\(\int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx\) [453]

Optimal result

Integrand size = 26, antiderivative size = 96 \[ \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {a} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c \sqrt {e} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

2*a^(1/2)*(d*x+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2) 
/e^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/e^(1/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/ 
(b*x+a))^(1/2)
 

Mathematica [A] (verified)

Time = 3.48 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {\frac {b+\frac {a}{x}}{b}} \sqrt {\frac {d+\frac {c}{x}}{d}} x^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )}{\sqrt {-\frac {a}{b}} \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

Integrate[1/(Sqrt[e*x]*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
 

Output:

(-2*Sqrt[(b + a/x)/b]*Sqrt[(d + c/x)/d]*x^(3/2)*EllipticF[ArcSin[Sqrt[-(a/ 
b)]/Sqrt[x]], (b*c)/(a*d)])/(Sqrt[-(a/b)]*Sqrt[e*x]*Sqrt[a + b*x]*Sqrt[c + 
 d*x])
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {127, 126}

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.

\(\displaystyle \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {\sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{\sqrt {a+b x} \sqrt {c+d x}}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}\)

Input:

Int[1/(Sqrt[e*x]*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
 

Output:

(2*Sqrt[-a]*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[ArcSin[(Sqrt[b]* 
Sqrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/(b*c)])/(Sqrt[b]*Sqrt[e]*Sqrt[a + b* 
x]*Sqrt[c + d*x])
 

Defintions of rubi rules used

rule 126

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* 
Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & 
& GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x 
]))   Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free 
Q[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18

method	result	size
default	\(\frac {2 \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) \sqrt {-\frac {x d}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {\frac {x d +c}{c}}\, c \sqrt {x d +c}\, \sqrt {b x +a}}{d \left (b d \,x^{2}+a d x +b c x +a c \right ) \sqrt {e x}}\)	\(113\)
elliptic	\(\frac {2 \sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}\, d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\)	\(154\)

Input:

int(1/(e*x)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2*EllipticF(((d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2))*(-1/c*x*d)^(1/2)*(d* 
(b*x+a)/(a*d-b*c))^(1/2)*((d*x+c)/c)^(1/2)*c*(d*x+c)^(1/2)*(b*x+a)^(1/2)/d 
/(b*d*x^2+a*d*x+b*c*x+a*c)/(e*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 \, \sqrt {b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )}{b d e} \] Input:

integrate(1/(e*x)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

2*sqrt(b*d*e)*weierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d 
^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^ 
3), 1/3*(3*b*d*x + b*c + a*d)/(b*d))/(b*d*e)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{\sqrt {e x} \sqrt {a + b x} \sqrt {c + d x}}\, dx \] Input:

integrate(1/(e*x)**(1/2)/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)
 

Output:

Integral(1/(sqrt(e*x)*sqrt(a + b*x)*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {e x}} \,d x } \] Input:

integrate(1/(e*x)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {e x}} \,d x } \] Input:

integrate(1/(e*x)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{\sqrt {e\,x}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \] Input:

int(1/((e*x)^(1/2)*(a + b*x)^(1/2)*(c + d*x)^(1/2)),x)
 

Output:

int(1/((e*x)^(1/2)*(a + b*x)^(1/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {d x +c}\, \sqrt {b x +a}}{b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}d x \right )}{e} \] Input:

int(1/(e*x)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x)
 

Output:

(sqrt(e)*int((sqrt(x)*sqrt(c + d*x)*sqrt(a + b*x))/(a*c*x + a*d*x**2 + b*c 
*x**2 + b*d*x**3),x))/e