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

Optimal result

Integrand size = 35, antiderivative size = 463 \[ \int \frac {\sqrt {a+b x^2}}{x^2 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\left (b-\frac {a d (2 d e-c f)}{c (d e-c f)}\right ) x \sqrt {a+b x^2}}{a c e \sqrt {c+d x^2}}-\frac {\left (a+b x^2\right )^{3/2}}{a c e x \sqrt {c+d x^2}}+\frac {b (2 d e-c f) x \sqrt {c+d x^2}}{c^2 e (d e-c f) \sqrt {a+b x^2}}-\frac {\sqrt {a} \sqrt {b} (2 d e-c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{c^2 e (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} \sqrt {b} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{c^2 e \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e^2 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

(b-a*d*(-c*f+2*d*e)/c/(-c*f+d*e))*x*(b*x^2+a)^(1/2)/a/c/e/(d*x^2+c)^(1/2)- 
(b*x^2+a)^(3/2)/a/c/e/x/(d*x^2+c)^(1/2)+b*(-c*f+2*d*e)*x*(d*x^2+c)^(1/2)/c 
^2/e/(-c*f+d*e)/(b*x^2+a)^(1/2)-a^(1/2)*b^(1/2)*(-c*f+2*d*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))/c^2/e/ 
(-c*f+d*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+a^(1/2)*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^2/e/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+a^(3/2)*f^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^2/(-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 = 4.52 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {a+b x^2}}{x^2 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {a \sqrt {\frac {b}{a}} c d e^2-a \sqrt {\frac {b}{a}} c^2 e f+b \sqrt {\frac {b}{a}} c d e^2 x^2+2 a \sqrt {\frac {b}{a}} d^2 e^2 x^2-b \sqrt {\frac {b}{a}} c^2 e f x^2-a \sqrt {\frac {b}{a}} c d e f x^2+2 b \sqrt {\frac {b}{a}} d^2 e^2 x^4-b \sqrt {\frac {b}{a}} c d e f x^4-i b c e (-2 d e+c f) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c e (-d e+c f) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-i b c^2 e f x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+i a c^2 f^2 x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c^2 e^2 (-d e+c f) x \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(a*Sqrt[b/a]*c*d*e^2 - a*Sqrt[b/a]*c^2*e*f + b*Sqrt[b/a]*c*d*e^2*x^2 + 2*a 
*Sqrt[b/a]*d^2*e^2*x^2 - b*Sqrt[b/a]*c^2*e*f*x^2 - a*Sqrt[b/a]*c*d*e*f*x^2 
 + 2*b*Sqrt[b/a]*d^2*e^2*x^4 - b*Sqrt[b/a]*c*d*e*f*x^4 - I*b*c*e*(-2*d*e + 
 c*f)*x*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*e*(-(d*e) + c*f)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 
 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c^2*e*f 
*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSi 
nh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*a*c^2*f^2*x*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)])/ 
(Sqrt[b/a]*c^2*e^2*(-(d*e) + c*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^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.

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

\(\Big \downarrow \) 450

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 450

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

Maple [A] (verified)

Time = 19.75 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.27

method	result	size
default	\(\frac {\left (-\sqrt {-\frac {b}{a}}\, b c d e f \,x^{4}+2 \sqrt {-\frac {b}{a}}\, b \,d^{2} e^{2} x^{4}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} e f x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d \,e^{2} x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} e f x -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d \,e^{2} x -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a \,c^{2} f^{2} x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b \,c^{2} e f x -\sqrt {-\frac {b}{a}}\, a c d e f \,x^{2}+2 \sqrt {-\frac {b}{a}}\, a \,d^{2} e^{2} x^{2}-\sqrt {-\frac {b}{a}}\, b \,c^{2} e f \,x^{2}+\sqrt {-\frac {b}{a}}\, b c d \,e^{2} x^{2}-\sqrt {-\frac {b}{a}}\, a \,c^{2} e f +\sqrt {-\frac {b}{a}}\, a c d \,e^{2}\right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{\left (c f -d e \right ) \sqrt {-\frac {b}{a}}\, x \,e^{2} c^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(586\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{c^{2} e x}+\frac {\left (-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {c e \left (a d -b c \right ) d \left (\frac {\left (b d \,x^{2}+a d \right ) x}{c \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{\left (a d -b c \right ) c}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{c f -d e}-\frac {c^{2} \left (a f -b e \right ) f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{\left (c f -d e \right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{e \,c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(621\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+a d \right ) d x}{c^{2} \left (c f -d e \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}-\frac {\sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{c^{2} e x}-\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, d b \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, \left (c f -d e \right )}-\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, b \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, e}+\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, b \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, e}-\frac {f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a}{\left (c f -d e \right ) e^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b}{\left (c f -d e \right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(631\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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