Integrand size = 28, antiderivative size = 218 \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=-\frac {107 \sqrt {1-2 x} (2+3 x)^{7/2}}{1815 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{9/2}}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {4553 \sqrt {1-2 x} (2+3 x)^{5/2}}{99825 \sqrt {3+5 x}}+\frac {17427983 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{8318750}+\frac {380188 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{831875}+\frac {604915631 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3781250 \sqrt {33}}+\frac {18177329 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3781250 \sqrt {33}} \]
604915631/124781250*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))* 33^(1/2)+18177329/124781250*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155 ^(1/2))*33^(1/2)+7/11*(2+3*x)^(9/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)-107/1815*( 2+3*x)^(7/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2)-4553/99825*(2+3*x)^(5/2)*(1-2*x)^ (1/2)/(3+5*x)^(1/2)+380188/831875*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2 )+17427983/8318750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {\frac {5 \sqrt {2+3 x} \left (904528061+2667846028 x+1267558775 x^2-1255998150 x^3-242574750 x^4\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}}-i \sqrt {33} \left (604915631 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-623092960 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{124781250} \]
((5*Sqrt[2 + 3*x]*(904528061 + 2667846028*x + 1267558775*x^2 - 1255998150* x^3 - 242574750*x^4))/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - I*Sqrt[33]*(604915 631*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 623092960*EllipticF[I*Ar cSinh[Sqrt[9 + 15*x]], -2/33]))/124781250
Time = 0.31 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {109, 27, 167, 27, 167, 27, 171, 27, 171, 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.
| \(\displaystyle \int \frac {(3 x+2)^{11/2}}{(1-2 x)^{3/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {1}{11} \int \frac {(3 x+2)^{7/2} (624 x+353)}{2 \sqrt {1-2 x} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {1}{22} \int \frac {(3 x+2)^{7/2} (624 x+353)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (-\frac {2}{165} \int \frac {(3 x+2)^{5/2} (64986 x+38081)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (-\frac {1}{165} \int \frac {(3 x+2)^{5/2} (64986 x+38081)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{165} \left (-\frac {2}{55} \int \frac {3 (3 x+2)^{3/2} (760376 x+453799)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {9106 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{165} \left (-\frac {3}{55} \int \frac {(3 x+2)^{3/2} (760376 x+453799)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {9106 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{165} \left (-\frac {3}{55} \left (-\frac {1}{25} \int -\frac {3 \sqrt {3 x+2} (17427983 x+10731550)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {760376}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {9106 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{165} \left (-\frac {3}{55} \left (\frac {3}{25} \int \frac {\sqrt {3 x+2} (17427983 x+10731550)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {760376}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {9106 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{165} \left (-\frac {3}{55} \left (\frac {3}{25} \left (-\frac {1}{15} \int -\frac {1209831262 x+765888881}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {17427983}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {760376}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {9106 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{165} \left (-\frac {3}{55} \left (\frac {3}{25} \left (\frac {1}{30} \int \frac {1209831262 x+765888881}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {17427983}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {760376}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {9106 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{165} \left (-\frac {3}{55} \left (\frac {3}{25} \left (\frac {1}{30} \left (\frac {199950619}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1209831262}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {17427983}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {760376}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {9106 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{165} \left (-\frac {3}{55} \left (\frac {3}{25} \left (\frac {1}{30} \left (\frac {199950619}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1209831262}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {17427983}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {760376}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {9106 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{165} \left (-\frac {3}{55} \left (\frac {3}{25} \left (\frac {1}{30} \left (-\frac {36354658}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1209831262}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {17427983}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {760376}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {9106 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {214 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\) |
(7*(2 + 3*x)^(9/2))/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((-214*Sqrt[1 - 2 *x]*(2 + 3*x)^(7/2))/(165*(3 + 5*x)^(3/2)) + ((-9106*Sqrt[1 - 2*x]*(2 + 3* x)^(5/2))/(55*Sqrt[3 + 5*x]) - (3*((-760376*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)* Sqrt[3 + 5*x])/25 + (3*((-17427983*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5* x])/15 + ((-1209831262*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] ], 35/33])/5 - (36354658*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2* x]], 35/33])/5)/30))/25))/55)/165)/22
3.30.35.3.1 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[(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[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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 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 = 1.44 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (2937438240 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-3024578155 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1762462944 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1814746893 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+3638621250 x^{5}+21265719750 x^{4}-6453400125 x^{3}-52693278170 x^{2}-40246381195 x -9045280610\right )}{124781250 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(229\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {243 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1250}+\frac {10881 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{12500}-\frac {765888881 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{873468750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {604915631 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{436734375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5671875 \left (x +\frac {3}{5}\right )^{2}}-\frac {1076 \left (-30 x^{2}-5 x +10\right )}{12478125 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {16807 \left (-30 x^{2}-38 x -12\right )}{10648 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(286\) |
1/124781250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(2937438240*5^(1/2)*7^(1/2)*Ellipt icF((10+15*x)^(1/2),1/35*70^(1/2))*x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^ (1/2)-3024578155*5^(1/2)*7^(1/2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))* x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)+1762462944*5^(1/2)*(2+3*x)^(1 /2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70 ^(1/2))-1814746893*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1 /2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+3638621250*x^5+21265719750*x^ 4-6453400125*x^3-52693278170*x^2-40246381195*x-9045280610)/(3+5*x)^(3/2)/( 6*x^2+x-2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.54 \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {225 \, {\left (242574750 \, x^{4} + 1255998150 \, x^{3} - 1267558775 \, x^{2} - 2667846028 \, x - 904528061\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 10275970066 \, \sqrt {-30} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 27221203395 \, \sqrt {-30} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{5615156250 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
1/5615156250*(225*(242574750*x^4 + 1255998150*x^3 - 1267558775*x^2 - 26678 46028*x - 904528061)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 10275970 066*sqrt(-30)*(50*x^3 + 35*x^2 - 12*x - 9)*weierstrassPInverse(1159/675, 3 8998/91125, x + 23/90) - 27221203395*sqrt(-30)*(50*x^3 + 35*x^2 - 12*x - 9 )*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 389 98/91125, x + 23/90)))/(50*x^3 + 35*x^2 - 12*x - 9)
Timed out. \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {11}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {11}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{11/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]