Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {62 (1-2 x)^{3/2} \sqrt {2+3 x}}{275 \sqrt {3+5 x}}-\frac {212 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1375}+\frac {496}{375} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {178}{375} \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:
-2/15*(1-2*x)^(3/2)*(2+3*x)^(3/2)/(3+5*x)^(3/2)-62/275*(1-2*x)^(3/2)*(2+3* x)^(1/2)/(3+5*x)^(1/2)-212/1375*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+ 496/1875*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-1 78/1875*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 = 6.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\frac {2 \left (-\frac {55 \sqrt {1-2 x} \sqrt {2+3 x} \left (437+800 x+150 x^2\right )}{(3+5 x)^{3/2}}-2728 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+1855 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{20625} \] Input:
Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^(3/2))/(3 + 5*x)^(5/2),x]
Output:
(2*((-55*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(437 + 800*x + 150*x^2))/(3 + 5*x)^(3 /2) - (2728*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (185 5*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/20625
Time = 0.29 (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, 25, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (3 x+2)^{3/2}}{(5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{15} \int -\frac {3 \sqrt {1-2 x} \sqrt {3 x+2} (12 x+1)}{2 (5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{5} \int \frac {\sqrt {1-2 x} \sqrt {3 x+2} (12 x+1)}{(5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{5} \left (-\frac {2}{5} \int -\frac {(25-267 x) \sqrt {3 x+2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \int \frac {(25-267 x) \sqrt {3 x+2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \left (\frac {89}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{15} \int \frac {3 (496 x+123)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \left (\frac {89}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{10} \int \frac {496 x+123}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {873}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {496}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {89}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {873}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {496}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {89}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {496}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {582}{5} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )\right )+\frac {89}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
Input:
Int[((1 - 2*x)^(3/2)*(2 + 3*x)^(3/2))/(3 + 5*x)^(5/2),x]
Output:
(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + ((-62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(5*Sqrt[3 + 5*x]) + (2*((89*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5 + ((496*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (582*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/10))/5)/5
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 = 0.36 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(-\frac {\left (13095 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2480 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+7857 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1488 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+9000 x^{4}+49500 x^{3}+31220 x^{2}-11630 x -8740\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}}{1875 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(220\) |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9375 \left (x +\frac {3}{5}\right )^{2}}-\frac {248 \left (-30 x^{2}-5 x +10\right )}{1875 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {41 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {496 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\) | \(255\) |
Input:
int((1-2*x)^(3/2)*(2+3*x)^(3/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/1875*(13095*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)+2480*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)+7857*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))+1488*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))+9000*x^4+49500*x^3+31220*x^2-1 1630*x-8740)*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=-\frac {450 \, {\left (150 \, x^{2} + 800 \, x + 437\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 169 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 22320 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{84375 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
-1/84375*(450*(150*x^2 + 800*x + 437)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2* x + 1) + 169*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassPInverse(1159/675, 3 8998/91125, x + 23/90) + 22320*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassZe ta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 2 3/90)))/(25*x^2 + 30*x + 9)
\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{\frac {3}{2}}}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((1-2*x)**(3/2)*(2+3*x)**(3/2)/(3+5*x)**(5/2),x)
Output:
Integral((1 - 2*x)**(3/2)*(3*x + 2)**(3/2)/(5*x + 3)**(5/2), x)
\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate((3*x + 2)^(3/2)*(-2*x + 1)^(3/2)/(5*x + 3)^(5/2), x)
\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="giac")
Output:
integrate((3*x + 2)^(3/2)*(-2*x + 1)^(3/2)/(5*x + 3)^(5/2), x)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int(((1 - 2*x)^(3/2)*(3*x + 2)^(3/2))/(5*x + 3)^(5/2),x)
Output:
int(((1 - 2*x)^(3/2)*(3*x + 2)^(3/2))/(5*x + 3)^(5/2), x)
\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\frac {-780 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}+2288 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -2092 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-2234325 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}-2681190 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x -804357 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )+1108650 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}+1330380 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x +399114 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )}{24375 x^{2}+29250 x +8775} \] Input:
int((1-2*x)^(3/2)*(2+3*x)^(3/2)/(3+5*x)^(5/2),x)
Output:
( - 780*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 2288*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 2092*sqrt(3*x + 2)*sqrt(5*x + 3)*s qrt( - 2*x + 1) - 2234325*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 - 2681190*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 - 804357*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) + 1108650*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 + 1330380*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 + 399114*int ((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(750*x**5 + 1475*x**4 + 78 5*x**3 - 153*x**2 - 243*x - 54),x))/(975*(25*x**2 + 30*x + 9))