\(\int \frac {A+B x^2}{(a+b x^2) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [14]

Optimal result

Integrand size = 39, antiderivative size = 226 \[ \int \frac {A+B x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {\sqrt {c} (B e-A f) \sqrt {e+f x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {c f}{d e}\right )}{\sqrt {d} e (b e-a f) \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {(A b-a B) e^{3/2} \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {b e}{a f},\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{a c \sqrt {f} (b e-a f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \] Output:

c^(1/2)*(-A*f+B*e)*(f*x^2+e)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2 
)),(1-c*f/d/e)^(1/2))/d^(1/2)/e/(-a*f+b*e)/(d*x^2+c)^(1/2)/(c*(f*x^2+e)/e/ 
(d*x^2+c))^(1/2)+(A*b-B*a)*e^(3/2)*(d*x^2+c)^(1/2)*EllipticPi(f^(1/2)*x/e^ 
(1/2)/(1+f*x^2/e)^(1/2),1-b*e/a/f,(1-d*e/c/f)^(1/2))/a/c/f^(1/2)/(-a*f+b*e 
)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.58 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=-\frac {i \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \left (a B \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )+(A b-a B) \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )}{a b \sqrt {\frac {d}{c}} \sqrt {c+d x^2} \sqrt {e+f x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(B*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e 
)/(c*f)])/(b*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^ 
2]) + ((A*b - a*B)*Sqrt[-c]*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Ellipt 
icPi[(b*c)/(a*d), ArcSin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(a*b*Sqrt[d] 
*Sqrt[c + d*x^2]*Sqrt[e + f*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 = 5.78 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\left (A \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) b +B \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a -B \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a \right ) \sqrt {\frac {f \,x^{2}+e}{e}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}{b a \sqrt {-\frac {d}{c}}\, \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\)	\(191\)
elliptic	\(\frac {\sqrt {\left (x^{2} d +c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {B \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) A}{a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) B}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}\)	\(325\)

Input:

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

Output:

(A*EllipticPi(x*(-1/c*d)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-1/c*d)^(1/2))*b+B*El 
lipticF(x*(-1/c*d)^(1/2),(c*f/d/e)^(1/2))*a-B*EllipticPi(x*(-1/c*d)^(1/2), 
b*c/a/d,(-f/e)^(1/2)/(-1/c*d)^(1/2))*a)*((f*x^2+e)/e)^(1/2)*((d*x^2+c)/c)^ 
(1/2)/b*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a/(-1/c*d)^(1/2)/(d*f*x^4+c*f*x^2+ 
d*e*x^2+c*e)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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