Integrand size = 34, antiderivative size = 858 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\frac {\left (38 c e-\frac {22 a d e}{b}+\frac {3 d e^2}{f}-\frac {22 a c f}{b}+\frac {3 c^2 f}{d}+\frac {15 a^2 d f}{b^2}\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{48 \sqrt {a+b x^2}}-\frac {(5 a d f-7 b (d e+c f)) x \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{24 b^2}+\frac {d f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{6 b}-\frac {\sqrt {b c-a d} e \left (15 a^2 d^2 f^2-22 a b d f (d e+c f)+b^2 \left (3 d^2 e^2+38 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{48 b^3 \sqrt {c} d f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {b c-a d} \left (48 b^3 c d e^2-15 a^3 d^2 f^2+6 a^2 b d f (7 d e+2 c f)-a b^2 \left (31 d^2 e^2+44 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{48 b^4 \sqrt {c} d \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a (b d e+b c f-a d f) \left (5 a^2 d^2 f^2-4 a b d f (d e+c f)-b^2 \left (d^2 e^2-10 c d e f+c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{16 b^4 \sqrt {c} d \sqrt {b c-a d} f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:
1/48*(38*c*e-22*a*d*e/b+3*d*e^2/f-22*a*c*f/b+3*c^2*f/d+15*a^2*d*f/b^2)*x*( d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(1/2)-1/24*(5*a*d*f-7*b*(c*f+d*e) )*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b^2+1/6*d*f*x^3*(b*x^2 +a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b-1/48*(-a*d+b*c)^(1/2)*e*(15*a^ 2*d^2*f^2-22*a*b*d*f*(c*f+d*e)+b^2*(3*c^2*f^2+38*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^3/c^(1/2)/d/f/( a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/48*(-a*d+b*c)^(1/2)*(48*b ^3*c*d*e^2-15*a^3*d^2*f^2+6*a^2*b*d*f*(2*c*f+7*d*e)-a*b^2*(-3*c^2*f^2+44*c *d*e*f+31*d^2*e^2))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*Ellipt icF((-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^4/c^(1/2)/d/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/16 *a*(-a*d*f+b*c*f+b*d*e)*(5*a^2*d^2*f^2-4*a*b*d*f*(c*f+d*e)-b^2*(c^2*f^2-10 *c*d*e*f+d^2*e^2))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*Ellipti cPi((-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^4/c^(1/2)/d/(-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)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx \] Input:
Integrate[((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/Sqrt[a + b*x^2],x]
Output:
Integrate[((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/Sqrt[a + b*x^2], x]
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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 434 |
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}}dx\) |
Input:
Int[((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/Sqrt[a + b*x^2],x]
Output:
$Aborted
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]
\[\int \frac {\left (x^{2} d +c \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\sqrt {b \,x^{2}+a}}d x\]
Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2),x)
Output:
int((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2),x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}{\sqrt {a + b x^{2}}}\, dx \] Input:
integrate((d*x**2+c)**(3/2)*(f*x**2+e)**(3/2)/(b*x**2+a)**(1/2),x)
Output:
Integral((c + d*x**2)**(3/2)*(e + f*x**2)**(3/2)/sqrt(a + b*x**2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="ma xima")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)/sqrt(b*x^2 + a), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="gi ac")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)/sqrt(b*x^2 + a), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{3/2}}{\sqrt {b\,x^2+a}} \,d x \] Input:
int(((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(a + b*x^2)^(1/2),x)
Output:
int(((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2))/(a + b*x^2)^(1/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\text {too large to display} \] Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)^(3/2)/(b*x^2+a)^(1/2),x)
Output:
( - 5*sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f*x + 7*sqrt( e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*f*x + 7*sqrt(e + f*x**2) *sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*x + 4*sqrt(e + f*x**2)*sqrt(c + d *x**2)*sqrt(a + b*x**2)*b*d*f*x**3 + 15*int((sqrt(e + f*x**2)*sqrt(c + d*x **2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a**2*d**2*f**2 - 2 2*int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a *c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*b*c*d*f**2 - 22*int((sqrt(e + f*x**2)*sqrt(c + d*x**2) *sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b* c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*b*d**2*e*f + 3*int(( sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x* *2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d* f*x**6),x)*b**2*c**2*f**2 + 38*int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt (a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x **2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b**2*c*d*e*f + 3*int((sqrt( e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x** 6),x)*b**2*d**2*e**2 + 10*int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**...