\(\int \sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^{3/2} \, dx\) [420]

Optimal result

Integrand size = 34, antiderivative size = 850 \[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2} \, dx=\frac {\left (8 a e+\frac {8 b c e}{d}+\frac {3 b e^2}{f}-\frac {3 a^2 f}{b}-\frac {3 b c^2 f}{d^2}+\frac {2 a c f}{d}\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{48 \sqrt {a+b x^2}}+\frac {(7 b d e+b c f+a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{24 b d}+\frac {1}{6} f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}+\frac {\sqrt {b c-a d} e \left (3 a^2 d^2 f^2-2 a b d f (4 d e+c f)-b^2 \left (3 d^2 e^2+8 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{48 b^2 \sqrt {c} d^2 f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \sqrt {b c-a d} \left (12 a b d^2 e f-3 a^2 d^2 f^2-b^2 \left (17 d^2 e^2+10 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{48 b^3 \sqrt {c} d^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a \left (a^3 d^3 f^3-b^3 (d e-c f)^3-a^2 b d^2 f^2 (3 d e+c f)+a b^2 d f \left (3 d^2 e^2+6 c d e f-c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{16 b^3 \sqrt {c} d^2 \sqrt {b c-a d} f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/48*(8*a*e+8*b*c*e/d+3*b*e^2/f-3*a^2*f/b-3*b*c^2*f/d^2+2*a*c*f/d)*x*(d*x^ 
2+c)^(1/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(1/2)+1/24*(a*d*f+b*c*f+7*b*d*e)*x*(b 
*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b/d+1/6*f*x^3*(b*x^2+a)^(1/2 
)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)+1/48*(-a*d+b*c)^(1/2)*e*(3*a^2*d^2*f^2-2 
*a*b*d*f*(c*f+4*d*e)-b^2*(-3*c^2*f^2+8*c*d*e*f+3*d^2*e^2))*(d*x^2+c)^(1/2) 
*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticE((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x 
^2+a)^(1/2),(c*(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/b^2/c^(1/2)/d^2/f/(a*(d*x^2 
+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)-1/48*a*(-a*d+b*c)^(1/2)*(12*a*b*d^2 
*e*f-3*a^2*d^2*f^2-b^2*(-3*c^2*f^2+10*c*d*e*f+17*d^2*e^2))*(d*x^2+c)^(1/2) 
*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticF((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x 
^2+a)^(1/2),(c*(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/b^3/c^(1/2)/d^2/(a*(d*x^2+c 
)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/16*a*(a^3*d^3*f^3-b^3*(-c*f+d*e)^3- 
a^2*b*d^2*f^2*(c*f+3*d*e)+a*b^2*d*f*(-c^2*f^2+6*c*d*e*f+3*d^2*e^2))*(d*x^2 
+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticPi((-a*d+b*c)^(1/2)*x/c^ 
(1/2)/(b*x^2+a)^(1/2),b*c/(-a*d+b*c),(c*(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/b^ 
3/c^(1/2)/d^2/(-a*d+b*c)^(1/2)/f/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e) 
^(1/2)
 

Mathematica [F]

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

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

\(\Big \downarrow \) 434

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

Input:

Int[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 434

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2)^(r_.), x_Symbol] :> Unintegrable[(a + b*x^2)^p*(c + d*x^2)^q*(e + f* 
x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x]
 

Maple [F]

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

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]

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

Sympy [F]

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

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

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

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