\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{(e+f x^2)^{3/2}} \, dx\) [326]

Optimal result

Integrand size = 34, antiderivative size = 481 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx=\frac {\sqrt {a} \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{e f \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {a} b (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{c f^2 \sqrt {-b e+a f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {a} b d e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{c f^2 \sqrt {-b e+a f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \] Output:

a^(1/2)*(a*f-b*e)^(1/2)*(d*x^2+c)^(1/2)*(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)*El 
lipticE((a*f-b*e)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b* 
e))^(1/2))/e/f/(b*x^2+a)^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)-a^(1/2)*b*( 
-c*f+d*e)*(d*x^2+c)^(1/2)*(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)*EllipticF((a*f-b 
*e)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/c/f 
^2/(a*f-b*e)^(1/2)/(b*x^2+a)^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)+a^(1/2) 
*b*d*e*(d*x^2+c)^(1/2)*(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)*EllipticPi((a*f-b*e 
)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),-a*f/(-a*f+b*e),(a*(-c*f+d*e)/c/(-a*f+b* 
e))^(1/2))/c/f^2/(a*f-b*e)^(1/2)/(b*x^2+a)^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e)) 
^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 748, normalized size of antiderivative = 1.56, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {432, 428, 412, 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.

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

\(\Big \downarrow \) 432

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

\(\Big \downarrow \) 428

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

\(\Big \downarrow \) 412

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

\(\Big \downarrow \) 429

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

\(\Big \downarrow \) 324

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

-(((b*e - a*f)*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)*S 
qrt[e + f*x^2]*Sqrt[1 + ((d*e - c*f)*x^2)/(c*(e + f*x^2))]) - (a*Sqrt[c]*S 
qrt[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))]*Ellipt 
icF[ArcTan[(Sqrt[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*f*Sqrt[a + b*x^2]*Sqrt[(e*(c + d*x^2))/(c*(e + 
f*x^2))])) + (b*c*Sqrt[e]*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x 
^2))]*EllipticPi[(d*e)/(d*e - c*f), ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sq 
rt[c + d*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(a*f*Sqrt[d*e - c*f] 
*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*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]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 428

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

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]
 

rule 432

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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