\(\int \frac {A+B x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^{3/2}} \, dx\) [33]

Optimal result

Integrand size = 41, antiderivative size = 332 \[ \int \frac {A+B x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {c (B e-A f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} e \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {(B c-A d) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

c*(-A*f+B*e)*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticE((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))/ 
a^(1/2)/e/(a*f-b*e)^(1/2)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2 
+e))^(1/2)-(-A*d+B*c)*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*Elli 
pticF((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))/a^(1/2)/(a*f-b*e)^(1/2)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a 
/(f*x^2+e))^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

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

Input:

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

Fricas [F]

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

integrate((B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, alg 
orithm="fricas")
 

Output:

integral((B*x^2 + A)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b*d* 
f^2*x^8 + (2*b*d*e*f + (b*c + a*d)*f^2)*x^6 + (b*d*e^2 + a*c*f^2 + 2*(b*c 
+ a*d)*e*f)*x^4 + a*c*e^2 + (2*a*c*e*f + (b*c + a*d)*e^2)*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, alg 
orithm="maxima")
 

Output:

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

Giac [F]

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

integrate((B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, alg 
orithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Reduce [F]

\[ \int \frac {A+B x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d \,f^{2} x^{6}+c \,f^{2} x^{4}+2 d e f \,x^{4}+2 c e f \,x^{2}+d \,e^{2} x^{2}+c \,e^{2}}d x \] Input:

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

Output:

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