Integrand size = 28, antiderivative size = 218 \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {3218 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac {217 (2+3 x)^{7/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {110519 \sqrt {1-2 x} (2+3 x)^{3/2}}{1098075 \sqrt {3+5 x}}-\frac {5199979 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3660250}-\frac {90397364 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{831875 \sqrt {33}}-\frac {5442127 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1663750 \sqrt {33}} \]
7/33*(2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2)-90397364/27451875*EllipticE (1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-5442127/54903750*Ell ipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-217/121*(2+3*x )^(7/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)+3218/19965*(2+3*x)^(5/2)*(1-2*x)^(1/2) /(3+5*x)^(3/2)+110519/1098075*(2+3*x)^(3/2)*(1-2*x)^(1/2)/(3+5*x)^(1/2)-51 99979/3660250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.71 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {-\frac {5 \sqrt {2+3 x} \left (246962693+89252928 x-1696384053 x^2-1825153850 x^3+177888150 x^4\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}}+i \sqrt {33} \left (180794728 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-186236855 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{54903750} \]
((-5*Sqrt[2 + 3*x]*(246962693 + 89252928*x - 1696384053*x^2 - 1825153850*x ^3 + 177888150*x^4))/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + I*Sqrt[33]*(18079 4728*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 186236855*EllipticF[I*A rcSinh[Sqrt[9 + 15*x]], -2/33]))/54903750
Time = 0.29 (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, 25, 167, 27, 167, 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.
\(\displaystyle \int \frac {(3 x+2)^{11/2}}{(1-2 x)^{5/2} (5 x+3)^{5/2}} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{33} \int \frac {3 (3 x+2)^{7/2} (204 x+115)}{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{22} \int \frac {(3 x+2)^{7/2} (204 x+115)}{(1-2 x)^{3/2} (5 x+3)^{5/2}}dx\) |
\(\Big \downarrow \) 167 |
\(\displaystyle \frac {1}{22} \left (-\frac {1}{11} \int -\frac {(3 x+2)^{5/2} (13131 x+7235)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \int \frac {(3 x+2)^{5/2} (13131 x+7235)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 167 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{165} \int \frac {(3 x+2)^{3/2} (1357893 x+792632)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \int \frac {(3 x+2)^{3/2} (1357893 x+792632)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 167 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {2}{55} \int \frac {9 \sqrt {3 x+2} (5199979 x+3208775)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \int \frac {\sqrt {3 x+2} (5199979 x+3208775)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \left (-\frac {1}{15} \int -\frac {361589456 x+228926353}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5199979}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \left (\frac {1}{30} \int \frac {361589456 x+228926353}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5199979}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \left (\frac {1}{30} \left (\frac {59863397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {361589456}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {5199979}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \left (\frac {1}{30} \left (\frac {59863397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {361589456}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5199979}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 129 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \left (\frac {1}{30} \left (-\frac {10884254}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {361589456}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5199979}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
(7*(2 + 3*x)^(9/2))/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + ((-434*(2 + 3*x )^(7/2))/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((6436*Sqrt[1 - 2*x]*(2 + 3* x)^(5/2))/(165*(3 + 5*x)^(3/2)) + ((221038*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/ (55*Sqrt[3 + 5*x]) + (9*((-5199979*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5* x])/15 + ((-361589456*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]] , 35/33])/5 - (10884254*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x ]], 35/33])/5)/30))/55)/165)/11)/22
3.30.91.3.1 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 = 4.72 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {31513109}{181500000}+\frac {52521907 x}{181500000}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (-\frac {2810713793}{4392300000}-\frac {46518287 x}{54903750}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}-\frac {81 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{500}+\frac {228926353 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{384326250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {180794728 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{192163125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(247\) |
default | \(-\frac {\sqrt {1-2 x}\, \left (1755947490 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1807947280 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+175594749 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-180794728 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-526784247 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+542384184 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+2668322250 x^{5}-25598426250 x^{4}-43697299295 x^{3}-15625046610 x^{2}+4596969675 x +2469626930\right )}{54903750 \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {2+3 x}}\) | \(316\) |
(-(-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 )*((31513109/181500000+52521907/181500000*x)*(-30*x^3-23*x^2+7*x+6)^(1/2)/ (-3/10+x^2+1/10*x)^2-2*(-20-30*x)*(-2810713793/4392300000-46518287/5490375 0*x)/((-3/10+x^2+1/10*x)*(-20-30*x))^(1/2)-81/500*(-30*x^3-23*x^2+7*x+6)^( 1/2)+228926353/384326250*(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))+18079 4728/192163125*(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*Ellip ticF((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 = 133, normalized size of antiderivative = 0.61 \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {450 \, {\left (177888150 \, x^{4} - 1825153850 \, x^{3} - 1696384053 \, x^{2} + 89252928 \, x + 246962693\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 6143407141 \, \sqrt {-30} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 16271525520 \, \sqrt {-30} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{4941337500 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
-1/4941337500*(450*(177888150*x^4 - 1825153850*x^3 - 1696384053*x^2 + 8925 2928*x + 246962693)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 614340714 1*sqrt(-30)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*weierstrassPInverse(1159 /675, 38998/91125, x + 23/90) - 16271525520*sqrt(-30)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInver se(1159/675, 38998/91125, x + 23/90)))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
Timed out. \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {11}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {11}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{11/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]