\(\int \frac {\sqrt {e x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx\) [452]

Optimal result

Integrand size = 26, antiderivative size = 127 \[ \int \frac {\sqrt {e x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {e x} \sqrt {c+d x}}{d \sqrt {a+b x}}-\frac {2 \sqrt {a} \sqrt {e} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.72 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {e x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 i c \sqrt {e x} \sqrt {1+\frac {d x}{c}} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {b x}{a}}\right )|\frac {a d}{b c}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b x}{a}}\right ),\frac {a d}{b c}\right )\right )}{d \sqrt {\frac {b x}{a+b x}} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {124, 123}

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.

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 123

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])
 

rule 124

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d 
*x]*Sqrt[b*((e + f*x)/(b*e - a*f))]))   Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x 
/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && Gt 
Q[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]
 

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.46

method	result	size
default	\(-\frac {2 \left (a \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) d -\operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a d +\operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b c \right ) \sqrt {-\frac {x d}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {\frac {x d +c}{c}}\, c \sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}{b \,d^{2} x \left (b d \,x^{2}+a d x +b c x +a c \right )}\)	\(186\)
elliptic	\(\frac {2 \sqrt {e x}\, \sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{x \sqrt {x d +c}\, \sqrt {b x +a}\, d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\)	\(214\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (112) = 224\).

Time = 0.10 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {e x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b d e} b d {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) + \sqrt {b d e} {\left (b c + a d\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )}}{3 \, b^{2} d^{2}} \] Input:

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

Output:

-2/3*(3*sqrt(b*d*e)*b*d*weierstrassZeta(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), 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))) + sqrt(b*d*e)*(b*c + a*d)*weierstrassPIn 
verse(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^2*d^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((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 + a*d*x + b*c*x + b 
*d*x**2),x)