\(\int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [178]

Optimal result

Integrand size = 33, antiderivative size = 209 \[ \int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {e \sqrt {c+d x^2}}{c x \sqrt {a+b x^2}}-\frac {\sqrt {b} e \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} c \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} f \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-e*(d*x^2+c)^(1/2)/c/x/(b*x^2+a)^(1/2)-b^(1/2)*e*(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))/a^(1/2)/c/(b*x^2+a 
)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+a^(1/2)*f*(d*x^2+c)^(1/2)*InverseJ 
acobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/(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 = 0.22 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.95 \[ \int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (-\sqrt {\frac {b}{a}} e \left (a+b x^2\right ) \left (c+d x^2\right )-i b c e x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b e+a f) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{b c x \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {445, 25, 406, 320, 388, 313}

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.

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 406

\(\displaystyle \frac {b d e \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{a c x}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {b d e \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{a c x}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {b d e \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{a c x}\)

\(\Big \downarrow \) 313

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

Input:

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

Output:

-((e*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x)) + (b*d*e*((x*Sqrt[a + b*x^2 
])/(b*Sqrt[c + d*x^2]) - (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])) + (c^(3/2)*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(S 
qrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c 
+ d*x^2))]*Sqrt[c + d*x^2]))/(a*c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 5.53 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.35

method	result	size
default	\(\frac {\left (-\sqrt {-\frac {b}{a}}\, b d e \,x^{4}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c f x -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c e x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c e x -\sqrt {-\frac {b}{a}}\, a d e \,x^{2}-\sqrt {-\frac {b}{a}}\, b c e \,x^{2}-\sqrt {-\frac {b}{a}}\, a c e \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{\sqrt {-\frac {b}{a}}\, x c a \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(283\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{a c x}+\frac {f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(285\)
risch	\(-\frac {e \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{a c x}+\frac {\left (\frac {a c f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b e c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{a c \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(286\)

Input:

int((f*x^2+e)/x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(-(-b/a)^(1/2)*b*d*e*x^4+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF 
(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*c*f*x-((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c) 
^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c*e*x+((b*x^2+a)/a)^(1/ 
2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c*e*x-( 
-b/a)^(1/2)*a*d*e*x^2-(-b/a)^(1/2)*b*c*e*x^2-(-b/a)^(1/2)*a*c*e)*(d*x^2+c) 
^(1/2)*(b*x^2+a)^(1/2)/(-b/a)^(1/2)/x/c/a/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.60 \[ \int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {a c} b^{2} e x \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a b e - {\left (b^{2} e + a^{2} f\right )} \sqrt {a c} x \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c})}{a^{2} b c x} \] Input:

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

Output:

(sqrt(a*c)*b^2*e*x*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) 
- sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a*b*e - (b^2*e + a^2*f)*sqrt(a*c)*x*sqrt 
(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)))/(a^2*b*c*x)
 

Sympy [F]

\[ \int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {e + f x^{2}}{x^{2} \sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c*x**2 + a*d*x**4 + b*c*x**4 + 
b*d*x**6),x)*e + 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)*f