\(\int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2} (e+f x^2)} \, dx\) [128]

Optimal result

Integrand size = 35, antiderivative size = 498 \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {b (b c e-2 a d e-3 a c f) x \sqrt {c+d x^2}}{3 a c^2 e^2 \sqrt {a+b x^2}}-\frac {b (2 d e+3 c f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c^2 e^2}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a c e x^3}+\frac {(2 d e+3 c f) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a c^2 e^2 x}-\frac {\sqrt {b} (b c e-2 a d e-3 a c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} c^2 e^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {a} \sqrt {b} (d e+3 c f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 c^2 e^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/3*b*(-3*a*c*f-2*a*d*e+b*c*e)*x*(d*x^2+c)^(1/2)/a/c^2/e^2/(b*x^2+a)^(1/2) 
-1/3*b*(3*c*f+2*d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c^2/e^2-1/3*(b*x^ 
2+a)^(3/2)*(d*x^2+c)^(1/2)/a/c/e/x^3+1/3*(3*c*f+2*d*e)*(b*x^2+a)^(3/2)*(d* 
x^2+c)^(1/2)/a/c^2/e^2/x-1/3*b^(1/2)*(-3*a*c*f-2*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^2/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+d*e)*(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^2/(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^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 = 3.72 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} e \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-b c e x^2+a \left (-c e+2 d e x^2+3 c f x^2\right )\right )+i b c e (-b c e+2 a d e+3 a c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c e (-b c e+a d e+3 a c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 i a c^2 f (-b e+a f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b c^2 e^3 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[Sqrt[a + b*x^2]/(x^4*Sqrt[c + d*x^2]*(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*(-(c*e) 
+ 2*d*e*x^2 + 3*c*f*x^2)) + I*b*c*e*(-(b*c*e) + 2*a*d*e + 3*a*c*f)*x^3*Sqr 
t[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) + a*d*e + 3*a*c*f)*x^3*Sqrt[1 + (b*x^2)/a]*S 
qrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (3*I)* 
a*c^2*f*(-(b*e) + a*f)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipti 
cPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b*c^2*e^3*x^3*S 
qrt[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.

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

\(\Big \downarrow \) 450

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

Input:

Int[Sqrt[a + b*x^2]/(x^4*Sqrt[c + d*x^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 = 8.65 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-3 a c f \,x^{2}-2 a d e \,x^{2}+b c e \,x^{2}+a c e \right )}{3 a \,c^{2} e^{2} x^{3}}-\frac {\left (-\frac {b \left (3 a c f +2 a d e -b c e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {a c d e b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {3 a \,c^{2} f \left (a f -b e \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 c^{2} e^{2} a \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(448\)
default	\(\frac {\left (3 \sqrt {-\frac {b}{a}}\, a b c d e f \,x^{6}+2 \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}+\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}-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 ) 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 b \,c^{2} e f \,x^{3}+3 \sqrt {-\frac {b}{a}}\, a^{2} c d e f \,x^{4}+2 \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}-\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}+\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 ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{3 \sqrt {-\frac {b}{a}}\, a \,x^{3} e^{3} c^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(806\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 c e \,x^{3}}+\frac {\left (3 a c f +2 a d e -b c e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a \,c^{2} e^{2} x}+\frac {d 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 )}{3 c e \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}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right ) f}{e^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b^{2} \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 )}{3 a e \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}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right ) f}{e^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {2 b \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 ) d}{3 c e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b^{2} \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 )}{3 a e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\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}{e^{3} \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}{e^{2} \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}}\)	\(893\)

Input:

int((b*x^2+a)^(1/2)/x^4/(d*x^2+c)^(1/2)/(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^2-2*a*d*e*x^2+b*c*e*x^2+a 
*c*e)/a/c^2/e^2/x^3-1/3/c^2/e^2/a*(-b*(3*a*c*f+2*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) 
*(EllipticF(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^2*f*(a*f-b*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)))*((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. \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((b*x^2+a)^(1/2)/x^4/(d*x^2+c)^(1/2)/(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]

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

integrate((b*x^2+a)^(1/2)/x^4/(d*x^2+c)^(1/2)/(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. \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {b\,x^2+a}}{x^4\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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