\(\int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx\) [1489]

Optimal result

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx=\frac {2 i \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {\frac {d \left (\sqrt {a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {a} d}} \sqrt {c+d x} \left (E\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{d \left (-\sqrt {a}+\sqrt {b} x\right )}} \sqrt {a-b x^2}} \] Input:

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

Output:

((2*I)*(Sqrt[b]*c - Sqrt[a]*d)*Sqrt[(d*(Sqrt[a] + Sqrt[b]*x))/(-(Sqrt[b]*c 
) + Sqrt[a]*d)]*Sqrt[c + d*x]*(EllipticE[I*ArcSinh[Sqrt[-((Sqrt[b]*(c + d* 
x))/(Sqrt[b]*c + Sqrt[a]*d))]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[ 
a]*d)] - EllipticF[I*ArcSinh[Sqrt[-((Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[ 
a]*d))]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)]))/(Sqrt[b]*d*Sq 
rt[(Sqrt[b]*(c + d*x))/(d*(-Sqrt[a] + Sqrt[b]*x))]*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 = {509, 508, 327}

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.

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
 

rule 508

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

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(378\) vs. \(2(105)=210\).

Time = 0.46 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.87

method	result	size
default	\(\frac {2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \left (b c -d \sqrt {a b}\right ) \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \left (\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) d +c \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b -\sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) d -\operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b c \right )}{d \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right ) b^{2}}\)	\(379\)
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {2 c \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 d \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(539\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {-b d} c {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) - 3 \, \sqrt {-b d} d {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right )\right )}}{3 \, b d} \] Input:

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

Output:

-2/3*(2*sqrt(-b*d)*c*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) - 3*sqrt(-b*d)*d*weier 
strassZeta(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 
), 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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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