\(\int \frac {\sqrt {a x}}{\sqrt {d+e x} \sqrt {e+f x}} \, dx\) [533]

Optimal result

Integrand size = 26, antiderivative size = 114 \[ \int \frac {\sqrt {a x}}{\sqrt {d+e x} \sqrt {e+f x}} \, dx=\frac {2 \sqrt {-e^2+d f} \sqrt {a x} \sqrt {\frac {e (e+f x)}{e^2-d f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {-e^2+d f}}\right )|1-\frac {e^2}{d f}\right )}{e \sqrt {f} \sqrt {-\frac {e x}{d}} \sqrt {e+f x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a x}}{\sqrt {d+e x} \sqrt {e+f x}} \, dx=-\frac {2 i e \sqrt {a x} \sqrt {1+\frac {f x}{e}} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {e x}{d}}\right )|\frac {d f}{e^2}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e x}{d}}\right ),\frac {d f}{e^2}\right )\right )}{f \sqrt {\frac {e x}{d+e x}} \sqrt {d+e x} \sqrt {e+f x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 124

\(\displaystyle \frac {\sqrt {a x} \sqrt {\frac {e (e+f x)}{e^2-d f}} \int \frac {\sqrt {-\frac {e x}{d}}}{\sqrt {d+e x} \sqrt {\frac {e^2}{e^2-d f}+\frac {f x e}{e^2-d f}}}dx}{\sqrt {-\frac {e x}{d}} \sqrt {e+f x}}\)

\(\Big \downarrow \) 123

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

Input:

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

Output:

(2*Sqrt[-e^2 + d*f]*Sqrt[a*x]*Sqrt[(e*(e + f*x))/(e^2 - d*f)]*EllipticE[Ar 
cSin[(Sqrt[f]*Sqrt[d + e*x])/Sqrt[-e^2 + d*f]], 1 - e^2/(d*f)])/(e*Sqrt[f] 
*Sqrt[-((e*x)/d)]*Sqrt[e + f*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 = 0.79 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.68

method	result	size
default	\(-\frac {2 \left (d \operatorname {EllipticF}\left (\sqrt {\frac {f x +e}{e}}, \sqrt {-\frac {e^{2}}{d f -e^{2}}}\right ) f -\operatorname {EllipticE}\left (\sqrt {\frac {f x +e}{e}}, \sqrt {-\frac {e^{2}}{d f -e^{2}}}\right ) d f +\operatorname {EllipticE}\left (\sqrt {\frac {f x +e}{e}}, \sqrt {-\frac {e^{2}}{d f -e^{2}}}\right ) e^{2}\right ) \sqrt {-\frac {f x}{e}}\, \sqrt {\frac {\left (e x +d \right ) f}{d f -e^{2}}}\, \sqrt {\frac {f x +e}{e}}\, \sqrt {x a}\, \sqrt {e x +d}\, \sqrt {f x +e}}{f^{2} x \left (e f \,x^{2}+x d f +e^{2} x +d e \right )}\)	\(191\)
elliptic	\(\frac {2 \sqrt {x a}\, \sqrt {\left (e x +d \right ) \left (f x +e \right ) x a}\, e \sqrt {\frac {\left (x +\frac {e}{f}\right ) f}{e}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {e}{f}+\frac {d}{e}}}\, \sqrt {-\frac {f x}{e}}\, \left (\left (-\frac {e}{f}+\frac {d}{e}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {e}{f}\right ) f}{e}}, \sqrt {-\frac {e}{f \left (-\frac {e}{f}+\frac {d}{e}\right )}}\right )-\frac {d \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {e}{f}\right ) f}{e}}, \sqrt {-\frac {e}{f \left (-\frac {e}{f}+\frac {d}{e}\right )}}\right )}{e}\right )}{\sqrt {e x +d}\, \sqrt {f x +e}\, x f \sqrt {a e f \,x^{3}+a d f \,x^{2}+a \,e^{2} x^{2}+a d e x}}\)	\(215\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (97) = 194\).

Time = 0.17 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.40 \[ \int \frac {\sqrt {a x}}{\sqrt {d+e x} \sqrt {e+f x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {a e f} e f {\rm weierstrassZeta}\left (\frac {4 \, {\left (e^{4} - d e^{2} f + d^{2} f^{2}\right )}}{3 \, e^{2} f^{2}}, -\frac {4 \, {\left (2 \, e^{6} - 3 \, d e^{4} f - 3 \, d^{2} e^{2} f^{2} + 2 \, d^{3} f^{3}\right )}}{27 \, e^{3} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (e^{4} - d e^{2} f + d^{2} f^{2}\right )}}{3 \, e^{2} f^{2}}, -\frac {4 \, {\left (2 \, e^{6} - 3 \, d e^{4} f - 3 \, d^{2} e^{2} f^{2} + 2 \, d^{3} f^{3}\right )}}{27 \, e^{3} f^{3}}, \frac {3 \, e f x + e^{2} + d f}{3 \, e f}\right )\right ) + \sqrt {a e f} {\left (e^{2} + d f\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (e^{4} - d e^{2} f + d^{2} f^{2}\right )}}{3 \, e^{2} f^{2}}, -\frac {4 \, {\left (2 \, e^{6} - 3 \, d e^{4} f - 3 \, d^{2} e^{2} f^{2} + 2 \, d^{3} f^{3}\right )}}{27 \, e^{3} f^{3}}, \frac {3 \, e f x + e^{2} + d f}{3 \, e f}\right )\right )}}{3 \, e^{2} f^{2}} \] Input:

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

Output:

-2/3*(3*sqrt(a*e*f)*e*f*weierstrassZeta(4/3*(e^4 - d*e^2*f + d^2*f^2)/(e^2 
*f^2), -4/27*(2*e^6 - 3*d*e^4*f - 3*d^2*e^2*f^2 + 2*d^3*f^3)/(e^3*f^3), we 
ierstrassPInverse(4/3*(e^4 - d*e^2*f + d^2*f^2)/(e^2*f^2), -4/27*(2*e^6 - 
3*d*e^4*f - 3*d^2*e^2*f^2 + 2*d^3*f^3)/(e^3*f^3), 1/3*(3*e*f*x + e^2 + d*f 
)/(e*f))) + sqrt(a*e*f)*(e^2 + d*f)*weierstrassPInverse(4/3*(e^4 - d*e^2*f 
 + d^2*f^2)/(e^2*f^2), -4/27*(2*e^6 - 3*d*e^4*f - 3*d^2*e^2*f^2 + 2*d^3*f^ 
3)/(e^3*f^3), 1/3*(3*e*f*x + e^2 + d*f)/(e*f)))/(e^2*f^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*int((sqrt(x)*sqrt(e + f*x)*sqrt(d + e*x))/(d*e + d*f*x + e**2*x + 
e*f*x**2),x)