\(\int \frac {\sqrt {a+b x^2}}{x^4 (c+d x^2)^{3/2} (e+f x^2)} \, dx\) [153]

Optimal result

Integrand size = 35, antiderivative size = 626 \[ \int \frac {\sqrt {a+b x^2}}{x^4 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=-\frac {\left (b c \left (5 d^2 e^2-2 c d e f-3 c^2 f^2\right )-a d \left (8 d^2 e^2-2 c d e f-3 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{3 a c^3 e^2 (d e-c f) \sqrt {c+d x^2}}-\frac {\left (a+b x^2\right )^{3/2}}{3 a c e x^3 \sqrt {c+d x^2}}+\frac {(4 d e+3 c f) \left (a+b x^2\right )^{3/2}}{3 a c^2 e^2 x \sqrt {c+d x^2}}+\frac {b \left (b c-\frac {a \left (8 d^2 e^2-2 c d e f-3 c^2 f^2\right )}{e (d e-c f)}\right ) x \sqrt {c+d x^2}}{3 a c^3 e \sqrt {a+b x^2}}-\frac {\sqrt {b} \left (b c e (d e-c f)-a \left (8 d^2 e^2-2 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} c^3 e^2 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {a} \sqrt {b} (4 d e+3 c f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 c^3 e^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} f^3 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e^3 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/3*(b*c*(-3*c^2*f^2-2*c*d*e*f+5*d^2*e^2)-a*d*(-3*c^2*f^2-2*c*d*e*f+8*d^2 
*e^2))*x*(b*x^2+a)^(1/2)/a/c^3/e^2/(-c*f+d*e)/(d*x^2+c)^(1/2)-1/3*(b*x^2+a 
)^(3/2)/a/c/e/x^3/(d*x^2+c)^(1/2)+1/3*(3*c*f+4*d*e)*(b*x^2+a)^(3/2)/a/c^2/ 
e^2/x/(d*x^2+c)^(1/2)+1/3*b*(b*c-a*(-3*c^2*f^2-2*c*d*e*f+8*d^2*e^2)/e/(-c* 
f+d*e))*x*(d*x^2+c)^(1/2)/a/c^3/e/(b*x^2+a)^(1/2)-1/3*b^(1/2)*(b*c*e*(-c*f 
+d*e)-a*(-3*c^2*f^2-2*c*d*e*f+8*d^2*e^2))*(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^3/e^2/(-c*f+d*e 
)/(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+4*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^3/e^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-a^(3/ 
2)*f^3*(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/(-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 = 5.53 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a+b x^2}}{x^4 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} \left (-\sqrt {\frac {b}{a}} e \left (a+b x^2\right ) \left (b c e (-d e+c f) x^2 \left (c+d x^2\right )+a \left (8 d^3 e^2 x^4+c^3 f \left (e-3 f x^2\right )+2 c d^2 e x^2 \left (2 e-f x^2\right )-c^2 d \left (e^2+e f x^2+3 f^2 x^4\right )\right )\right )+i b c e \left (b c e (d e-c f)+a \left (-8 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right ) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c e (-d e+c f) (-b c e+4 a d e+3 a c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 i a c^3 f^2 (-b e+a f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b c^3 e^3 (-d e+c f) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[Sqrt[a + b*x^2]/(x^4*(c + d*x^2)^(3/2)*(e + f*x^2)),x]
 

Output:

(Sqrt[b/a]*(-(Sqrt[b/a]*e*(a + b*x^2)*(b*c*e*(-(d*e) + c*f)*x^2*(c + d*x^2 
) + a*(8*d^3*e^2*x^4 + c^3*f*(e - 3*f*x^2) + 2*c*d^2*e*x^2*(2*e - f*x^2) - 
 c^2*d*(e^2 + e*f*x^2 + 3*f^2*x^4)))) + I*b*c*e*(b*c*e*(d*e - c*f) + a*(-8 
*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))*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*(-(d*e) + c* 
f)*(-(b*c*e) + 4*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^3*f^2*(-(b* 
e) + a*f)*x^3*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)]))/(3*b*c^3*e^3*(-(d*e) + c*f)*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.

\(\displaystyle \int \frac {\sqrt {a+b x^2}}{x^4 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {\sqrt {a+b x^2}}{x^4 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}dx\)

Input:

Int[Sqrt[a + b*x^2]/(x^4*(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 = 21.52 (sec) , antiderivative size = 772, normalized size of antiderivative = 1.23

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-3 a c f \,x^{2}-5 a d e \,x^{2}+b c e \,x^{2}+a c e \right )}{3 a \,c^{3} e^{2} x^{3}}-\frac {\left (-\frac {b \left (3 a c f +5 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^{3} f^{2} \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 )}{\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}}+\frac {3 a c \,e^{2} d^{2} \left (a d -b c \right ) \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}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 e^{2} c^{3} a \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(772\)
elliptic	\(\text {Expression too large to display}\)	\(1081\)
default	\(\text {Expression too large to display}\)	\(1279\)

Input:

int((b*x^2+a)^(1/2)/x^4/(d*x^2+c)^(3/2)/(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^2-5*a*d*e*x^2+b*c*e*x^2+a 
*c*e)/a/c^3/e^2/x^3-1/3/e^2/c^3/a*(-b*(3*a*c*f+5*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^3*f^2*(a*f-b*e)/(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))+3*a*c*e^2*d 
^2*(a*d-b*c)/(c*f-d*e)*((b*d*x^2+a*d)/c/(a*d-b*c)*x/((x^2+c/d)*(b*d*x^2+a* 
d))^(1/2)+(1/c-1/(a*d-b*c)/c*a*d)/(-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))+b/(a*d-b*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))) 
))*((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. \[ \int \frac {\sqrt {a+b x^2}}{x^4 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(1/2)/x^4/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

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

integrate((b*x**2+a)**(1/2)/x**4/(d*x**2+c)**(3/2)/(f*x**2+e),x)
 

Output:

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

Maxima [F]

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

integrate((b*x^2+a)^(1/2)/x^4/(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^4), x)
 

Giac [F]

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

integrate((b*x^2+a)^(1/2)/x^4/(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^4), x)
 

Mupad [F(-1)]

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

int((a + b*x^2)^(1/2)/(x^4*(c + d*x^2)^(3/2)*(e + f*x^2)),x)
 

Output:

int((a + b*x^2)^(1/2)/(x^4*(c + d*x^2)^(3/2)*(e + f*x^2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2}}{x^4 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d^{2} f \,x^{10}+2 c d f \,x^{8}+d^{2} e \,x^{8}+c^{2} f \,x^{6}+2 c d e \,x^{6}+c^{2} e \,x^{4}}d x \] Input:

int((b*x^2+a)^(1/2)/x^4/(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**4 + c**2*f*x**6 + 2*c*d 
*e*x**6 + 2*c*d*f*x**8 + d**2*e*x**8 + d**2*f*x**10),x)