420505*(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)*(-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)))-14/19683*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x) ^5)
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 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\frac {2 \, {\left (135 \, {\left (1726266492 \, x^{4} + 4661331894 \, x^{3} + 4722182964 \, x^{2} + 2127363207 \, x + 359554583\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 362073752 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 959036940 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{131274675 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
2/131274675*(135*(1726266492*x^4 + 4661331894*x^3 + 4722182964*x^2 + 21273 63207*x + 359554583)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 36207375 2*sqrt(-30)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*weierstr assPInverse(1159/675, 38998/91125, x + 23/90) + 959036940*sqrt(-30)*(243*x ^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*weierstrassZeta(1159/675, 38998/91125, 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. \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{11/2}} \,d x \]