Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=-\frac {1310 (1-2 x)^{5/2} \sqrt {3+5 x}}{3969 (2+3 x)^{3/2}}+\frac {1250 (1-2 x)^{3/2} \sqrt {3+5 x}}{189 \sqrt {2+3 x}}+\frac {181180 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35721}-\frac {74 (1-2 x)^{5/2} (3+5 x)^{3/2}}{441 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}-\frac {904798 \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{15309}+\frac {306904 \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{15309} \] Output:
-1310/3969*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+1250/189*(1-2*x)^(3/2 )*(3+5*x)^(1/2)/(2+3*x)^(1/2)+181180/35721*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+ 5*x)^(1/2)-74/441*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(5/2)-2/21*(1-2*x)^( 5/2)*(3+5*x)^(5/2)/(2+3*x)^(7/2)-904798/107163*EllipticE(1/11*55^(1/2)*(1- 2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+306904/107163*EllipticF(1/11*55^(1/2) *(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
Result contains complex when optimal does not.
Time = 8.51 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (2337569+11107911 x+17788023 x^2+9846603 x^3+396900 x^4\right )}{(2+3 x)^{7/2}}+i \sqrt {33} \left (452399 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-317065 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{107163} \] Input:
Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(9/2),x]
Output:
(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2337569 + 11107911*x + 17788023*x^2 + 9846603*x^3 + 396900*x^4))/(2 + 3*x)^(7/2) + I*Sqrt[33]*(452399*EllipticE[ I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 317065*EllipticF[I*ArcSinh[Sqrt[9 + 15 *x]], -2/33])))/107163
Time = 0.31 (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, 171, 25, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{21} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 (3 x+2)^{7/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{21} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{(3 x+2)^{7/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{21} \left (-\frac {2}{15} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (655 x+283)}{(3 x+2)^{5/2}}dx-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{21} \left (-\frac {2}{15} \left (-\frac {2}{9} \int -\frac {(3809-17760 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)^{3/2}}dx-\frac {922 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{21} \left (-\frac {2}{15} \left (\frac {1}{9} \int \frac {(3809-17760 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^{3/2}}dx-\frac {922 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{21} \left (-\frac {2}{15} \left (\frac {1}{9} \left (\frac {2}{21} \int \frac {15 (22083-135334 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {31298 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 \sqrt {3 x+2}}\right )-\frac {922 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{21} \left (-\frac {2}{15} \left (\frac {1}{9} \left (\frac {5}{7} \int \frac {(22083-135334 x) \sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {31298 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 \sqrt {3 x+2}}\right )-\frac {922 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{21} \left (-\frac {2}{15} \left (\frac {1}{9} \left (\frac {5}{7} \left (\frac {135334}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{9} \int -\frac {452399 x+122572}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {31298 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 \sqrt {3 x+2}}\right )-\frac {922 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5}{21} \left (-\frac {2}{15} \left (\frac {1}{9} \left (\frac {5}{7} \left (\frac {1}{9} \int \frac {452399 x+122572}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {135334}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {31298 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 \sqrt {3 x+2}}\right )-\frac {922 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {5}{21} \left (-\frac {2}{15} \left (\frac {1}{9} \left (\frac {5}{7} \left (\frac {1}{9} \left (\frac {452399}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {744337}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {135334}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {31298 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 \sqrt {3 x+2}}\right )-\frac {922 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle -\frac {5}{21} \left (-\frac {2}{15} \left (\frac {1}{9} \left (\frac {5}{7} \left (\frac {1}{9} \left (-\frac {744337}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {452399}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {135334}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {31298 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 \sqrt {3 x+2}}\right )-\frac {922 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle -\frac {5}{21} \left (-\frac {2}{15} \left (\frac {1}{9} \left (\frac {5}{7} \left (\frac {1}{9} \left (\frac {135334}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {452399}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {135334}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {31298 \sqrt {1-2 x} (5 x+3)^{3/2}}{7 \sqrt {3 x+2}}\right )-\frac {922 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}\) |
Input:
Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(9/2),x]
Output:
(-2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(21*(2 + 3*x)^(7/2)) - (5*((-74*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(15*(2 + 3*x)^(5/2)) - (2*((-922*Sqrt[1 - 2*x] *(3 + 5*x)^(5/2))/(9*(2 + 3*x)^(3/2)) + ((-31298*Sqrt[1 - 2*x]*(3 + 5*x)^( 3/2))/(7*Sqrt[2 + 3*x]) + (5*((135334*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-452399*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (135334*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9))/7)/9))/15))/21
