Integrand size = 34, antiderivative size = 866 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {\left (3 a^2 d^2 f^2+2 a b d f (19 d e-11 c f)+b^2 \left (3 d^2 e^2-22 c d e f+15 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{48 d^3 f \sqrt {a+b x^2}}+\frac {(7 b d e-5 b c f+7 a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{24 d^2}+\frac {b f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{6 d}-\frac {\sqrt {b c-a d} e \left (3 a^2 d^2 f^2+2 a b d f (19 d e-11 c f)+b^2 \left (3 d^2 e^2-22 c d e f+15 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{48 b \sqrt {c} d^3 f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \sqrt {b c-a d} \left (3 a^2 d^2 f^2-6 a b d f (5 d e-2 c f)-b^2 \left (17 d^2 e^2-32 c d e f+15 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{48 b^2 \sqrt {c} d^3 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \left (a^3 d^3 f^3-9 a b^2 d f (d e-c f)^2-3 a^2 b d^2 f^2 (3 d e-c f)+b^3 (d e-c f)^2 (d e+5 c f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{16 b^2 \sqrt {c} d^3 \sqrt {b c-a d} f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:
1/48*(3*a^2*d^2*f^2+2*a*b*d*f*(-11*c*f+19*d*e)+b^2*(15*c^2*f^2-22*c*d*e*f+ 3*d^2*e^2))*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d^3/f/(b*x^2+a)^(1/2)+1/24*( 7*a*d*f-5*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) /d^2+1/6*b*f*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d-1/48*(- a*d+b*c)^(1/2)*e*(3*a^2*d^2*f^2+2*a*b*d*f*(-11*c*f+19*d*e)+b^2*(15*c^2*f^2 -22*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)*El lipticE((-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/c^(1/2)/d^3/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)*(3*a^2*d^2*f^2-6*a*b*d*f*(-2*c*f+5*d*e)-b^2*(15* c^2*f^2-32*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^2/c^(1/2)/d^3/(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^3-9*a*b^2*d*f*(-c*f+d*e)^2-3*a^2*b*d^2*f^2*(-c *f+3*d*e)+b^3*(-c*f+d*e)^2*(5*c*f+d*e))*(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^2/c^(1/2)/d^3/(-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)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx \] Input:
Integrate[((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/Sqrt[c + d*x^2],x]
Output:
Integrate[((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/Sqrt[c + d*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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 434 |
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}}dx\) |
Input:
Int[((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/Sqrt[c + d*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]
\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\sqrt {x^{2} d +c}}d x\]
Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x)
Output:
int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(f*x**2+e)**(3/2)/(d*x**2+c)**(1/2),x)
Output:
Integral((a + b*x**2)**(3/2)*(e + f*x**2)**(3/2)/sqrt(c + d*x**2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="ma xima")
Output:
integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^(3/2)/sqrt(d*x^2 + c), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="gi ac")
Output:
integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^(3/2)/sqrt(d*x^2 + c), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{3/2}}{\sqrt {d\,x^2+c}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(1/2),x)
Output:
int(((a + b*x^2)^(3/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(1/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x)
Output:
(7*sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f*x - 5*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)*sq rt(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 + 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)*a**2*d**2*f**2 - 22*in t((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 + 38*int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqr t(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 + 15*int((sqr t(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 - 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)*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 - 14*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**2 +...