\(\int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx\) [1510]

Optimal result

Integrand size = 22, antiderivative size = 132 \[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \sqrt {a} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 i \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \sqrt {a-b x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

(-2*Sqrt[a]*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b* 
x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((S 
qrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*Sqrt[c + d*x]*Sqrt[a - b*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 = 1.59 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.47

method	result	size
default	\(\frac {2 \left (b c -d \sqrt {a b}\right ) \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}{b d \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right )}\)	\(194\)
elliptic	\(\frac {2 \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\)	\(237\)

Input:

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

Output:

2*(b*c-d*(a*b)^(1/2))*EllipticF((-(d*x+c)*b/(d*(a*b)^(1/2)-b*c))^(1/2),(-( 
d*(a*b)^(1/2)-b*c)/(d*(a*b)^(1/2)+b*c))^(1/2))*((b*x+(a*b)^(1/2))*d/(d*(a* 
b)^(1/2)-b*c))^(1/2)*((-b*x+(a*b)^(1/2))*d/(d*(a*b)^(1/2)+b*c))^(1/2)*(-(d 
*x+c)*b/(d*(a*b)^(1/2)-b*c))^(1/2)*(-b*x^2+a)^(1/2)*(d*x+c)^(1/2)/b/d/(-b* 
d*x^3-b*c*x^2+a*d*x+a*c)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \, \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )}{b d} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral(1/(sqrt(a - b*x**2)*sqrt(c + d*x)), x)
 

Maxima [F]

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

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

Output:

integrate(1/(sqrt(-b*x^2 + a)*sqrt(d*x + c)), x)
 

Giac [F]

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

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

Output:

integrate(1/(sqrt(-b*x^2 + a)*sqrt(d*x + c)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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