\(\int \frac {\sqrt {2+b x^2}}{\sqrt {3+d x^2}} \, dx\) [232]

Optimal result

Integrand size = 23, antiderivative size = 182 \[ \int \frac {\sqrt {2+b x^2}}{\sqrt {3+d x^2}} \, dx=\frac {x \sqrt {2+b x^2}}{\sqrt {3+d x^2}}-\frac {\sqrt {2} \sqrt {2+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}+\frac {\sqrt {2} \sqrt {2+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right ),1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}} \] Output:

x*(b*x^2+2)^(1/2)/(d*x^2+3)^(1/2)-2^(1/2)*(b*x^2+2)^(1/2)*EllipticE(d^(1/2 
)*x*3^(1/2)/(3*d*x^2+9)^(1/2),1/2*(4-6*b/d)^(1/2))/d^(1/2)/((b*x^2+2)/(d*x 
^2+3))^(1/2)/(d*x^2+3)^(1/2)+2^(1/2)*(b*x^2+2)^(1/2)*InverseJacobiAM(arcta 
n(1/3*d^(1/2)*x*3^(1/2)),1/2*(4-6*b/d)^(1/2))/d^(1/2)/((b*x^2+2)/(d*x^2+3) 
)^(1/2)/(d*x^2+3)^(1/2)
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {2+b x^2}}{\sqrt {3+d x^2}} \, dx=\frac {\sqrt {2} E\left (\arcsin \left (\frac {\sqrt {-d} x}{\sqrt {3}}\right )|\frac {3 b}{2 d}\right )}{\sqrt {-d}} \] Input:

Integrate[Sqrt[2 + b*x^2]/Sqrt[3 + d*x^2],x]
 

Output:

(Sqrt[2]*EllipticE[ArcSin[(Sqrt[-d]*x)/Sqrt[3]], (3*b)/(2*d)])/Sqrt[-d]
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {324, 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.

\(\displaystyle \int \frac {\sqrt {b x^2+2}}{\sqrt {d x^2+3}} \, dx\)

\(\Big \downarrow \) 324

\(\displaystyle 2 \int \frac {1}{\sqrt {b x^2+2} \sqrt {d x^2+3}}dx+b \int \frac {x^2}{\sqrt {b x^2+2} \sqrt {d x^2+3}}dx\)

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

Int[Sqrt[2 + b*x^2]/Sqrt[3 + d*x^2],x]
 

Output:

b*((x*Sqrt[2 + b*x^2])/(b*Sqrt[3 + d*x^2]) - (Sqrt[2]*Sqrt[2 + b*x^2]*Elli 
pticE[ArcTan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(b*Sqrt[d]*Sqrt[(2 + 
b*x^2)/(3 + d*x^2)]*Sqrt[3 + d*x^2])) + (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticF 
[ArcTan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(Sqrt[d]*Sqrt[(2 + b*x^2)/ 
(3 + d*x^2)]*Sqrt[3 + d*x^2])
 

Defintions of rubi rules used

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 324

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b   Int[x^2/(Sqr 
t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c 
] && PosQ[b/a]
 

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]
 

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.20

method	result	size
default	\(\frac {\operatorname {EllipticE}\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {b}{d}}}{2}\right ) \sqrt {2}}{\sqrt {-d}}\)	\(37\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+2\right ) \left (x^{2} d +3\right )}\, \left (\frac {\sqrt {3 x^{2} d +9}\, \sqrt {2 b \,x^{2}+4}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )}{\sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 x^{2} d +6}}-\frac {\sqrt {3 x^{2} d +9}\, \sqrt {2 b \,x^{2}+4}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )\right )}{\sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 x^{2} d +6}}\right )}{\sqrt {b \,x^{2}+2}\, \sqrt {x^{2} d +3}}\)	\(219\)

Input:

int((b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

EllipticE(1/3*x*3^(1/2)*(-d)^(1/2),1/2*3^(1/2)*2^(1/2)*(b/d)^(1/2))*2^(1/2 
)/(-d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {2+b x^2}}{\sqrt {3+d x^2}} \, dx=-\frac {9 \, \sqrt {3} \sqrt {b d} b x \sqrt {-\frac {1}{d}} E(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{d}}}{x}\right )\,|\,\frac {2 \, d}{3 \, b}) - \sqrt {3} \sqrt {b d} {\left (2 \, d^{2} + 9 \, b\right )} x \sqrt {-\frac {1}{d}} F(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{d}}}{x}\right )\,|\,\frac {2 \, d}{3 \, b}) - 3 \, \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3} b d}{3 \, b d^{2} x} \] Input:

integrate((b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x, algorithm="fricas")
 

Output:

-1/3*(9*sqrt(3)*sqrt(b*d)*b*x*sqrt(-1/d)*elliptic_e(arcsin(sqrt(3)*sqrt(-1 
/d)/x), 2/3*d/b) - sqrt(3)*sqrt(b*d)*(2*d^2 + 9*b)*x*sqrt(-1/d)*elliptic_f 
(arcsin(sqrt(3)*sqrt(-1/d)/x), 2/3*d/b) - 3*sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3 
)*b*d)/(b*d^2*x)
 

Sympy [F]

\[ \int \frac {\sqrt {2+b x^2}}{\sqrt {3+d x^2}} \, dx=\int \frac {\sqrt {b x^{2} + 2}}{\sqrt {d x^{2} + 3}}\, dx \] Input:

integrate((b*x**2+2)**(1/2)/(d*x**2+3)**(1/2),x)
 

Output:

Integral(sqrt(b*x**2 + 2)/sqrt(d*x**2 + 3), x)
 

Maxima [F]

\[ \int \frac {\sqrt {2+b x^2}}{\sqrt {3+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + 2}}{\sqrt {d x^{2} + 3}} \,d x } \] Input:

integrate((b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*x^2 + 2)/sqrt(d*x^2 + 3), x)
 

Giac [F]

\[ \int \frac {\sqrt {2+b x^2}}{\sqrt {3+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + 2}}{\sqrt {d x^{2} + 3}} \,d x } \] Input:

integrate((b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*x^2 + 2)/sqrt(d*x^2 + 3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2+b x^2}}{\sqrt {3+d x^2}} \, dx=\int \frac {\sqrt {b\,x^2+2}}{\sqrt {d\,x^2+3}} \,d x \] Input:

int((b*x^2 + 2)^(1/2)/(d*x^2 + 3)^(1/2),x)
 

Output:

int((b*x^2 + 2)^(1/2)/(d*x^2 + 3)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {2+b x^2}}{\sqrt {3+d x^2}} \, dx=\int \frac {\sqrt {d \,x^{2}+3}\, \sqrt {b \,x^{2}+2}}{d \,x^{2}+3}d x \] Input:

int((b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int((sqrt(d*x**2 + 3)*sqrt(b*x**2 + 2))/(d*x**2 + 3),x)