Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {178 (1-2 x)^{5/2} \sqrt {2+3 x}}{825 \sqrt {3+5 x}}-\frac {1136 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}-\frac {2836 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{20625}+\frac {9206 \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{3125}-\frac {2958 \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{3125} \] Output:
-2/15*(1-2*x)^(5/2)*(2+3*x)^(3/2)/(3+5*x)^(3/2)-178/825*(1-2*x)^(5/2)*(2+3 *x)^(1/2)/(3+5*x)^(1/2)-1136/3125*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2 )-2836/20625*(1-2*x)^(3/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+9206/15625*Elliptic E(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-2958/15625*Ellipti cF(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 = 7.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.54 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {5 \sqrt {1-2 x} \sqrt {2+3 x} \left (-25421-48650 x-9450 x^2+4500 x^3\right )}{(3+5 x)^{3/2}}-13809 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+9940 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{46875} \] Input:
Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^(3/2))/(3 + 5*x)^(5/2),x]
Output:
(2*((5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-25421 - 48650*x - 9450*x^2 + 4500*x^3 ))/(3 + 5*x)^(3/2) - (13809*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x] ], -2/33] + (9940*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]) )/46875
Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {108, 27, 167, 27, 171, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^{3/2}}{(5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{15} \int -\frac {(1-2 x)^{3/2} \sqrt {3 x+2} (48 x+11)}{2 (5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{5/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{15} \int \frac {(1-2 x)^{3/2} \sqrt {3 x+2} (48 x+11)}{(5 x+3)^{3/2}}dx-\frac {2 (3 x+2)^{3/2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{15} \left (-\frac {2}{5} \int \frac {3 \sqrt {1-2 x} \sqrt {3 x+2} (429 x+97)}{\sqrt {5 x+3}}dx-\frac {178 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \int \frac {\sqrt {1-2 x} \sqrt {3 x+2} (429 x+97)}{\sqrt {5 x+3}}dx-\frac {178 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {2}{75} \int -\frac {3 (1150-13311 x) \sqrt {3 x+2}}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {286}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {178 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {286}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}-\frac {1}{25} \int \frac {(1150-13311 x) \sqrt {3 x+2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {178 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{15} \int \frac {3 (27618 x+8059)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4437}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {286}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {178 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{10} \int \frac {27618 x+8059}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4437}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {286}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {178 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{10} \left (\frac {27618}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {42559}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {4437}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {286}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {178 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{10} \left (-\frac {42559}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9206}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {4437}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {286}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {178 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{15} \left (-\frac {6}{5} \left (\frac {1}{25} \left (\frac {1}{10} \left (\frac {7738}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {9206}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {4437}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {286}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {178 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
Input:
Int[((1 - 2*x)^(5/2)*(2 + 3*x)^(3/2))/(3 + 5*x)^(5/2),x]
Output:
(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + ((-178*(1 - 2* x)^(3/2)*(2 + 3*x)^(3/2))/(5*Sqrt[3 + 5*x]) - (6*((286*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/25 + ((-4437*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5 + ((-9206*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35 /33])/5 + (7738*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3 3])/5)/10)/25))/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 = 0.54 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.18
| method | result | size |
| default | \(-\frac {\left (638385 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+138090 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+383031 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+82854 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-270000 x^{5}+522000 x^{4}+3103500 x^{3}+1822760 x^{2}-718790 x -508420\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{46875 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(225\) |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {24 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{625}-\frac {396 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3125}-\frac {8059 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{65625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {9206 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{21875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{46875 \left (x +\frac {3}{5}\right )^{2}}-\frac {2596 \left (-30 x^{2}-5 x +10\right )}{9375 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\) | \(275\) |
Input:
int((1-2*x)^(5/2)*(2+3*x)^(3/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/46875*(638385*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)+138090*2^(1/2)*EllipticE(1/7*(28+4 2*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+3830 31*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))+82854*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))-270000*x^5+522000*x^4+3 103500*x^3+1822760*x^2-718790*x-508420)*(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 = 98, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (75 \, {\left (4500 \, x^{3} - 9450 \, x^{2} - 48650 \, x - 25421\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 7508 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 207135 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{703125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
2/703125*(75*(4500*x^3 - 9450*x^2 - 48650*x - 25421)*sqrt(5*x + 3)*sqrt(3* x + 2)*sqrt(-2*x + 1) + 7508*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassPInv erse(1159/675, 38998/91125, x + 23/90) - 207135*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 3 8998/91125, x + 23/90)))/(25*x^2 + 30*x + 9)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(5/2)*(2+3*x)**(3/2)/(3+5*x)**(5/2),x)
Output:
Timed out
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate((3*x + 2)^(3/2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2), x)
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="giac")
Output:
integrate((3*x + 2)^(3/2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2), x)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int(((1 - 2*x)^(5/2)*(3*x + 2)^(3/2))/(5*x + 3)^(5/2),x)
Output:
int(((1 - 2*x)^(5/2)*(3*x + 2)^(3/2))/(5*x + 3)^(5/2), x)
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\frac {46800 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}-98280 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}+212108 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -200722 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-220684200 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}-264821040 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x -79446312 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )+109739025 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}+131686830 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x +39506049 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )}{1218750 x^{2}+1462500 x +438750} \] Input:
int((1-2*x)^(5/2)*(2+3*x)^(3/2)/(3+5*x)^(5/2),x)
Output:
(46800*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 98280*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 212108*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 200722*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) - 220684200*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(75 0*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x)*x**2 - 264821040 *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 - 79446312*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) + 109739025*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqr t( - 2*x + 1))/(750*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x )*x**2 + 131686830*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 + 39506049*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))/(48750*(25*x**2 + 30*x + 9))