Integrand size = 28, antiderivative size = 98 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\frac {2}{9} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {37}{27} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )+\frac {4}{27} \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:
2/9*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-37/135*EllipticE(1/11*55^(1/ 2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+4/135*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 = 0.96 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\frac {1}{135} \left (30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+37 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-35 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \] Input:
Integrate[(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/Sqrt[2 + 3*x],x]
Output:
(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + (37*I)*Sqrt[33]*EllipticE[ I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (35*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sq rt[9 + 15*x]], -2/33])/135
Time = 0.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {112, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{\sqrt {3 x+2}} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {2}{9} \int -\frac {37 x+20}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {37 x+20}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{9} \left (\frac {37}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {11}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {2}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{9} \left (-\frac {11}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {37}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {2}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {37}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {2}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\) |
Input:
Int[(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/Sqrt[2 + 3*x],x]
Output:
(2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-37*Sqrt[11/3]*Ellipti cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (2*Sqrt[11/3]*EllipticF[Ar cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9
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*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 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 = 0.37 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.41
| method | result | size |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (33 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+37 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+900 x^{3}+690 x^{2}-210 x -180\right )}{4050 x^{3}+3105 x^{2}-945 x -810}\) | \(138\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9}+\frac {20 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{189 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {37 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{189 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(192\) |
| risch | \(-\frac {2 \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{9 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {4 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{99 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {37 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{495 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(242\) |
Input:
int((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/135*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(33*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))+3 7*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))+900*x^3+690*x^2-210*x-180)/(30*x^3+23*x^2-7*x-6)
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\frac {2}{9} \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {949}{12150} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {37}{135} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="fricas")
Output:
2/9*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 949/12150*sqrt(-30)*weier strassPInverse(1159/675, 38998/91125, x + 23/90) + 37/135*sqrt(-30)*weiers trassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125 , x + 23/90))
\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\sqrt {3 x + 2}}\, dx \] Input:
integrate((1-2*x)**(1/2)*(3+5*x)**(1/2)/(2+3*x)**(1/2),x)
Output:
Integral(sqrt(1 - 2*x)*sqrt(5*x + 3)/sqrt(3*x + 2), x)
\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{\sqrt {3 \, x + 2}} \,d x } \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/sqrt(3*x + 2), x)
\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{\sqrt {3 \, x + 2}} \,d x } \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/sqrt(3*x + 2), x)
Timed out. \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int \frac {\sqrt {1-2\,x}\,\sqrt {5\,x+3}}{\sqrt {3\,x+2}} \,d x \] Input:
int(((1 - 2*x)^(1/2)*(5*x + 3)^(1/2))/(3*x + 2)^(1/2),x)
Output:
int(((1 - 2*x)^(1/2)*(5*x + 3)^(1/2))/(3*x + 2)^(1/2), x)
\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{23}+\frac {185 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{23}-\frac {131 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{46} \] Input:
int((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2),x)
Output:
(2*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 370*int((sqrt(3*x + 2)*s qrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(30*x**3 + 23*x**2 - 7*x - 6),x) - 131 *int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(30*x**3 + 23*x**2 - 7 *x - 6),x))/46