\(\int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} (e+f x^2)^3} \, dx\) [176]

Optimal result

Integrand size = 32, antiderivative size = 606 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=-\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 e (d e-c f) \left (e+f x^2\right )^2}+\frac {(b e (5 d e-2 c f)-3 a f (2 d e-c f)) x \sqrt {a+b x^2}}{8 e^2 (b e-a f) (d e-c f) \sqrt {c+d x^2} \left (e+f x^2\right )}-\frac {\sqrt {c} \sqrt {d} (b e (5 d e-2 c f)-3 a f (2 d e-c f)) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{8 e^2 (b e-a f) (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {d} \left (3 b c d e^2-a \left (8 d^2 e^2-8 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{8 a e^2 (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} \left (3 b^2 d^2 e^4-2 a b e f \left (6 d^2 e^2-5 c d e f+2 c^2 f^2\right )+a^2 f^2 \left (8 d^2 e^2-8 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{8 a \sqrt {d} e^3 (b e-a f) (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

-1/4*f*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/(-c*f+d*e)/(f*x^2+e)^2+1/8*(b*e 
*(-2*c*f+5*d*e)-3*a*f*(-c*f+2*d*e))*x*(b*x^2+a)^(1/2)/e^2/(-a*f+b*e)/(-c*f 
+d*e)/(d*x^2+c)^(1/2)/(f*x^2+e)-1/8*c^(1/2)*d^(1/2)*(b*e*(-2*c*f+5*d*e)-3* 
a*f*(-c*f+2*d*e))*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^ 
(1/2),(1-b*c/a/d)^(1/2))/e^2/(-a*f+b*e)/(-c*f+d*e)^2/(c*(b*x^2+a)/a/(d*x^2 
+c))^(1/2)/(d*x^2+c)^(1/2)-1/8*c^(1/2)*d^(1/2)*(3*b*c*d*e^2-a*(3*c^2*f^2-8 
*c*d*e*f+8*d^2*e^2))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1 
/2)),(1-b*c/a/d)^(1/2))/a/e^2/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2) 
/(d*x^2+c)^(1/2)+1/8*c^(3/2)*(3*b^2*d^2*e^4-2*a*b*e*f*(2*c^2*f^2-5*c*d*e*f 
+6*d^2*e^2)+a^2*f^2*(3*c^2*f^2-8*c*d*e*f+8*d^2*e^2))*(b*x^2+a)^(1/2)*Ellip 
ticPi(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d 
^(1/2)/e^3/(-a*f+b*e)/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+ 
c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.66 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\frac {-\sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (2 e (b e-a f) (d e-c f)+(b e (5 d e-2 c f)+3 a f (-2 d e+c f)) \left (e+f x^2\right )\right )+i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right )^2 \left (b c e f (-3 a f (-2 d e+c f)+b e (-5 d e+2 c f)) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-b e (d e-c f) (b e (3 d e-2 c f)+a f (-4 d e+3 c f)) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+\left (3 b^2 d^2 e^4-2 a b e f \left (6 d^2 e^2-5 c d e f+2 c^2 f^2\right )+a^2 f^2 \left (8 d^2 e^2-8 c d e f+3 c^2 f^2\right )\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{8 \sqrt {\frac {b}{a}} e^3 f (b e-a f) (d e-c f)^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \] Input:

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

Output:

(-(Sqrt[b/a]*e*f^2*x*(a + b*x^2)*(c + d*x^2)*(2*e*(b*e - a*f)*(d*e - c*f) 
+ (b*e*(5*d*e - 2*c*f) + 3*a*f*(-2*d*e + c*f))*(e + f*x^2))) + I*Sqrt[1 + 
(b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)^2*(b*c*e*f*(-3*a*f*(-2*d*e + c* 
f) + b*e*(-5*d*e + 2*c*f))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] 
- b*e*(d*e - c*f)*(b*e*(3*d*e - 2*c*f) + a*f*(-4*d*e + 3*c*f))*EllipticF[I 
*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (3*b^2*d^2*e^4 - 2*a*b*e*f*(6*d^2*e^ 
2 - 5*c*d*e*f + 2*c^2*f^2) + a^2*f^2*(8*d^2*e^2 - 8*c*d*e*f + 3*c^2*f^2))* 
EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(8*Sqrt[b/a 
]*e^3*f*(b*e - a*f)*(d*e - c*f)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x 
^2)^2)
 

