Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{63 (2+3 x)^{7/2}}+\frac {832 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {112436 \sqrt {1-2 x} \sqrt {3+5 x}}{11907 (2+3 x)^{3/2}}+\frac {7810384 \sqrt {1-2 x} \sqrt {3+5 x}}{83349 \sqrt {2+3 x}}-\frac {7810384 \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{35721}+\frac {224872 \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{35721} \] Output:
-2/27*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)+10/63*(1-2*x)^(3/2)*(3+5*x )^(1/2)/(2+3*x)^(7/2)+832/567*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+11 2436/11907*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+7810384/83349*(1-2*x) ^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)-7810384/250047*EllipticE(1/11*55^(1/2)* (1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+224872/250047*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 = 7.59 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.48 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (65886031+389804925 x+865270206 x^2+854146674 x^3+316320552 x^4\right )}{4 (2+3 x)^{9/2}}+i \sqrt {33} \left (976298 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1005655 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{250047} \] Input:
Integrate[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^(11/2),x]
Output:
(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(65886031 + 389804925*x + 865270206*x^2 + 854146674*x^3 + 316320552*x^4))/(4*(2 + 3*x)^(9/2)) + I*Sqrt[33]*(97629 8*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 1005655*EllipticF[I*ArcSin h[Sqrt[9 + 15*x]], -2/33])))/250047
Time = 0.30 (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, 27, 167, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} \sqrt {5 x+3}}{(3 x+2)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{27} \int -\frac {5 (1-2 x)^{3/2} (12 x+5)}{2 (3 x+2)^{9/2} \sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{5/2} \sqrt {5 x+3}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \int \frac {(1-2 x)^{3/2} (12 x+5)}{(3 x+2)^{9/2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{21} \int \frac {3 (58-17 x) \sqrt {1-2 x}}{(3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \int \frac {(58-17 x) \sqrt {1-2 x}}{(3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {416 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}-\frac {2}{15} \int -\frac {5323-6070 x}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \int \frac {5323-6070 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {416 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {2}{21} \int \frac {231736-140545 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {56218 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {416 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {5 (1952596 x+1236143)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1952596 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {56218 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {416 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {2}{21} \left (\frac {5}{7} \int \frac {1952596 x+1236143}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1952596 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {56218 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {416 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {322927}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1952596}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {1952596 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {56218 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {416 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {322927}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1952596}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {1952596 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {56218 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {416 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle -\frac {5}{27} \left (-\frac {2}{7} \left (\frac {1}{15} \left (\frac {2}{21} \left (\frac {5}{7} \left (-\frac {58714}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1952596}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {1952596 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {56218 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {416 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\right )-\frac {6 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{27 (3 x+2)^{9/2}}\) |
Input:
Int[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^(11/2),x]
Output:
(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) - (5*((-6*(1 - 2*x )^(3/2)*Sqrt[3 + 5*x])/(7*(2 + 3*x)^(7/2)) - (2*((416*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + ((56218*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*( 2 + 3*x)^(3/2)) + (2*((1952596*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3* x]) + (5*((-1952596*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (58714*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3 5/33])/5))/7))/21)/15))/7))/27
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 = 0.43 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.34
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{59049 \left (\frac {2}{3}+x \right )^{5}}+\frac {146 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19683 \left (\frac {2}{3}+x \right )^{4}}+\frac {2260 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45927 \left (\frac {2}{3}+x \right )^{3}}+\frac {112436 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{107163 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {78103840}{83349} x^{2}-\frac {7810384}{83349} x +\frac {7810384}{27783}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {24722860 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1750329 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {39051920 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1750329 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(297\) |
| default | \(-\frac {2 \left (156942522 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-316320552 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+418513392 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-843521472 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+418513392 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-843521472 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+186005952 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-374898432 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-9489616560 x^{6}+31000992 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-62483072 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-26573361876 x^{5}-25673661234 x^{4}-6602638302 x^{3}+4641436149 x^{2}+3310586232 x +592974279\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{250047 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(494\) |
Input:
int((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(11/2),x,method=_RETURNVERBOSE)
Output:
(-(3+5*x)*(-1+2*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*(-98/59049*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^5+146/19683*(-30*x^3-23* x^2+7*x+6)^(1/2)/(2/3+x)^4+2260/45927*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x) ^3+112436/107163*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+7810384/250047*(-3 0*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+24722860/1750329*(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))+39051920/1750329*(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))))
Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\frac {2 \, {\left (135 \, {\left (316320552 \, x^{4} + 854146674 \, x^{3} + 865270206 \, x^{2} + 389804925 \, x + 65886031\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 66343162 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 175733640 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{11252115 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(11/2),x, algorithm="fricas" )
Output:
2/11252115*(135*(316320552*x^4 + 854146674*x^3 + 865270206*x^2 + 389804925 *x + 65886031)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 66343162*sqrt( -30)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*weierstrassPInv erse(1159/675, 38998/91125, x + 23/90) + 175733640*sqrt(-30)*(243*x^5 + 81 0*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*weierstrassZeta(1159/675, 38998/9 1125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(243*x^5 + 8 10*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(5/2)*(3+5*x)**(1/2)/(2+3*x)**(11/2),x)
Output:
Timed out
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(11/2),x, algorithm="maxima" )
Output:
integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2), x)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(11/2),x, algorithm="giac")
Output:
integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2), x)
Timed out. \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{11/2}} \,d x \] Input:
int(((1 - 2*x)^(5/2)*(5*x + 3)^(1/2))/(3*x + 2)^(11/2),x)
Output:
int(((1 - 2*x)^(5/2)*(5*x + 3)^(1/2))/(3*x + 2)^(11/2), x)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(11/2),x)
Output:
( - 3200*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 576*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 3534*sqrt(3*x + 2)*sqrt(5*x + 3)*s qrt( - 2*x + 1) + 110644218*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(7290*x**8 + 29889*x**7 + 49329*x**6 + 39312*x**5 + 11340*x**4 - 5040*x**3 - 5264*x**2 - 1664*x - 192),x)*x**5 + 368814060*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(7290*x**8 + 29889*x**7 + 49329*x **6 + 39312*x**5 + 11340*x**4 - 5040*x**3 - 5264*x**2 - 1664*x - 192),x)*x **4 + 491752080*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(7 290*x**8 + 29889*x**7 + 49329*x**6 + 39312*x**5 + 11340*x**4 - 5040*x**3 - 5264*x**2 - 1664*x - 192),x)*x**3 + 327834720*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(7290*x**8 + 29889*x**7 + 49329*x**6 + 39312* x**5 + 11340*x**4 - 5040*x**3 - 5264*x**2 - 1664*x - 192),x)*x**2 + 109278 240*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(7290*x**8 + 2 9889*x**7 + 49329*x**6 + 39312*x**5 + 11340*x**4 - 5040*x**3 - 5264*x**2 - 1664*x - 192),x)*x + 14570432*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2* x + 1)*x**2)/(7290*x**8 + 29889*x**7 + 49329*x**6 + 39312*x**5 + 11340*x** 4 - 5040*x**3 - 5264*x**2 - 1664*x - 192),x) - 39102831*int((sqrt(3*x + 2) *sqrt(5*x + 3)*sqrt( - 2*x + 1))/(7290*x**8 + 29889*x**7 + 49329*x**6 + 39 312*x**5 + 11340*x**4 - 5040*x**3 - 5264*x**2 - 1664*x - 192),x)*x**5 - 13 0342770*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(7290*x**8 +...