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

Optimal result

Integrand size = 39, antiderivative size = 227 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {\sqrt {a} (B c-A d) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {c^{3/2} (B e-A f) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

-a^(1/2)*(-A*d+B*c)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/ 
2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/(-c*f+d*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c 
/(b*x^2+a))^(1/2)+c^(3/2)*(-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/(-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 = 8.31 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (B e \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+(-B e+A f) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} e f \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 204, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(B*Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c 
)/(a*d)])/(a*Sqrt[d]*f*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]*Sqrt[1 + (d*x^2)/c]*Ellipt 
icPi[(a*f)/(b*e), ArcSin[(Sqrt[b]*x)/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*e*f 
*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 = 5.80 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.84

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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