Rubi [F]

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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 424

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

\(\Big \downarrow \) 433

\(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (b e (3 d e-2 c f)-a f (2 d e-c f)) \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{2 \sqrt {b} e^2 \sqrt {a+b x^2} \sqrt {c+d x^2} (b e-a f) (d e-c f)}-\frac {b d \left (\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{f}-\frac {(b e-a f) \int \left (-\frac {f^{3/2}}{8 (-e)^{3/2} \left (\sqrt {-e} \sqrt {f}-f x\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {f^{3/2}}{8 (-e)^{3/2} \left (f x+\sqrt {-e} \sqrt {f}\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {3 f}{16 e^2 \left (\sqrt {-e} \sqrt {f}-f x\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {3 f}{16 e^2 \left (f x+\sqrt {-e} \sqrt {f}\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {3 f}{8 e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} \left (-f^2 x^2-e f\right )}\right )dx}{f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {\sqrt {-a} (b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{2 \sqrt {b} e^2 (b e-a f) (d e-c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}\right )}{f}-\frac {(b e-a f) \left (-\frac {\int \frac {1}{\left (\sqrt {-e} \sqrt {f}-f x\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx f^{3/2}}{8 (-e)^{3/2}}-\frac {\int \frac {1}{\left (f x+\sqrt {-e} \sqrt {f}\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx f^{3/2}}{8 (-e)^{3/2}}-\frac {3 \int \frac {1}{\left (\sqrt {-e} \sqrt {f}-f x\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx f}{16 e^2}-\frac {3 \int \frac {1}{\left (f x+\sqrt {-e} \sqrt {f}\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx f}{16 e^2}+\frac {3 \sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{8 \sqrt {b} e^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}\right )}{f}\)

Input:

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

Output:

$Aborted
 

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 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]
 

rule 433

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

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2504\) vs. \(2(574)=1148\).

Time = 9.19 (sec) , antiderivative size = 2505, normalized size of antiderivative = 4.13

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2505\)
default	\(\text {Expression too large to display}\)	\(4175\)

Input:

int((b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,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)*(1/4*f/(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)^2+1/8*f*(3*a*c*f^2-6* 
a*d*e*f-2*b*c*e*f+5*b*d*e^2)/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e^2/(c*f-d* 
e)*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+e)-3/8*b^2*d^2/(a*c*f^2-a* 
d*e*f-b*c*e*f+b*d*e^2)*e/(c*f-d*e)/f/(-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))-1/4*b^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e/(c*f- 
d*e)*c^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))*f+5 
/8*b^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/(c*f-d*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))*d+1/4*b^2/(a*c*f^2-a*d*e*f-b*c*e* 
f+b*d*e^2)/e/(c*f-d*e)*c^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)*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b* 
c)/c/b)^(1/2))*f-5/8*b^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/(c*f-d*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))*d-7/8*b*d/(a*c* 
f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e/(c*f-d*e)*f/(-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*c+3/4*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d \,f^{3} x^{8}+c \,f^{3} x^{6}+3 d e \,f^{2} x^{6}+3 c e \,f^{2} x^{4}+3 d \,e^{2} f \,x^{4}+3 c \,e^{2} f \,x^{2}+d \,e^{3} x^{2}+c \,e^{3}}d x \] Input:

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

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c*e**3 + 3*c*e**2*f*x**2 + 3*c*e* 
f**2*x**4 + c*f**3*x**6 + d*e**3*x**2 + 3*d*e**2*f*x**4 + 3*d*e*f**2*x**6 
+ d*f**3*x**8),x)