\(\int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx\) [1511]

Optimal result

Integrand size = 25, antiderivative size = 124 \[ \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-2*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*((-b*x^2+a)/a)^(1/2)*Elli 
pticPi(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 = 21.90 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.16 \[ \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 i \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x) \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-\operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{c \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \sqrt {a-b x^2}} \] Input:

Integrate[1/(x*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) 
] - EllipticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + (S 
qrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sq 
rt[a]*d)]))/(c*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*Sqrt[a - b*x^2])
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {633, 632, 186, 413, 412}

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.

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

\(\Big \downarrow \) 633

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

\(\Big \downarrow \) 632

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

\(\Big \downarrow \) 186

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 186

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] && GtQ[(d*e - c*f)/d, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 632

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

rule 633

Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : 
> Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(x*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. \(212\) vs. \(2(101)=202\).

Time = 2.14 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.72

method	result	size
default	\(-\frac {2 \left (b c -d \sqrt {a b}\right ) \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \frac {b c -d \sqrt {a b}}{b c}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}{b c \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right )}\)	\(213\)
elliptic	\(-\frac {2 \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, d \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}\, c}\)	\(263\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/x/(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), x)
 

Giac [F]

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

integrate(1/x/(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), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{4}-b c \,x^{3}+a d \,x^{2}+a c x}d x \] Input:

int(1/x/(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*x + a*d*x**2 - b*c*x**3 - b*d*x* 
*4),x)