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[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 = 0.55 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.36
| method | result | size |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19683 \left (\frac {2}{3}+x \right )^{4}}+\frac {74 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2187 \left (\frac {2}{3}+x \right )^{3}}-\frac {26882 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45927 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {6509780}{35721} x^{2}-\frac {650978}{35721} x +\frac {650978}{11907}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {1225720 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{750141 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4523990 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{750141 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {200 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(303\) |
| default | \(\frac {2 \left (60291297 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+12214773 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+120582594 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+24429546 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+80388396 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+16286364 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+11907000 x^{6}+17864088 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+3619192 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+296588790 x^{5}+559608399 x^{4}+297981972 x^{3}-56641404 x^{2}-92958492 x -21038121\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{107163 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\) | \(406\) |
Input:
int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(9/2),x,method=_RETURNVERBOSE)
Output:
-(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-(3+5*x)*(-1+2*x)*(2+3*x))^(1/2)/(10*x^2+x-3 )/(2+3*x)^(1/2)*(-14/19683*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+74/2187* (-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3-26882/45927*(-30*x^3-23*x^2+7*x+6)^ (1/2)/(2/3+x)^2+650978/107163*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1 /2)+1225720/750141*(28+42*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^ 3-23*x^2+7*x+6)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+4523990/ 750141*(28+42*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x +6)^(1/2)*(-1/15*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-3/5*EllipticF (1/7*(28+42*x)^(1/2),1/2*70^(1/2)))+200/729*(-30*x^3-23*x^2+7*x+6)^(1/2))
Time = 0.07 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\frac {270 \, {\left (396900 \, x^{4} + 9846603 \, x^{3} + 17788023 \, x^{2} + 11107911 \, x + 2337569\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 626303 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 40715910 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{4822335 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(9/2),x, algorithm="fricas")
Output:
1/4822335*(270*(396900*x^4 + 9846603*x^3 + 17788023*x^2 + 11107911*x + 233 7569)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 626303*sqrt(-30)*(81*x^ 4 + 216*x^3 + 216*x^2 + 96*x + 16)*weierstrassPInverse(1159/675, 38998/911 25, x + 23/90) + 40715910*sqrt(-30)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 1 6)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38 998/91125, x + 23/90)))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(9/2),x)
Output:
Timed out
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(9/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2), x)
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(9/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2), x)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{9/2}} \,d x \] Input:
int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^(9/2),x)
Output:
int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^(9/2), x)
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(9/2),x)
Output:
(237600*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 1417680*sqrt(3 *x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 6869544*sqrt(3*x + 2)*sqrt(5 *x + 3)*sqrt( - 2*x + 1)*x**2 - 9147908*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 13738710*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 117 109075698*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(2430*x* *7 + 8343*x**6 + 10881*x**5 + 5850*x**4 - 120*x**3 - 1600*x**2 - 688*x - 9 6),x)*x**4 + 312290868528*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1 )*x**2)/(2430*x**7 + 8343*x**6 + 10881*x**5 + 5850*x**4 - 120*x**3 - 1600* x**2 - 688*x - 96),x)*x**3 + 312290868528*int((sqrt(3*x + 2)*sqrt(5*x + 3) *sqrt( - 2*x + 1)*x**2)/(2430*x**7 + 8343*x**6 + 10881*x**5 + 5850*x**4 - 120*x**3 - 1600*x**2 - 688*x - 96),x)*x**2 + 138795941568*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(2430*x**7 + 8343*x**6 + 10881*x** 5 + 5850*x**4 - 120*x**3 - 1600*x**2 - 688*x - 96),x)*x + 23132656928*int( (sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(2430*x**7 + 8343*x**6 + 10881*x**5 + 5850*x**4 - 120*x**3 - 1600*x**2 - 688*x - 96),x) - 406364 20767*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(2430*x**7 + 8343 *x**6 + 10881*x**5 + 5850*x**4 - 120*x**3 - 1600*x**2 - 688*x - 96),x)*x** 4 - 108363788712*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(2430* x**7 + 8343*x**6 + 10881*x**5 + 5850*x**4 - 120*x**3 - 1600*x**2 - 688*x - 96),x)*x**3 - 108363788712*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*...