\(\int \frac {A+B x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [5]

Optimal result

Integrand size = 42, antiderivative size = 357 \[ \int \frac {A+B x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {A \sqrt {a+b x^2}}{a e x \sqrt {c+d x^2}}-\frac {A \sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} e \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} \sqrt {d} (B e-A f) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a e (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} f (B e-A f) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e^2 (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

-A*(b*x^2+a)^(1/2)/a/e/x/(d*x^2+c)^(1/2)-A*d^(1/2)*(b*x^2+a)^(1/2)*Ellipti 
cE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/a/c^(1/2)/e/(c*( 
b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+c^(1/2)*d^(1/2)*(-A*f+B*e)*(b* 
x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/ 
a/e/(-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*( 
-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^2/(-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 = 9.31 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} \left (-a A \sqrt {\frac {b}{a}} c e-A b \sqrt {\frac {b}{a}} c e x^2-a A \sqrt {\frac {b}{a}} d e x^2-A b \sqrt {\frac {b}{a}} d e x^4-i A b c e x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i A b c e x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-i a B c e x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+i a A c f x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{b c e^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*(-(a*A*Sqrt[b/a]*c*e) - A*b*Sqrt[b/a]*c*e*x^2 - a*A*Sqrt[b/a]*d 
*e*x^2 - A*b*Sqrt[b/a]*d*e*x^4 - I*A*b*c*e*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*A*b*c*e*x*Sq 
rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a 
*d)/(b*c)] - I*a*B*c*e*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticP 
i[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*a*A*c*f*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)]))/(b*c*e^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.76, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(A*d*x*Sqrt[a + b*x^2])/(a*c*e*Sqrt[c + d*x^2]) - (A*Sqrt[a + b*x^2]*Sqrt[ 
c + d*x^2])/(a*c*e*x) - (A*Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[ 
d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[c]*e*Sqrt[(c*(a + b*x^2))/(a*(c 
+ d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[-a]*(B*e - A*f)*Sqrt[1 + (b*x^2)/a]*Sq 
rt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), ArcSin[(Sqrt[b]*x)/Sqrt[-a]], (a 
*d)/(b*c)])/(Sqrt[b]*e^2*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 = 9.99 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {A \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{a c e x}+\frac {\left (-\frac {A b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {a c \left (A f -B e \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{a c e \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(311\)
default	\(\frac {\left (-A \sqrt {-\frac {b}{a}}\, b d e \,x^{4}-A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c e x +A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c e x -A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a c f x +B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a c e x -A \sqrt {-\frac {b}{a}}\, a d e \,x^{2}-A \sqrt {-\frac {b}{a}}\, b c e \,x^{2}-A \sqrt {-\frac {b}{a}}\, a c e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{\sqrt {-\frac {b}{a}}\, x \,e^{2} c a \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(378\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {A \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{a c e x}-\frac {b A \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{a e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b A \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{a e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) A}{e^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) B}{e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(464\)

Input:

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

Output:

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

integrate((B*x^2+A)/x^2/(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]

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

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

Output:

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

Maxima [F]

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

integrate((B*x^2+A)/x^2/(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^2), x 
)
 

Giac [F]

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

integrate((B*x^2+A)/x^2/(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^2), x 
)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((B*x^2+A)/x^2/(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**2 + c*f*x**4 + d*e*x**4 + 
d*f*x**6),x)