\(\int \frac {(a+b x^2)^{3/2}}{\sqrt {c+d x^2} (e+f x^2)^2} \, dx\) [152]

Optimal result

Integrand size = 32, antiderivative size = 493 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=-\frac {b (b e-a f) x \sqrt {c+d x^2}}{2 e f (d e-c f) \sqrt {a+b x^2}}+\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e (d e-c f)}-\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 e (d e-c f) \left (e+f x^2\right )}+\frac {\sqrt {a} \sqrt {b} (b e-a f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{2 e f (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \sqrt {b} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 c e f \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} (b e (d e-2 c f)+a f (2 d e-c f)) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c e^2 f (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/2*b*(-a*f+b*e)*x*(d*x^2+c)^(1/2)/e/f/(-c*f+d*e)/(b*x^2+a)^(1/2)+1/2*b*x 
*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/(-c*f+d*e)-1/2*f*x*(b*x^2+a)^(3/2)*(d*x 
^2+c)^(1/2)/e/(-c*f+d*e)/(f*x^2+e)+1/2*a^(1/2)*b^(1/2)*(-a*f+b*e)*(d*x^2+c 
)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/e 
/f/(-c*f+d*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/2*a^(3/2)* 
b^(1/2)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b 
/c)^(1/2))/c/e/f/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/2*a^(3/ 
2)*(b*e*(-2*c*f+d*e)+a*f*(-c*f+2*d*e))*(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^2/f/( 
-c*f+d*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 = 6.22 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {-i b c e f (-b e+a f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b e (b e+a f) (-d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+(b e-a f) \left (\sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right )-i (a f (-2 d e+c f)+b e (-d e+2 c f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{2 \sqrt {\frac {b}{a}} e^2 f^2 (d e-c f) \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.39, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {425, 414, 425, 413, 413, 412, 424, 406, 320, 388, 313, 413, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 414

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

\(\Big \downarrow \) 424

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

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 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 424

Int[1/(((a_) + (b_.)*(x_)^2)^2*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)* 
(x_)^2]), x_Symbol] :> Simp[b^2*x*Sqrt[c + d*x^2]*(Sqrt[e + f*x^2]/(2*a*(b* 
c - a*d)*(b*e - a*f)*(a + b*x^2))), x] + (Simp[(b^2*c*e + 3*a^2*d*f - 2*a*b 
*(d*e + c*f))/(2*a*(b*c - a*d)*(b*e - a*f))   Int[1/((a + b*x^2)*Sqrt[c + d 
*x^2]*Sqrt[e + f*x^2]), x], x] - Simp[d*(f/(2*a*(b*c - a*d)*(b*e - a*f))) 
 Int[(a + b*x^2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x]) /; FreeQ[{a, b, 
 c, d, e, f}, x]
 

rule 425

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1480\) vs. \(2(455)=910\).

Time = 6.57 (sec) , antiderivative size = 1481, normalized size of antiderivative = 3.00

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1481\)
default	\(\text {Expression too large to display}\)	\(1842\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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