Integrand size = 28, antiderivative size = 156 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}+\frac {532}{9} \sqrt {35} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {536 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{9 \sqrt {35}} \] Output:
14/9*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+88/3*(1-2*x)^(1/2)/(2+3*x)^ (1/2)/(3+5*x)^(1/2)-2660/9*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)+532/9 *EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-536/315*E llipticF(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 = 6.74 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {2}{9} \left (-\frac {3 \sqrt {1-2 x} \left (1683+5188 x+3990 x^2\right )}{(2+3 x)^{3/2} \sqrt {3+5 x}}-2 i \sqrt {33} \left (133 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-137 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right ) \] Input:
Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)),x]
Output:
(2*((-3*Sqrt[1 - 2*x]*(1683 + 5188*x + 3990*x^2))/((2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) - (2*I)*Sqrt[33]*(133*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 137*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/9
Time = 0.24 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {109, 27, 169, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^{5/2} (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{9} \int \frac {11 (8-9 x)}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{9} \int \frac {8-9 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {22}{9} \left (\frac {2}{7} \int \frac {7 (97-60 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {12 \sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{9} \left (\int \frac {97-60 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {12 \sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {22}{9} \left (-\frac {2}{11} \int \frac {3 (665 x+421)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1330 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}+\frac {12 \sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{9} \left (-\frac {6}{11} \int \frac {665 x+421}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1330 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}+\frac {12 \sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {22}{9} \left (-\frac {6}{11} \left (22 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+133 \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {1330 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}+\frac {12 \sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {22}{9} \left (-\frac {6}{11} \left (22 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-133 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1330 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}+\frac {12 \sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {22}{9} \left (-\frac {6}{11} \left (-4 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-133 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1330 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}+\frac {12 \sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}\) |
Input:
Int[(1 - 2*x)^(3/2)/((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)),x]
Output:
(14*Sqrt[1 - 2*x])/(9*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (22*((12*Sqrt[1 - 2 *x])/(Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (1330*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1 1*Sqrt[3 + 5*x]) - (6*(-133*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] - 4*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3 5/33]))/11))/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 + 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 = 0.46 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (396 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-798 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+264 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-532 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-23940 x^{3}-19158 x^{2}+5466 x +5049\right )}{9 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\) | \(215\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27 \left (\frac {2}{3}+x \right )^{2}}-\frac {334 \left (-30 x^{2}-3 x +9\right )}{9 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {1684 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{63 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {380 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{9 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
Input:
int((1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/9*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(396*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)-798*2^(1/2)*El lipticE(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)+264*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellipt icF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-532*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))-23940*x^3-1 9158*x^2+5466*x+5049)/(2+3*x)^(3/2)/(10*x^2+x-3)
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (135 \, {\left (3990 \, x^{2} + 5188 \, x + 1683\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 4519 \, \sqrt {-30} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 11970 \, \sqrt {-30} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{405 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Output:
-2/405*(135*(3990*x^2 + 5188*x + 1683)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2 *x + 1) - 4519*sqrt(-30)*(45*x^3 + 87*x^2 + 56*x + 12)*weierstrassPInverse (1159/675, 38998/91125, x + 23/90) + 11970*sqrt(-30)*(45*x^3 + 87*x^2 + 56 *x + 12)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/6 75, 38998/91125, x + 23/90)))/(45*x^3 + 87*x^2 + 56*x + 12)
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((1-2*x)**(3/2)/(2+3*x)**(5/2)/(3+5*x)**(3/2),x)
Output:
Integral((1 - 2*x)**(3/2)/((3*x + 2)**(5/2)*(5*x + 3)**(3/2)), x)
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="maxima")
Output:
integrate((-2*x + 1)^(3/2)/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)), x)
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="giac")
Output:
integrate((-2*x + 1)^(3/2)/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)), x)
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:
int((1 - 2*x)^(3/2)/((3*x + 2)^(5/2)*(5*x + 3)^(3/2)),x)
Output:
int((1 - 2*x)^(3/2)/((3*x + 2)^(5/2)*(5*x + 3)^(3/2)), x)
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+2970 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x^{3}+5742 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x^{2}+3696 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x +792 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right )-2475 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x^{3}-4785 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x^{2}-3080 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right ) x -660 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1350 x^{6}+3645 x^{5}+3366 x^{4}+769 x^{3}-638 x^{2}-420 x -72}d x \right )}{270 x^{3}+522 x^{2}+336 x +72} \] Input:
int((1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x)
Output:
(2*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 2970*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**3 + 5742*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 + 3696*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 - 6 38*x**2 - 420*x - 72),x)*x + 792*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) - 2475*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 - 4785*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**2 - 3080* 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 - 660*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(1350*x**6 + 3645*x**5 + 3366*x**4 + 7 69*x**3 - 638*x**2 - 420*x - 72),x))/(6*(45*x**3 + 87*x**2 + 56*x + 12))