Integrand size = 28, antiderivative size = 86 \[ \int \frac {\sqrt {2+3 x} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {\sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\sqrt {35} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{\sqrt {35}} \] Output:
(2+3*x)^(1/2)*(3+5*x)^(1/2)/(1-2*x)^(1/2)+EllipticE(1/11*55^(1/2)*(1-2*x)^ (1/2),1/35*1155^(1/2))*35^(1/2)-1/35*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 = 2.52 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {2+3 x} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {\sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}}-i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+\frac {34 i \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{\sqrt {33}} \] Input:
Integrate[(Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(1 - 2*x)^(3/2),x]
Output:
(Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] - I*Sqrt[33]*EllipticE[I*ArcSi nh[Sqrt[9 + 15*x]], -2/33] + ((34*I)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/Sqrt[33]
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {108, 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.
| \(\displaystyle \int \frac {\sqrt {3 x+2} \sqrt {5 x+3}}{(1-2 x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {\sqrt {3 x+2} \sqrt {5 x+3}}{\sqrt {1-2 x}}-\int \frac {30 x+19}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {3 x+2} \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {1}{2} \int \frac {30 x+19}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-6 \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {\sqrt {3 x+2} \sqrt {5 x+3}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{2} \left (2 \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {\sqrt {3 x+2} \sqrt {5 x+3}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{\sqrt {33}}+2 \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {\sqrt {3 x+2} \sqrt {5 x+3}}{\sqrt {1-2 x}}\) |
Input:
Int[(Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(1 - 2*x)^(3/2),x]
Output:
(Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (2*Sqrt[33]*EllipticE[ArcSin [Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] + (2*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33])/2
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[((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 = 0.35 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.53
| method | result | size |
| default | \(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (\sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-30 x^{2}-38 x -12\right )}{60 x^{3}+46 x^{2}-14 x -12}\) | \(132\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {-30 x^{2}-38 x -12}{2 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {19 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{42 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {5 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(201\) |
Input:
int((2+3*x)^(1/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(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))-2*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))-30*x^2-38*x-12)/(30*x^3+23*x^2-7*x-6)
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {2+3 x} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {17 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 45 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) - 45 \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{45 \, {\left (2 \, x - 1\right )}} \] Input:
integrate((2+3*x)^(1/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x, algorithm="fricas")
Output:
1/45*(17*sqrt(-30)*(2*x - 1)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 45*sqrt(-30)*(2*x - 1)*weierstrassZeta(1159/675, 38998/91125, w eierstrassPInverse(1159/675, 38998/91125, x + 23/90)) - 45*sqrt(5*x + 3)*s qrt(3*x + 2)*sqrt(-2*x + 1))/(2*x - 1)
\[ \int \frac {\sqrt {2+3 x} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int \frac {\sqrt {3 x + 2} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((2+3*x)**(1/2)*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)
Output:
Integral(sqrt(3*x + 2)*sqrt(5*x + 3)/(1 - 2*x)**(3/2), x)
\[ \int \frac {\sqrt {2+3 x} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((2+3*x)^(1/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(5*x + 3)*sqrt(3*x + 2)/(-2*x + 1)^(3/2), x)
\[ \int \frac {\sqrt {2+3 x} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((2+3*x)^(1/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(5*x + 3)*sqrt(3*x + 2)/(-2*x + 1)^(3/2), x)
Timed out. \[ \int \frac {\sqrt {2+3 x} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int \frac {\sqrt {3\,x+2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}} \,d x \] Input:
int(((3*x + 2)^(1/2)*(5*x + 3)^(1/2))/(1 - 2*x)^(3/2),x)
Output:
int(((3*x + 2)^(1/2)*(5*x + 3)^(1/2))/(1 - 2*x)^(3/2), x)
\[ \int \frac {\sqrt {2+3 x} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {-38 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+2040 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{60 x^{4}+16 x^{3}-37 x^{2}-5 x +6}d x \right ) x -1020 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{60 x^{4}+16 x^{3}-37 x^{2}-5 x +6}d x \right )-818 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{60 x^{4}+16 x^{3}-37 x^{2}-5 x +6}d x \right ) x +409 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{60 x^{4}+16 x^{3}-37 x^{2}-5 x +6}d x \right )}{60 x -30} \] Input:
int((2+3*x)^(1/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x)
Output:
( - 38*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 2040*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(60*x**4 + 16*x**3 - 37*x**2 - 5* x + 6),x)*x - 1020*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2) /(60*x**4 + 16*x**3 - 37*x**2 - 5*x + 6),x) - 818*int((sqrt(3*x + 2)*sqrt( 5*x + 3)*sqrt( - 2*x + 1))/(60*x**4 + 16*x**3 - 37*x**2 - 5*x + 6),x)*x + 409*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(60*x**4 + 16*x**3 - 37*x**2 - 5*x + 6),x))/(30*(2*x - 1))