\(\int \frac {\sqrt {c+d x^2}}{(a-b x^2)^{3/2} (e+f x^2)} \, dx\) [126]

Optimal result
Mathematica [C] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [F(-1)]
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 33, antiderivative size = 358 \[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {b x \sqrt {c+d x^2}}{a (b e+a f) \sqrt {a-b x^2}}-\frac {\sqrt {b} c \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{a^{3/2} (b e+a f) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2}}+\frac {(b c+a d) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{a^{3/2} \sqrt {b} (b e+a f) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2}}-\frac {\sqrt {a} (d e-c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.21 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {b \sqrt {-\frac {b}{a}} c e x+b \sqrt {-\frac {b}{a}} d e x^3+i b c e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i (b c+a d) e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+i a d e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )-i a c f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{a \sqrt {-\frac {b}{a}} e (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[Sqrt[c + d*x^2]/((a - b*x^2)^(3/2)*(e + f*x^2)),x]
 

Output:

(b*Sqrt[-(b/a)]*c*e*x + b*Sqrt[-(b/a)]*d*e*x^3 + I*b*c*e*Sqrt[1 - (b*x^2)/ 
a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c)) 
] - I*(b*c + a*d)*e*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*Ar 
cSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] + I*a*d*e*Sqrt[1 - (b*x^2)/a]*Sqrt[ 
1 + (d*x^2)/c]*EllipticPi[-((a*f)/(b*e)), I*ArcSinh[Sqrt[-(b/a)]*x], -((a* 
d)/(b*c))] - I*a*c*f*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[-( 
(a*f)/(b*e)), I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(a*Sqrt[-(b/a)]* 
e*(b*e + a*f)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.27, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {421, 401, 27, 399, 323, 323, 321, 331, 330, 327, 410, 331, 330, 327, 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.

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

\(\Big \downarrow \) 421

\(\displaystyle \frac {f^2 \int \frac {\sqrt {a-b x^2} \sqrt {d x^2+c}}{f x^2+e}dx}{(a f+b e)^2}+\frac {b \int \frac {\sqrt {d x^2+c} \left (-b f x^2+b e+2 a f\right )}{\left (a-b x^2\right )^{3/2}}dx}{(a f+b e)^2}\)

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 410

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 414

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

Input:

Int[Sqrt[c + d*x^2]/((a - b*x^2)^(3/2)*(e + f*x^2)),x]
 

Output:

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

Defintions of rubi rules used

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 323
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

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 330
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 331
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]
 

rule 399
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 401
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 410
Int[(Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*(x_ 
)^2), x_Symbol] :> Simp[d/b   Int[Sqrt[e + f*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*c - a*d)/b   Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x] 
, x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !SimplerSqrtQ[-f/e, -d/c]
 

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]
 

rule 421
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2   Int[(c + d*x^2)^(q + 2)*((e + 
 f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2   Int[(c + d*x^2)^q*( 
e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} 
, x] && LtQ[q, -1]
 
Maple [A] (verified)

Time = 7.00 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.09

method result size
default \(\frac {\left (\sqrt {\frac {b}{a}}\, b d e \,x^{3}+\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a d e +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c e -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c e +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a c f -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a d e +\sqrt {\frac {b}{a}}\, b c e x \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{e \sqrt {\frac {b}{a}}\, a \left (a f +b e \right ) \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\) \(391\)
elliptic \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}-b c \right ) x}{a \left (a f +b e \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {d \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\left (a f +b e \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{a \left (a f +b e \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{a \left (a f +b e \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {f \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) c}{\left (a f +b e \right ) e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) d}{\left (a f +b e \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) \(613\)

Input:

int((d*x^2+c)^(1/2)/(-b*x^2+a)^(3/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

((b/a)^(1/2)*b*d*e*x^3+((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF( 
x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*a*d*e+((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^( 
1/2)*EllipticF(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b*c*e-((-b*x^2+a)/a)^(1/2)* 
((d*x^2+c)/c)^(1/2)*EllipticE(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b*c*e+((-b*x 
^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c 
*d)^(1/2)/(b/a)^(1/2))*a*c*f-((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Elli 
pticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2))*a*d*e+(b/a)^(1/2 
)*b*c*e*x)*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/(b/a)^(1/2)/a/(a*f+b*e)/(-b* 
d*x^4+a*d*x^2-b*c*x^2+a*c)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((d*x^2+c)^(1/2)/(-b*x^2+a)^(3/2)/(f*x^2+e),x, algorithm="fricas" 
)
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\left (a - b x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}\, dx \] Input:

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

Output:

Integral(sqrt(c + d*x**2)/((a - b*x**2)**(3/2)*(e + f*x**2)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((d*x^2+c)^(1/2)/(-b*x^2+a)^(3/2)/(f*x^2+e),x, algorithm="maxima" 
)
 

Output:

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

Giac [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate((d*x^2+c)^(1/2)/(-b*x^2+a)^(3/2)/(f*x^2+e),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d\,x^2+c}}{{\left (a-b\,x^2\right )}^{3/2}\,\left (f\,x^2+e\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {c+d x^2}}{\left (a-b x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b^{2} f \,x^{6}-2 a b f \,x^{4}+b^{2} e \,x^{4}+a^{2} f \,x^{2}-2 a b e \,x^{2}+a^{2} e}d x \] Input:

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

Output:

int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*e + a**2*f*x**2 - 2*a*b*e*x* 
*2 - 2*a*b*f*x**4 + b**2*e*x**4 + b**2*f*x**6),x)