Integrand size = 37, antiderivative size = 616 \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b d f \sqrt {e+f x^2}}-\frac {c (a f (d e-c f)-b e (3 d e-c f)) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{2 \sqrt {a} b d f^2 \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {e (b e (3 d e-5 c f)+a f (d e-c f)) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{2 \sqrt {a} b f^3 \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {e (3 b d e+b c f+a d f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{2 \sqrt {a} b d f^3 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:
1/2*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d/f/(f*x^2+e)^(1/2)-1/2*c*(a*f*(-c *f+d*e)-b*e*(-c*f+3*d*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)/b/d/f^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)+1/2*e*(b*e*(-5*c*f+3*d*e)+a*f*(-c*f+d*e))*(b *x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticF((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)/b/f^3/( 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)- 1/2*e*(a*d*f+b*c*f+3*b*d*e)*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2 )*EllipticPi((a*f-b*e)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),-a*f/(-a*f+b*e),(a* (-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(1/2)/b/d/f^3/(a*f-b*e)^(1/2)/(d*x^2+c)^ (1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx \] Input:
Integrate[x^6/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]
Output:
Integrate[x^6/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)), 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.
| \(\displaystyle \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}dx\) |
Input:
Int[x^6/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]
Output:
$Aborted
Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ (q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q, r}, x]
\[\int \frac {x^{6}}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x\]
Input:
int(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
Output:
int(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
Timed out. \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm ="fricas")
Output:
Timed out
\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {x^{6}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**6/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(3/2),x)
Output:
Integral(x**6/(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**(3/2)), x)
\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {x^{6}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm ="maxima")
Output:
integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)), x)
\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {x^{6}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm ="giac")
Output:
integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)), x)
Timed out. \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {x^6}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \] Input:
int(x^6/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(3/2)),x)
Output:
int(x^6/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(3/2)), x)
\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{6}}{b d \,f^{2} x^{8}+a d \,f^{2} x^{6}+b c \,f^{2} x^{6}+2 b d e f \,x^{6}+a c \,f^{2} x^{4}+2 a d e f \,x^{4}+2 b c e f \,x^{4}+b d \,e^{2} x^{4}+2 a c e f \,x^{2}+a d \,e^{2} x^{2}+b c \,e^{2} x^{2}+a c \,e^{2}}d x \] Input:
int(x^6/(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)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2 *x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e**2*x**4 + 2 *b*d*e*f*x**6 + b*d*f**2*x**8),x)