\(\int \frac {A+B x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [6]

Optimal result

Integrand size = 42, antiderivative size = 476 \[ \int \frac {A+B x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {(2 A b c e-3 a B c e+2 a A d e+3 a A c f) \sqrt {a+b x^2}}{3 a^2 c e^2 x \sqrt {c+d x^2}}-\frac {A \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c e x^3}-\frac {\sqrt {d} (3 a B c e-A (2 b c e+2 a d e+3 a c f)) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 c^{3/2} e^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {d} \left (3 a B c e f-3 a A c f^2+A b e (d e-c f)\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} e^2 (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} f^2 (B e-A f) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e^3 (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/3*(3*A*a*c*f+2*A*a*d*e+2*A*b*c*e-3*B*a*c*e)*(b*x^2+a)^(1/2)/a^2/c/e^2/x/ 
(d*x^2+c)^(1/2)-1/3*A*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e/x^3-1/3*d^(1/2 
)*(3*B*a*c*e-A*(3*a*c*f+2*a*d*e+2*b*c*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))/a^2/c^(3/2)/e^2/(c*(b*x^2 
+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*d^(1/2)*(3*a*B*c*e*f-3*a*A*c*f^ 
2+A*b*e*(-c*f+d*e))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/ 
2)),(1-b*c/a/d)^(1/2))/a^2/c^(1/2)/e^2/(-c*f+d*e)/(c*(b*x^2+a)/a/(d*x^2+c) 
)^(1/2)/(d*x^2+c)^(1/2)+c^(3/2)*f^2*(-A*f+B*e)*(b*x^2+a)^(1/2)*EllipticPi( 
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/(-c*f+d*e)/(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 = 11.29 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} e \left (a+b x^2\right ) \left (c+d x^2\right ) \left (2 A b c e x^2-3 a B c e x^2+a A \left (-c e+2 d e x^2+3 c f x^2\right )\right )+i b c e (-3 a B c e+A (2 b c e+2 a d e+3 a c f)) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c e (-3 a B c e+A (2 b c e+a d e+3 a c f)) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+3 i a^2 c^2 f (B e-A f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 e^3 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*e*(a + b*x^2)*(c + d*x^2)*(2*A*b*c*e*x^2 - 3*a*B*c*e*x^2 + a*A* 
(-(c*e) + 2*d*e*x^2 + 3*c*f*x^2)) + I*b*c*e*(-3*a*B*c*e + A*(2*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]*EllipticE[I*A 
rcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*e*(-3*a*B*c*e + A*(2*b*c*e + a*d 
*e + 3*a*c*f))*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*Arc 
Sinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (3*I)*a^2*c^2*f*(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*a^2*Sqrt[b/a]*c^2*e^3*x^3*Sqrt[a + b*x^2]*Sqrt[c + 
d*x^2])
 

Rubi [A] (verified)

Time = 2.15 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {7276, 2009}

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-2*A*d*(b*c + a*d)*x*Sqrt[a + b*x^2])/(3*a^2*c^2*e*Sqrt[c + d*x^2]) + (d* 
(B*e - A*f)*x*Sqrt[a + b*x^2])/(a*c*e^2*Sqrt[c + d*x^2]) - (A*Sqrt[a + b*x 
^2]*Sqrt[c + d*x^2])/(3*a*c*e*x^3) + (2*A*(b*c + a*d)*Sqrt[a + b*x^2]*Sqrt 
[c + d*x^2])/(3*a^2*c^2*e*x) - ((B*e - A*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2 
])/(a*c*e^2*x) + (2*A*Sqrt[d]*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan 
[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*c^(3/2)*e*Sqrt[(c*(a + b*x 
^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[d]*(B*e - A*f)*Sqrt[a + b*x 
^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[c]*e^ 
2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (A*b*Sqrt[d]*Sq 
rt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3* 
a^2*Sqrt[c]*e*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sq 
rt[-a]*f*(B*e - A*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a 
*f)/(b*e), ArcSin[(Sqrt[b]*x)/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*e^3*Sqrt[a 
 + b*x^2]*Sqrt[c + d*x^2])
 

Defintions of rubi rules used

rule 2009

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

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 10.35 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-3 A a c f \,x^{2}-2 A a d e \,x^{2}-2 A b c e \,x^{2}+3 B a c e \,x^{2}+A a c e \right )}{3 a^{2} c^{2} e^{2} x^{3}}-\frac {\left (-\frac {b \left (3 A a c f +2 A a d e +2 A b c e -3 B a c e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {a b c d e A \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {3 a^{2} c^{2} f \left (A f -B e \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 a^{2} c^{2} e^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(474\)
default	\(\text {Expression too large to display}\)	\(1061\)
elliptic	\(\text {Expression too large to display}\)	\(1113\)

Input:

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

Output:

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

integrate((B*x^2+A)/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algor 
ithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((B*x^2+A)/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algor 
ithm="maxima")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((B*x^2+A)/x^4/(b*x^2+a)^(1/2)/(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)