Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {46 (1-2 x)^{3/2} \sqrt {2+3 x}}{75 \sqrt {3+5 x}}-\frac {76}{375} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {338 \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{1125}+\frac {916 \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{1125} \] Output:
-2/15*(1-2*x)^(5/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)-46/75*(1-2*x)^(3/2)*(2+3*x )^(1/2)/(3+5*x)^(1/2)-76/375*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+338 /5625*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+916/ 5625*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 = 7.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \left (-712-925 x+100 x^2\right )}{(3+5 x)^{3/2}}-169 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+665 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{5625} \] Input:
Integrate[((1 - 2*x)^(5/2)*Sqrt[2 + 3*x])/(3 + 5*x)^(5/2),x]
Output:
(2*((15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-712 - 925*x + 100*x^2))/(3 + 5*x)^(3 /2) - (169*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (665* I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/5625
Time = 0.24 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {108, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} \sqrt {3 x+2}}{(5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{15} \int -\frac {(1-2 x)^{3/2} (36 x+17)}{2 \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{5/2} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{15} \int \frac {(1-2 x)^{3/2} (36 x+17)}{\sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {2 \sqrt {3 x+2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{15} \left (-\frac {2}{5} \int \frac {3 \sqrt {1-2 x} (57 x+52)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {46 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \int \frac {\sqrt {1-2 x} (57 x+52)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {46 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {2}{45} \int \frac {3 (169 x+647)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {38}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {46 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {1}{15} \int \frac {169 x+647}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {38}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {46 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {1}{15} \left (\frac {2728}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {169}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {38}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {46 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {1}{15} \left (\frac {2728}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {169}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {38}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {46 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {1}{15} \left (-\frac {496}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {169}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {38}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {46 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}}\) |
Input:
Int[((1 - 2*x)^(5/2)*Sqrt[2 + 3*x])/(3 + 5*x)^(5/2),x]
Output:
(-2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x])/(15*(3 + 5*x)^(3/2)) + ((-46*(1 - 2*x)^ (3/2)*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) - (6*((38*Sqrt[1 - 2*x]*Sqrt[2 + 3* x]*Sqrt[3 + 5*x])/15 + ((-169*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (496*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/15))/5)/15
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 = 0.50 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(\frac {2 \left (40920 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-845 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+24552 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-507 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+9000 x^{4}-81750 x^{3}-80955 x^{2}+17070 x +21360\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{5625 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(220\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9375 \left (x +\frac {3}{5}\right )^{2}}-\frac {418 \left (-30 x^{2}-5 x +10\right )}{1875 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {1294 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{7875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {338 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{375}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(244\) |
Input:
int((1-2*x)^(5/2)*(2+3*x)^(1/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/5625*(40920*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)-845*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)+24552*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))-507*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*El lipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+9000*x^4-81750*x^3-80955*x^2+170 70*x+21360)*(2+3*x)^(1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\frac {1350 \, {\left (100 \, x^{2} - 925 \, x - 712\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 54343 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 15210 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{253125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
1/253125*(1350*(100*x^2 - 925*x - 712)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2 *x + 1) + 54343*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassPInverse(1159/675 , 38998/91125, x + 23/90) - 15210*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstras sZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(25*x^2 + 30*x + 9)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {3 x + 2}}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((1-2*x)**(5/2)*(2+3*x)**(1/2)/(3+5*x)**(5/2),x)
Output:
Integral((1 - 2*x)**(5/2)*sqrt(3*x + 2)/(5*x + 3)**(5/2), x)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(3*x + 2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2), x)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(3*x + 2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2), x)
Timed out. \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int(((1 - 2*x)^(5/2)*(3*x + 2)^(1/2))/(5*x + 3)^(5/2),x)
Output:
int(((1 - 2*x)^(5/2)*(3*x + 2)^(1/2))/(5*x + 3)^(5/2), x)
\[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\frac {3120 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}-20072 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +21598 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+25091550 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}+30109860 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x +9032958 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )-12212475 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}-14654970 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x -4396491 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )}{146250 x^{2}+175500 x +52650} \] Input:
int((1-2*x)^(5/2)*(2+3*x)^(1/2)/(3+5*x)^(5/2),x)
Output:
(3120*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 20072*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 21598*sqrt(3*x + 2)*sqrt(5*x + 3)*s qrt( - 2*x + 1) + 25091550*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(750*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x)*x**2 + 30109860*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(750*x **5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x)*x + 9032958*int((sq rt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(750*x**5 + 1475*x**4 + 7 85*x**3 - 153*x**2 - 243*x - 54),x) - 12212475*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(750*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243* x - 54),x)*x**2 - 14654970*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(750*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x)*x - 43964 91*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(750*x**5 + 1475*x** 4 + 785*x**3 - 153*x**2 - 243*x - 54),x))/(5850*(25*x**2 + 30*x + 9))