\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 (e+f x^2)} \, dx\) [74]

Optimal result

Integrand size = 35, antiderivative size = 528 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )} \, dx=\frac {b (b c e+a d e-3 a c f) x \sqrt {c+d x^2}}{3 a c e^2 \sqrt {a+b x^2}}+\frac {d (b e-3 a f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c e^2}-\frac {b f x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{a c e^2}-\frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{3 a c e x^3}+\frac {f \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{a c e^2 x}-\frac {\sqrt {b} (b c e+a d e-3 a c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} c e^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} \sqrt {b} (2 d e-3 c f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 c e^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} f (d e-c f) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/3*b*(-3*a*c*f+a*d*e+b*c*e)*x*(d*x^2+c)^(1/2)/a/c/e^2/(b*x^2+a)^(1/2)+1/3 
*d*(-3*a*f+b*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e^2-b*f*x*(b*x^2+a)^ 
(1/2)*(d*x^2+c)^(3/2)/a/c/e^2-1/3*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)/a/c/e/x^ 
3+f*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)/a/c/e^2/x-1/3*b^(1/2)*(-3*a*c*f+a*d*e+ 
b*c*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))/a^(1/2)/c/e^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2 
)+1/3*a^(1/2)*b^(1/2)*(-3*c*f+2*d*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arcta 
n(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/c/e^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c) 
/c/(b*x^2+a))^(1/2)-a^(3/2)*f*(-c*f+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^3/( 
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 = 3.44 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} \left (-\sqrt {\frac {b}{a}} e \left (a+b x^2\right ) \left (c+d x^2\right ) \left (b c e x^2+a d e x^2+a c \left (e-3 f x^2\right )\right )+i b c e (-b c e-a d e+3 a c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c e (-b c e-2 a d e+3 a c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 i a c (-b e+a f) (-d e+c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b c e^3 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^4*(e + f*x^2)),x]
 

Output:

(Sqrt[b/a]*(-(Sqrt[b/a]*e*(a + b*x^2)*(c + d*x^2)*(b*c*e*x^2 + a*d*e*x^2 + 
 a*c*(e - 3*f*x^2))) + I*b*c*e*(-(b*c*e) - a*d*e + 3*a*c*f)*x^3*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*(-(b*c*e) - 2*a*d*e + 3*a*c*f)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 
 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (3*I)*a*c*( 
-(b*e) + a*f)*(-(d*e) + c*f)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*E 
llipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b*c*e^3*x 
^3*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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^4 \left (e+f x^2\right )}dx\)

Input:

Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^4*(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 = 8.49 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-3 a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e \right )}{3 a c \,e^{2} x^{3}}-\frac {\left (-\frac {b \left (3 a c f -a d e -b c e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {a c d e b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {3 \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 a c \,e^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(461\)
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (3 \sqrt {-\frac {b}{a}}\, a b c d e f \,x^{6}-\sqrt {-\frac {b}{a}}\, a b \,d^{2} e^{2} x^{6}-\sqrt {-\frac {b}{a}}\, b^{2} c d \,e^{2} x^{6}+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 b \,c^{2} e f \,x^{3}-2 \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 b c d \,e^{2} 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 ) b^{2} c^{2} e^{2} x^{3}-3 \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 ) a b \,c^{2} e f \,x^{3}+\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 ) a b c d \,e^{2} x^{3}+\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^{2} c^{2} e^{2} x^{3}+3 \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^{2} c^{2} f^{2} x^{3}-3 \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^{2} c d e f \,x^{3}-3 \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 b \,c^{2} e f \,x^{3}+3 \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 b c d \,e^{2} x^{3}+3 \sqrt {-\frac {b}{a}}\, a^{2} c d e f \,x^{4}-\sqrt {-\frac {b}{a}}\, a^{2} d^{2} e^{2} x^{4}+3 \sqrt {-\frac {b}{a}}\, a b \,c^{2} e f \,x^{4}-3 \sqrt {-\frac {b}{a}}\, a b c d \,e^{2} x^{4}-\sqrt {-\frac {b}{a}}\, b^{2} c^{2} e^{2} x^{4}+3 \sqrt {-\frac {b}{a}}\, a^{2} c^{2} e f \,x^{2}-2 \sqrt {-\frac {b}{a}}\, a^{2} c d \,e^{2} x^{2}-2 \sqrt {-\frac {b}{a}}\, a b \,c^{2} e^{2} x^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c^{2} e^{2}\right )}{3 \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) e^{3} x^{3} a c \sqrt {-\frac {b}{a}}}\)	\(977\)
elliptic	\(\text {Expression too large to display}\)	\(1090\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e),x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

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

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)/x**4/(f*x**2+e),x)
 

Output:

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

Maxima [F]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^4/(f*x^2+e),x, algorithm="giac 
")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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