\(\int \frac {\sqrt {c+d x}}{x \sqrt {a-b x^2}} \, dx\) [1490]

Optimal result

Integrand size = 25, antiderivative size = 259 \[ \int \frac {\sqrt {c+d x}}{x \sqrt {a-b x^2}} \, dx=-\frac {2 \sqrt {a} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.59 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {c+d x}}{x \sqrt {a-b x^2}} \, dx=-\frac {2 i \sqrt {\frac {d \left (\sqrt {a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {a} d}} \sqrt {c+d x} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-\operatorname {EllipticPi}\left (1+\frac {\sqrt {a} d}{\sqrt {b} c},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{\sqrt {\frac {\sqrt {b} (c+d x)}{d \left (-\sqrt {a}+\sqrt {b} x\right )}} \sqrt {a-b x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {637, 2009}

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.

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

\(\Big \downarrow \) 637

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 637

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 
/2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n 
 + 1/2] && IntegerQ[m]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.55

method	result	size
default	\(\frac {2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \left (c \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b -\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) d -\operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \frac {b c -d \sqrt {a b}}{b c}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b c +\sqrt {a b}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \frac {b c -d \sqrt {a b}}{b c}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) d \right )}{b \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right )}\)	\(402\)
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {2 d \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {2 \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, d \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(466\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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