Integrand size = 28, antiderivative size = 187 \[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {7 \sqrt {2+3 x}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {26 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {575 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 (3+5 x)^{3/2}}-\frac {2960 \sqrt {1-2 x} \sqrt {2+3 x}}{43923 \sqrt {3+5 x}}+\frac {592 \sqrt {35} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{43923}-\frac {8774 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{43923 \sqrt {35}} \] Output:
7/33*(2+3*x)^(1/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2)+26/121*(2+3*x)^(1/2)/(1-2*x )^(1/2)/(3+5*x)^(3/2)-575/3993*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)-2 960/43923*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)+592/43923*EllipticE(1/ 11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-8774/1537305*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 = 7.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {\sqrt {2+3 x} \left (1775+13572 x-810 x^2-29600 x^3\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}}-i \sqrt {33} \left (296 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-181 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{43923} \] Input:
Integrate[(2 + 3*x)^(3/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]
Output:
(2*((Sqrt[2 + 3*x]*(1775 + 13572*x - 810*x^2 - 29600*x^3))/((1 - 2*x)^(3/2 )*(3 + 5*x)^(3/2)) - I*Sqrt[33]*(296*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 181*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/43923
Time = 0.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {109, 27, 169, 27, 169, 25, 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.
| \(\displaystyle \int \frac {(3 x+2)^{3/2}}{(1-2 x)^{5/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{33} \int -\frac {3 (76 x+53)}{2 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \int \frac {76 x+53}{(1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{22} \left (\frac {52 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {2}{77} \int -\frac {7 (1170 x+817)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \int \frac {1170 x+817}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {52 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (-\frac {2}{33} \int -\frac {1725 x+1331}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {1150 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {52 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{33} \int \frac {1725 x+1331}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {1150 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {52 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{33} \left (-\frac {2}{11} \int \frac {3 (2960 x+511)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2960 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {1150 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {52 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{33} \left (-\frac {3}{11} \int \frac {2960 x+511}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2960 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {1150 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {52 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{33} \left (-\frac {3}{11} \left (592 \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-1265 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {2960 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {1150 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {52 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{33} \left (-\frac {3}{11} \left (-1265 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-592 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {2960 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {1150 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {52 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{33} \left (-\frac {3}{11} \left (230 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-592 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {2960 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {1150 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {52 \sqrt {3 x+2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
Input:
Int[(2 + 3*x)^(3/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]
Output:
(7*Sqrt[2 + 3*x])/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + ((52*Sqrt[2 + 3*x ])/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((-1150*Sqrt[1 - 2*x]*Sqrt[2 + 3*x ])/(33*(3 + 5*x)^(3/2)) + (2*((-2960*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt [3 + 5*x]) - (3*(-592*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]] , 35/33] + 230*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33 ]))/11))/33)/11)/22
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 = 0.81 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.25
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {2}{1815}+\frac {37 x}{18150}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (-\frac {43}{87846}+\frac {296 x}{43923}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}-\frac {73 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{43923 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2960 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{307461 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
| default | \(-\frac {\left (-894600-1043700 x +4473000 x^{3}+3429300 x^{2}+3729600 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{4}-1642032 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{2}-1363698 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x +796950 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2}+124320 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2}+79695 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x +12432 \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x -239085 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right )-37296 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2588460 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{3}\right ) \sqrt {1-2 x}}{922383 \sqrt {2+3 x}\, \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \left (-1+2 x \right )^{2} \left (3+5 x \right )^{\frac {3}{2}}}\) | \(485\) |
