\(\int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} (e+f x^2)} \, dx\) [95]

Optimal result

Integrand size = 32, antiderivative size = 102 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {a^{3/2} \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (b e \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+(-b e+a f) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} e f \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

((-I)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(b*e*EllipticF[I*ArcSinh[Sqr 
t[b/a]*x], (a*d)/(b*c)] + (-(b*e) + a*f)*EllipticPi[(a*f)/(b*e), I*ArcSinh 
[Sqrt[b/a]*x], (a*d)/(b*c)]))/(Sqrt[b/a]*e*f*Sqrt[a + b*x^2]*Sqrt[c + d*x^ 
2])
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {414}

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.

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

\(\Big \downarrow \) 414

\(\displaystyle \frac {a^{3/2} \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\)

Input:

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

Output:

(a^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (a*f)/(b*e), ArcTan[(Sqrt[b]*x)/Sq 
rt[a]], 1 - (a*d)/(b*c)])/(Sqrt[b]*c*e*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2) 
)/(c*(a + b*x^2))])
 

Defintions of rubi rules used

rule 414

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

Maple [A] (verified)

Time = 4.19 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.87

method	result	size
default	\(\frac {\left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b e +\operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a f -\operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b e \right ) \sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{f e \sqrt {-\frac {b}{a}}\, \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(191\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a}{e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(325\)

Input:

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

Output:

(EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*e+EllipticPi(x*(-b/a)^(1/2),a 
*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*f-EllipticPi(x*(-b/a)^(1/2),a*f/b/e, 
(-1/c*d)^(1/2)/(-b/a)^(1/2))*b*e)*((d*x^2+c)/c)^(1/2)*((b*x^2+a)/a)^(1/2)/ 
f*(d*x^2+c)^(1/2)*(b*x^2+a)^(1/2)/e/(-b/a)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+ 
a*c)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {b\,x^2+a}}{\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}d x \] Input:

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

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c*e + c*f*x**2 + d*e*x**2 + d*f*x 
**4),x)