Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1792 \sqrt {1-2 x}}{27 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {17804 \sqrt {1-2 x} \sqrt {2+3 x}}{27 \sqrt {3+5 x}}+\frac {17804}{27} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {512}{27} \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:
14/9*(1-2*x)^(3/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+1792/27*(1-2*x)^(1/2)/(2+3* x)^(1/2)/(3+5*x)^(1/2)-17804/27*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)+ 17804/135*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)- 512/135*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 = 7.71 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {1-2 x} \left (11265+34726 x+26706 x^2\right )}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {4}{45} i \sqrt {\frac {11}{3}} \left (4451 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-4585 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \] Input:
Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)),x]
Output:
(-2*Sqrt[1 - 2*x]*(11265 + 34726*x + 26706*x^2))/(9*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) - ((4*I)/45)*Sqrt[11/3]*(4451*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]] , -2/33] - 4585*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])
Time = 0.24 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {109, 167, 27, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^{5/2} (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{9} \int \frac {(130-29 x) \sqrt {1-2 x}}{(3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {2}{9} \left (\frac {896 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}-\frac {2}{3} \int -\frac {11 (649-402 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \left (\frac {11}{3} \int \frac {649-402 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {896 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{9} \left (\frac {11}{3} \left (-\frac {2}{11} \int \frac {3 (4451 x+2818)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {8902 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {896 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \left (\frac {11}{3} \left (-\frac {6}{11} \int \frac {4451 x+2818}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {8902 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {896 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{9} \left (\frac {11}{3} \left (-\frac {6}{11} \left (\frac {737}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {4451}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {8902 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {896 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{9} \left (\frac {11}{3} \left (-\frac {6}{11} \left (\frac {737}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4451}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {8902 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {896 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{9} \left (\frac {11}{3} \left (-\frac {6}{11} \left (-\frac {134}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {4451}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {8902 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {896 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
Input:
Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)),x]
Output:
(14*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (2*((896*Sqrt[1 - 2*x])/(3*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (11*((-8902*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) - (6*((-4451*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3 /7]*Sqrt[1 - 2*x]], 35/33])/5 - (134*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7] *Sqrt[1 - 2*x]], 35/33])/5))/11))/3))/9
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[(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 = 0.41 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.34
| method | result | size |
| default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (13266 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-26706 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+8844 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-17804 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-801180 x^{3}-641190 x^{2}+182940 x +168975\right )}{135 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\) | \(215\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81 \left (\frac {2}{3}+x \right )^{2}}-\frac {2254 \left (-30 x^{2}-3 x +9\right )}{27 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {11272 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{189 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {17804 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{189 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \left (-30 x^{2}-5 x +10\right )}{5 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
Input:
int((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/135*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13266*2^(1/2)*EllipticF(1/7*(28+42*x)^( 1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-26706*2^(1 /2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^( 1/2)*(1-2*x)^(1/2)+8844*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2) *EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-17804*2^(1/2)*(2+3*x)^(1/2)*( -3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-80 1180*x^3-641190*x^2+182940*x+168975)/(2+3*x)^(3/2)/(10*x^2+x-3)
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (675 \, {\left (26706 \, x^{2} + 34726 \, x + 11265\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 151247 \, \sqrt {-30} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 400590 \, \sqrt {-30} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{6075 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Output:
-2/6075*(675*(26706*x^2 + 34726*x + 11265)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqr t(-2*x + 1) - 151247*sqrt(-30)*(45*x^3 + 87*x^2 + 56*x + 12)*weierstrassPI nverse(1159/675, 38998/91125, x + 23/90) + 400590*sqrt(-30)*(45*x^3 + 87*x ^2 + 56*x + 12)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse (1159/675, 38998/91125, x + 23/90)))/(45*x^3 + 87*x^2 + 56*x + 12)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(5/2)/(2+3*x)**(5/2)/(3+5*x)**(3/2),x)
Output:
Timed out
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="maxima")
Output:
integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)), x)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="giac")
Output:
integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)), x)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:
int((1 - 2*x)^(5/2)/((3*x + 2)^(5/2)*(5*x + 3)^(3/2)),x)
Output:
int((1 - 2*x)^(5/2)/((3*x + 2)^(5/2)*(5*x + 3)^(3/2)), x)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {-96 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +230 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+519750 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x^{3}+1004850 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x^{2}+646800 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x +138600 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right )-287595 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x^{3}-556017 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x^{2}-357896 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x -76692 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right )}{8100 x^{3}+15660 x^{2}+10080 x +2160} \] Input:
int((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x)
Output:
( - 96*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 230*sqrt(3*x + 2)* sqrt(5*x + 3)*sqrt( - 2*x + 1) + 519750*int((sqrt(3*x + 2)*sqrt(5*x + 3)*s qrt( - 2*x + 1)*x**2)/(1350*x**6 + 3645*x**5 + 3366*x**4 + 769*x**3 - 638* x**2 - 420*x - 72),x)*x**3 + 1004850*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt ( - 2*x + 1)*x**2)/(1350*x**6 + 3645*x**5 + 3366*x**4 + 769*x**3 - 638*x** 2 - 420*x - 72),x)*x**2 + 646800*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(1350*x**6 + 3645*x**5 + 3366*x**4 + 769*x**3 - 638*x**2 - 420*x - 72),x)*x + 138600*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1 )*x**2)/(1350*x**6 + 3645*x**5 + 3366*x**4 + 769*x**3 - 638*x**2 - 420*x - 72),x) - 287595*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(1350* x**6 + 3645*x**5 + 3366*x**4 + 769*x**3 - 638*x**2 - 420*x - 72),x)*x**3 - 556017*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(1350*x**6 + 36 45*x**5 + 3366*x**4 + 769*x**3 - 638*x**2 - 420*x - 72),x)*x**2 - 357896*i nt((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(1350*x**6 + 3645*x**5 + 3366*x**4 + 769*x**3 - 638*x**2 - 420*x - 72),x)*x - 76692*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(1350*x**6 + 3645*x**5 + 3366*x**4 + 769*x**3 - 638*x**2 - 420*x - 72),x))/(180*(45*x**3 + 87*x**2 + 56*x + 12) )