Input:
int((2+3*x)^(3/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
(-(3+5*x)*(-1+2*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*((2/1815+37/18150*x)*(-30*x^3-23*x^2+7*x+6)^(1/2)/(-3/10+x^2+1/10*x)^2-2 *(-20-30*x)*(-43/87846+296/43923*x)/((-3/10+x^2+1/10*x)*(-20-30*x))^(1/2)- 73/43923*(28+42*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7 *x+6)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-2960/307461*(28+42 *x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-1 /15*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-3/5*EllipticF(1/7*(28+42*x )^(1/2),1/2*70^(1/2))))
Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {90 \, {\left (29600 \, x^{3} + 810 \, x^{2} - 13572 \, x - 1775\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 2209 \, \sqrt {-30} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 26640 \, \sqrt {-30} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{1976535 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((2+3*x)^(3/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
-1/1976535*(90*(29600*x^3 + 810*x^2 - 13572*x - 1775)*sqrt(5*x + 3)*sqrt(3 *x + 2)*sqrt(-2*x + 1) + 2209*sqrt(-30)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 26640*sqrt(-30 )*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*weierstrassZeta(1159/675, 38998/91 125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(100*x^4 + 20 *x^3 - 59*x^2 - 6*x + 9)
Timed out. \[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((2+3*x)**(3/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)
Output:
Timed out
\[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2+3*x)^(3/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate((3*x + 2)^(3/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)
\[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2+3*x)^(3/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")
Output:
integrate((3*x + 2)^(3/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)
Timed out. \[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int((3*x + 2)^(3/2)/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)),x)
Output:
int((3*x + 2)^(3/2)/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)), x)
\[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {4 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+11100 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{3000 x^{7}+2900 x^{6}-2010 x^{5}-2277 x^{4}+425 x^{3}+603 x^{2}-27 x -54}d x \right ) x^{4}+2220 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{3000 x^{7}+2900 x^{6}-2010 x^{5}-2277 x^{4}+425 x^{3}+603 x^{2}-27 x -54}d x \right ) x^{3}-6549 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{3000 x^{7}+2900 x^{6}-2010 x^{5}-2277 x^{4}+425 x^{3}+603 x^{2}-27 x -54}d x \right ) x^{2}-666 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{3000 x^{7}+2900 x^{6}-2010 x^{5}-2277 x^{4}+425 x^{3}+603 x^{2}-27 x -54}d x \right ) x +999 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{3000 x^{7}+2900 x^{6}-2010 x^{5}-2277 x^{4}+425 x^{3}+603 x^{2}-27 x -54}d x \right )-5400 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{3000 x^{7}+2900 x^{6}-2010 x^{5}-2277 x^{4}+425 x^{3}+603 x^{2}-27 x -54}d x \right ) x^{4}-1080 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{3000 x^{7}+2900 x^{6}-2010 x^{5}-2277 x^{4}+425 x^{3}+603 x^{2}-27 x -54}d x \right ) x^{3}+3186 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{3000 x^{7}+2900 x^{6}-2010 x^{5}-2277 x^{4}+425 x^{3}+603 x^{2}-27 x -54}d x \right ) x^{2}+324 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{3000 x^{7}+2900 x^{6}-2010 x^{5}-2277 x^{4}+425 x^{3}+603 x^{2}-27 x -54}d x \right ) x -486 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{3000 x^{7}+2900 x^{6}-2010 x^{5}-2277 x^{4}+425 x^{3}+603 x^{2}-27 x -54}d x \right )}{2100 x^{4}+420 x^{3}-1239 x^{2}-126 x +189} \] Input:
int((2+3*x)^(3/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)
Output:
(4*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 11100*int((sqrt(3*x + 2) *sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(3000*x**7 + 2900*x**6 - 2010*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 54),x)*x**4 + 2220*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(3000*x**7 + 2900*x**6 - 2010*x **5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 54),x)*x**3 - 6549*int((sqr t(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(3000*x**7 + 2900*x**6 - 2 010*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 54),x)*x**2 - 666*int( (sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(3000*x**7 + 2900*x**6 - 2010*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 54),x)*x + 999*int ((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(3000*x**7 + 2900*x** 6 - 2010*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 54),x) - 5400*int ((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(3000*x**7 + 2900*x**6 - 2 010*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 54),x)*x**4 - 1080*int ((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(3000*x**7 + 2900*x**6 - 2 010*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 54),x)*x**3 + 3186*int ((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(3000*x**7 + 2900*x**6 - 2 010*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 54),x)*x**2 + 324*int( (sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(3000*x**7 + 2900*x**6 - 20 10*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 54),x)*x - 486*int((sqr t(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(3000*x**7 + 2900*x**6 - 201...