\(\int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{(c+d x^2)^{3/2}} \, dx\) [385]

Optimal result

Integrand size = 34, antiderivative size = 478 \[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {a \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{c d \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {(b c-a d) f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{d^2 \sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {b c f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{d^2 \sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {432, 428, 412, 429, 326, 321, 327}

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.

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

\(\Big \downarrow \) 432

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

\(\Big \downarrow \) 428

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

\(\Big \downarrow \) 412

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

\(\Big \downarrow \) 429

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

-(((b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2]*(-((a 
*Sqrt[d*e - c*f]*EllipticE[ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d* 
x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/((b*c - a*d)*Sqrt[e])) + (c*( 
b*e - a*f)*EllipticF[ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])] 
, -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/((b*c - a*d)*Sqrt[e]*Sqrt[d*e - c*f 
])))/(c*d*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])) + (b*e*S 
qrt[a + b*x^2]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*EllipticPi[-((a*f)/(b 
*e - a*f)), ArcSin[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])], (a*( 
d*e - c*f))/(c*(b*e - a*f))])/(Sqrt[a]*d*Sqrt[-(b*e) + a*f]*Sqrt[c + d*x^2 
]*Sqrt[(e*(a + b*x^2))/(a*(e + f*x^2))])
 

Defintions of rubi rules used

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 326

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

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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