Integrand size = 28, antiderivative size = 191 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {1088}{17787 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {5314 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2377960 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}+\frac {475592 \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{195657}-\frac {17156 \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{195657} \] Output:
4/231/(1-2*x)^(3/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)+1088/17787/(1-2*x)^(1/2)/( 2+3*x)^(1/2)/(3+5*x)^(1/2)+5314/41503*(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^ (1/2)-2377960/1369599*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)+475592/136 9599*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-17156 /1369599*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 = 7.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (\frac {-2236533+5510400 x+5106644 x^2-14267760 x^3}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}-2 i \sqrt {33} \left (118898 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-121555 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{1369599} \] Input:
Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)),x]
Output:
(2*((-2236533 + 5510400*x + 5106644*x^2 - 14267760*x^3)/((1 - 2*x)^(3/2)*S qrt[2 + 3*x]*Sqrt[3 + 5*x]) - (2*I)*Sqrt[33]*(118898*EllipticE[I*ArcSinh[S qrt[9 + 15*x]], -2/33] - 121555*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33 ])))/1369599
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {115, 27, 169, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}-\frac {2}{231} \int -\frac {150 x+197}{2 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \int \frac {150 x+197}{(1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {1088}{77 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}-\frac {2}{77} \int -\frac {24480 x+18977}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \int \frac {24480 x+18977}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {1088}{77 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {2}{7} \int \frac {5 (18997-7971 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {15942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1088}{77 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {10}{7} \int \frac {18997-7971 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {15942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1088}{77 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {10}{7} \left (-\frac {2}{11} \int \frac {3 (237796 x+148523)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {237796 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {15942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1088}{77 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {10}{7} \left (-\frac {3}{11} \int \frac {237796 x+148523}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {237796 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {15942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1088}{77 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {10}{7} \left (-\frac {3}{11} \left (\frac {29227}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {237796}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {237796 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {15942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1088}{77 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {10}{7} \left (-\frac {3}{11} \left (\frac {29227}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {237796}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {237796 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {15942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1088}{77 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{231} \left (\frac {1}{77} \left (\frac {10}{7} \left (-\frac {3}{11} \left (-\frac {5314}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {237796}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {237796 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {15942 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1088}{77 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\) |
Input:
Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)),x]
Output:
4/(231*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (1088/(77*Sqrt[1 - 2 *x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + ((15942*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x] *Sqrt[3 + 5*x]) + (10*((-237796*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) - (3*((-237796*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (5314*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3 5/33])/5))/11))/7)/77)/231
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*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
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 = 0.87 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.14
| method | result | size |
| default | \(\frac {2 \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (175362 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-475592 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-87681 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+237796 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-14267760 x^{3}+5106644 x^{2}+5510400 x -2236533\right )}{1369599 \left (15 x^{2}+19 x +6\right ) \left (1-2 x \right )^{\frac {3}{2}}}\) | \(217\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (15-30 x \right ) \left (\frac {752183}{6847995}+\frac {78812 x}{456533}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}+\frac {4 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{17787 \left (x -\frac {1}{2}\right )^{2}}-\frac {2720 \left (-30 x^{2}-38 x -12\right )}{1369599 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {1485230 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{9587193 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2377960 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{9587193 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(253\) |
Input:
int(1/(1-2*x)^(5/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/1369599*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(175362*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)-47559 2*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)-87681*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))+237796*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))-14267760*x^3+5106644*x^2+5510400*x-2236533)/(15*x^2+19*x+6)/(1-2*x)^ (3/2)
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (45 \, {\left (14267760 \, x^{3} - 5106644 \, x^{2} - 5510400 \, x + 2236533\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 3948881 \, \sqrt {-30} {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 10700820 \, \sqrt {-30} {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{61631955 \, {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} \] Input:
integrate(1/(1-2*x)^(5/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="fricas ")
Output:
-2/61631955*(45*(14267760*x^3 - 5106644*x^2 - 5510400*x + 2236533)*sqrt(5* x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 3948881*sqrt(-30)*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 10700820*sqrt(-30)*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*weierstrassZeta(1 159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90 )))/(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(1-2*x)**(5/2)/(2+3*x)**(3/2)/(3+5*x)**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(1-2*x)^(5/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="maxima ")
Output:
integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(5/2)), x)
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(1-2*x)^(5/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="giac")
Output:
integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(5/2)), x)
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:
int(1/((1 - 2*x)^(5/2)*(3*x + 2)^(3/2)*(5*x + 3)^(3/2)),x)
Output:
int(1/((1 - 2*x)^(5/2)*(3*x + 2)^(3/2)*(5*x + 3)^(3/2)), x)
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{\left (-2 x +1\right )^{\frac {5}{2}} \left (3 x +2\right )^{\frac {3}{2}} \left (5 x +3\right )^{\frac {3}{2}}}d x \] Input:
int(1/(1-2*x)^(5/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2),x)
Output:
int(1/(1-2*x)^(5/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2),x)