\(\int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [12]

Optimal result

Integrand size = 35, antiderivative size = 293 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {(2 b c C-3 b B d+2 a C d) x \sqrt {c+d x^2}}{3 b d^2 \sqrt {a+b x^2}}+\frac {C x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d}+\frac {\sqrt {a} (2 b c C-3 b B d+2 a C d) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 b^{3/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {a} (a c C-3 A b d) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{3/2} c d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/3*(-3*B*b*d+2*C*a*d+2*C*b*c)*x*(d*x^2+c)^(1/2)/b/d^2/(b*x^2+a)^(1/2)+1/ 
3*C*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d+1/3*a^(1/2)*(-3*B*b*d+2*C*a*d+2* 
C*b*c)*(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))/b^(3/2)/d^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)- 
1/3*a^(1/2)*(-3*A*b*d+C*a*c)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2 
)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(3/2)/c/d/(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 = 6.78 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} C d x \left (a+b x^2\right ) \left (c+d x^2\right )+i c (2 b c C-3 b B d+2 a C d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i \left (a c C d+b \left (2 c^2 C-3 B c d+3 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 b \sqrt {\frac {b}{a}} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*C*d*x*(a + b*x^2)*(c + d*x^2) + I*c*(2*b*c*C - 3*b*B*d + 2*a*C* 
d)*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*c*C*d + b*(2*c^2*C - 3*B*c*d + 3*A*d^2))*Sqrt[1 + ( 
b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c 
)])/(3*b*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {7293, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(B*x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (2*C*(b*c + a*d)*x*Sqrt[a + b* 
x^2])/(3*b^2*d*Sqrt[c + d*x^2]) + (C*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3 
*b*d) - (B*Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 
1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c 
+ d*x^2]) + (2*Sqrt[c]*C*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqr 
t[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^2*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a 
*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*C*Sqrt[a + b*x^2]*EllipticF[Arc 
Tan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*d^(3/2)*Sqrt[(c*(a + b*x^ 
2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (A*Sqrt[c]*Sqrt[a + b*x^2]*Ellipti 
cF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + 
b*x^2))/(a*(c + d*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 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 6.60 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.07

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {C x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (A -\frac {a c C}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (B -\frac {C \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(314\)
risch	\(\frac {C x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 b d}+\frac {\left (-\frac {\left (3 B b d -2 C a d -2 C b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {3 A b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {C a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b d \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(393\)
default	\(\frac {\left (C \sqrt {-\frac {b}{a}}\, b \,d^{2} x^{5}+C \sqrt {-\frac {b}{a}}\, a \,d^{2} x^{3}+C \sqrt {-\frac {b}{a}}\, b c d \,x^{3}+3 A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,d^{2}-3 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d +3 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d +C \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d +2 C \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}-2 C \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d -2 C \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}+C \sqrt {-\frac {b}{a}}\, a c d x \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 \sqrt {-\frac {b}{a}}\, d^{2} b \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(504\)

Input:

int((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOS 
E)
 

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/3*C/b/d*x*( 
b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(A-1/3*a/b*c/d*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))-(B-1/3*C/b/d*(2*a*d+2*b*c))*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)/d*(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))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {{\left (2 \, C b c^{3} + {\left (2 \, C a - 3 \, B b\right )} c^{2} d\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, C b c^{3} + C a c d^{2} - 3 \, A b d^{3} + {\left (2 \, C a - 3 \, B b\right )} c^{2} d\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (C b c d^{2} x^{2} - 2 \, C b c^{2} d - {\left (2 \, C a - 3 \, B b\right )} c d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, b^{2} c d^{3} x} \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fr 
icas")
 

Output:

1/3*((2*C*b*c^3 + (2*C*a - 3*B*b)*c^2*d)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e 
(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (2*C*b*c^3 + C*a*c*d^2 - 3*A*b*d^3 + ( 
2*C*a - 3*B*b)*c^2*d)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/ 
x), a*d/(b*c)) + (C*b*c*d^2*x^2 - 2*C*b*c^2*d - (2*C*a - 3*B*b)*c*d^2)*sqr 
t(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*c*d^3*x)
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, c x -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a c d +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} d -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b \,c^{2}+3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b d -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,c^{2}}{3 b d} \] Input:

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

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*c*x - 2*int((sqrt(c + d*x**2)*sqrt(a + 
b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*c*d + 3*int((sqr 
t(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4 
),x)*b**2*d - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x* 
*2 + b*c*x**2 + b*d*x**4),x)*b*c**2 + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x 
**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b*d - int((sqrt(c + d*x* 
*2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*c**2)/(3 
*b*d)