\(\int \frac {1}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx\) [61]

Optimal result

Integrand size = 26, antiderivative size = 102 \[ \int \frac {1}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=-\frac {2 c \sqrt {\frac {d (e+f x)}{d e+c f}} \sqrt {1-\frac {d^2 x^2}{c^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {d x}{c}}}{\sqrt {2}}\right ),\frac {2 c f}{d e+c f}\right )}{d \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\frac {2 i \sqrt {\frac {f (-c+d x)}{d (e+f x)}} \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d e+c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d e-c f}{d e+c f}\right )}{f \sqrt {-\frac {d e+c f}{d}} \sqrt {c^2-d^2 x^2}} \] Input:

Integrate[1/(Sqrt[e + f*x]*Sqrt[c^2 - d^2*x^2]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {512, 511, 321}

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.

\(\displaystyle \int \frac {1}{\sqrt {c^2-d^2 x^2} \sqrt {e+f x}} \, dx\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\sqrt {1-\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {e+f x} \sqrt {1-\frac {d^2 x^2}{c^2}}}dx}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 511

\(\displaystyle -\frac {2 c \sqrt {1-\frac {d^2 x^2}{c^2}} \sqrt {\frac {d (e+f x)}{c f+d e}} \int \frac {1}{\sqrt {1-\frac {c f \left (1-\frac {d x}{c}\right )}{d e+c f}} \sqrt {\frac {1}{2} \left (\frac {d x}{c}-1\right )+1}}d\frac {\sqrt {1-\frac {d x}{c}}}{\sqrt {2}}}{d \sqrt {c^2-d^2 x^2} \sqrt {e+f x}}\)

\(\Big \downarrow \) 321

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

Input:

Int[1/(Sqrt[e + f*x]*Sqrt[c^2 - d^2*x^2]),x]
 

Output:

(-2*c*Sqrt[(d*(e + f*x))/(d*e + c*f)]*Sqrt[1 - (d^2*x^2)/c^2]*EllipticF[Ar 
cSin[Sqrt[1 - (d*x)/c]/Sqrt[2]], (2*c*f)/(d*e + c*f)])/(d*Sqrt[e + f*x]*Sq 
rt[c^2 - d^2*x^2])
 

Defintions of rubi rules used

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

Maple [A] (verified)

Time = 2.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.67

method	result	size
default	\(-\frac {2 \sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}\, \sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \sqrt {\frac {f \left (-d x +c \right )}{c f +d e}}\, \sqrt {\frac {f \left (d x +c \right )}{c f -d e}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right ) \left (c f -d e \right )}{d f \left (-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}\right )}\)	\(170\)
elliptic	\(\frac {2 \sqrt {\left (f x +e \right ) \left (-d^{2} x^{2}+c^{2}\right )}\, \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}\, \sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}\)	\(217\)

Input:

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

Output:

-2*(f*x+e)^(1/2)*(-d^2*x^2+c^2)^(1/2)/d/f*(-d*(f*x+e)/(c*f-d*e))^(1/2)*(f* 
(-d*x+c)/(c*f+d*e))^(1/2)*(f*(d*x+c)/(c*f-d*e))^(1/2)*EllipticF((-d*(f*x+e 
)/(c*f-d*e))^(1/2),(-(c*f-d*e)/(c*f+d*e))^(1/2))*(c*f-d*e)/(-d^2*f*x^3-d^2 
*e*x^2+c^2*f*x+c^2*e)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=-\frac {2 \, \sqrt {-d^{2} f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )}{d^{2} f} \] Input:

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

Output:

-2*sqrt(-d^2*f)*weierstrassPInverse(4/3*(d^2*e^2 + 3*c^2*f^2)/(d^2*f^2), - 
8/27*(d^2*e^3 - 9*c^2*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f)/(d^2*f)
 

Sympy [F]

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

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

Output:

Integral(1/(sqrt(-(-c + d*x)*(c + d*x))*sqrt(e + f*x)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} + c^{2}} \sqrt {f x + e}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(-d^2*x^2 + c^2)*sqrt(f*x + e)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} + c^{2}} \sqrt {f x + e}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(-d^2*x^2 + c^2)*sqrt(f*x + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,\sqrt {c^2-d^2\,x^2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}{-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +c^{2} e}d x \] Input:

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

Output:

int((sqrt(e + f*x)*sqrt(c**2 - d**2*x**2))/(c**2*e + c**2*f*x - d**2*e*x** 
2 - d**2*f*x**3),x)