Integrand size = 28, antiderivative size = 134 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 \sqrt {-b c+a d} \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {e+f x}} \] Output:
2*(a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2 )*EllipticF(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b* e))^(1/2))/b/d^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)
Time = 7.34 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a-\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )}{\sqrt {a-\frac {b c}{d}} d \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {e+f x}} \] Input:
Integrate[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
Output:
(-2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[ArcSin[Sqrt[ a - (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/(Sqrt[a - ( b*c)/d]*d*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[e + f*x])
Time = 0.31 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {131, 131, 130}
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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\sqrt {\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{\sqrt {c+d x}}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{\sqrt {c+d x} \sqrt {e+f x}}\) |
| \(\Big \downarrow \) 130 |
| \(\displaystyle \frac {2 \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {e+f x}}\) |
Input:
Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
Output:
(2*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/( b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[d]*Sqrt[c + d*x]*Sqrt[e + f*x])
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/(b*c - a*d), 0] && GtQ [b/(b*e - a*f), 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f *x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x] Int[1/(Sq rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && Simpler Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
Time = 3.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.45
| method | result | size |
| default | \(-\frac {2 \left (a d -b c \right ) \operatorname {EllipticF}\left (\sqrt {-\frac {b \left (x d +c \right )}{a d -b c}}, \sqrt {-\frac {\left (a d -b c \right ) f}{b \left (c f -d e \right )}}\right ) \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \sqrt {-\frac {b \left (x d +c \right )}{a d -b c}}\, \sqrt {f x +e}\, \sqrt {x d +c}\, \sqrt {b x +a}}{d b \left (b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e \right )}\) | \(194\) |
| elliptic | \(\frac {2 \sqrt {\left (f x +e \right ) \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {c}{d}-\frac {a}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )}{\sqrt {f x +e}\, \sqrt {b x +a}\, \sqrt {x d +c}\, \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\) | \(230\) |
Input:
int(1/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2*(a*d-b*c)*EllipticF((-b*(d*x+c)/(a*d-b*c))^(1/2),(-(a*d-b*c)*f/b/(c*f-d *e))^(1/2))*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-d*(f*x+e)/(c*f-d*e))^(1/2)*(-b*( d*x+c)/(a*d-b*c))^(1/2)/d/b*(f*x+e)^(1/2)*(d*x+c)^(1/2)*(b*x+a)^(1/2)/(b*d *f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)
Time = 0.08 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.74 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 \, \sqrt {b d f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} - {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2}\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{2} f - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e f^{2} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right )}{b d f} \] Input:
integrate(1/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="fricas ")
Output:
2*sqrt(b*d*f)*weierstrassPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e *f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^ 3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b *d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3) /(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f))/(b*d*f)
\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {1}{\sqrt {a + b x} \sqrt {c + d x} \sqrt {e + f x}}\, dx \] Input:
integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)
Output:
Integral(1/(sqrt(a + b*x)*sqrt(c + d*x)*sqrt(e + f*x)), x)
\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima ")
Output:
integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)), x)
\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \] Input:
int(1/((e + f*x)^(1/2)*(a + b*x)^(1/2)*(c + d*x)^(1/2)),x)
Output:
int(1/((e + f*x)^(1/2)*(a + b*x)^(1/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {f x +e}\, \sqrt {d x +c}\, \sqrt {b x +a}}{b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}d x \] Input:
int(1/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)
Output:
int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x))/(a*c*e + a*c*f*x + a*d*e*x + a*d*f*x**2 + b*c*e*x + b*c*f*x**2 + b*d*e*x**2 + b*d*f*x**3),x)