3.2.9 \(\int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} (e+f x^2)^{3/2}} \, dx\) [109]

3.2.9.1 Optimal result

Integrand size = 34, antiderivative size = 319 \[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {(d e-c f) x \sqrt {a+b x^2}}{e (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {\sqrt {c} \sqrt {d e-c f} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{e (b e-a f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

output

c^(3/2)*(1/(1+x^2*(-c*f+d*e)/c/(f*x^2+e)))^(1/2)*(1+x^2*(-c*f+d*e)/c/(f*x^ 
2+e))^(1/2)*EllipticF(x*(-c*f+d*e)^(1/2)/c^(1/2)/(f*x^2+e)^(1/2)/(1+x^2*(- 
c*f+d*e)/c/(f*x^2+e))^(1/2),(-(-a*d+b*c)*e/a/(-c*f+d*e))^(1/2))*(b*x^2+a)^ 
(1/2)/a/e/(-c*f+d*e)^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2) 
-(1/(1+x^2*(-c*f+d*e)/c/(f*x^2+e)))^(1/2)*(1+x^2*(-c*f+d*e)/c/(f*x^2+e))^( 
1/2)*EllipticE(x*(-c*f+d*e)^(1/2)/c^(1/2)/(f*x^2+e)^(1/2)/(1+x^2*(-c*f+d*e 
)/c/(f*x^2+e))^(1/2),(-(-a*d+b*c)*e/a/(-c*f+d*e))^(1/2))*c^(1/2)*(-c*f+d*e 
)^(1/2)*(b*x^2+a)^(1/2)/e/(-a*f+b*e)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^ 
2+c)^(1/2)+(-c*f+d*e)*x*(b*x^2+a)^(1/2)/e/(-a*f+b*e)/(d*x^2+c)^(1/2)/(f*x^ 
2+e)^(1/2)
 

3.2.9.2 Mathematica [A] (verified)

Time = 5.17 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {\sqrt {a} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (-d e+c f)}{c (-b e+a f)}\right )}{e \sqrt {-b e+a f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

input

Integrate[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)),x]
 

output

(Sqrt[a]*Sqrt[c + d*x^2]*Sqrt[(e*(a + b*x^2))/(a*(e + f*x^2))]*EllipticE[A 
rcSin[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])], (a*(-(d*e) + c*f) 
)/(c*(-(b*e) + a*f))])/(e*Sqrt[-(b*e) + a*f]*Sqrt[a + b*x^2]*Sqrt[(e*(c + 
d*x^2))/(c*(e + f*x^2))])
 

3.2.9.3 Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.78, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {429, 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.

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

\(\Big \downarrow \) 429

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

\(\Big \downarrow \) 324

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

input

Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)),x]
 

output

(Sqrt[c + d*x^2]*Sqrt[(e*(a + b*x^2))/(a*(e + f*x^2))]*(((d*e - c*f)*((a*x 
*Sqrt[1 + ((b*e - a*f)*x^2)/(a*(e + f*x^2))])/((b*e - a*f)*Sqrt[e + f*x^2] 
*Sqrt[1 + ((d*e - c*f)*x^2)/(c*(e + f*x^2))]) - (a*Sqrt[c]*Sqrt[1 + ((b*e 
- a*f)*x^2)/(a*(e + f*x^2))]*EllipticE[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c] 
*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/((b*e - a*f)*Sqrt 
[d*e - c*f]*Sqrt[(c*(a + ((b*e - a*f)*x^2)/(e + f*x^2)))/(a*(c + ((d*e - c 
*f)*x^2)/(e + f*x^2)))]*Sqrt[1 + ((d*e - c*f)*x^2)/(c*(e + f*x^2))])))/c + 
 (Sqrt[c]*Sqrt[1 + ((b*e - a*f)*x^2)/(a*(e + f*x^2))]*EllipticF[ArcTan[(Sq 
rt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c 
*f)))])/(Sqrt[d*e - c*f]*Sqrt[(c*(a + ((b*e - a*f)*x^2)/(e + f*x^2)))/(a*( 
c + ((d*e - c*f)*x^2)/(e + f*x^2)))]*Sqrt[1 + ((d*e - c*f)*x^2)/(c*(e + f* 
x^2))])))/(e*Sqrt[a + b*x^2]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))])
 

3.2.9.3.1 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]
 

rule 429

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_. 
)*(x_)^2]), x_Symbol] :> Simp[Sqrt[c + d*x^2]*(Sqrt[a*((e + f*x^2)/(e*(a + 
b*x^2)))]/(a*Sqrt[e + f*x^2]*Sqrt[a*((c + d*x^2)/(c*(a + b*x^2)))]))   Subs 
t[Int[Sqrt[1 - (b*c - a*d)*(x^2/c)]/Sqrt[1 - (b*e - a*f)*(x^2/e)], x], x, x 
/Sqrt[a + b*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x]
 

3.2.9.4 Maple [F]

\[\int \frac {\sqrt {d \,x^{2}+c}}{\sqrt {b \,x^{2}+a}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x\]

input

int((d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3/2),x)
 

output

int((d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3/2),x)
 

3.2.9.5 Fricas [F]

\[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="fr 
icas")
 

output

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

3.2.9.6 Sympy [F]

\[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\sqrt {a + b x^{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \]

input

integrate((d*x**2+c)**(1/2)/(b*x**2+a)**(1/2)/(f*x**2+e)**(3/2),x)
 

output

Integral(sqrt(c + d*x**2)/(sqrt(a + b*x**2)*(e + f*x**2)**(3/2)), x)
 

3.2.9.7 Maxima [F]

\[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="ma 
xima")
 

output

integrate(sqrt(d*x^2 + c)/(sqrt(b*x^2 + a)*(f*x^2 + e)^(3/2)), x)
 

3.2.9.8 Giac [F]

\[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="gi 
ac")
 

output

integrate(sqrt(d*x^2 + c)/(sqrt(b*x^2 + a)*(f*x^2 + e)^(3/2)), x)
 

3.2.9.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d\,x^2+c}}{\sqrt {b\,x^2+a}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \]

input

int((c + d*x^2)^(1/2)/((a + b*x^2)^(1/2)*(e + f*x^2)^(3/2)),x)
 

output

int((c + d*x^2)^(1/2)/((a + b*x^2)^(1/2)*(e + f*x^2)^(3/2)), x)