Integrand size = 34, antiderivative size = 650 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {(5 b d e-3 b c f+a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{8 d^2 \sqrt {a+b x^2}}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{4 d}-\frac {\sqrt {b c-a d} e (5 b d e-3 b c f+a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b \sqrt {c} d^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a \sqrt {b c-a d} f (7 b d e-3 b c f-a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b^2 \sqrt {c} d^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \left (a^2 d^2 f^2-3 b^2 (d e-c f)^2-2 a b d f (3 d e-c f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b^2 \sqrt {c} d^2 \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:
1/8*(a*d*f-3*b*c*f+5*b*d*e)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d^2/(b*x^2+a )^(1/2)+1/4*f*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d-1/8*(-a* d+b*c)^(1/2)*e*(a*d*f-3*b*c*f+5*b*d*e)*(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/c^(1/2)/d^2/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/( f*x^2+e)^(1/2)+1/8*a*(-a*d+b*c)^(1/2)*f*(-a*d*f-3*b*c*f+7*b*d*e)*(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^2/(a*(d *x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)-1/8*a*(a^2*d^2*f^2-3*b^2*(-c*f+ d*e)^2-2*a*b*d*f*(-c*f+3*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^2/(-a*d+b*c)^(1/2)/(a*(d* x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx \] Input:
Integrate[(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2))/Sqrt[c + d*x^2],x]
Output:
Integrate[(Sqrt[a + b*x^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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 434 |
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}}dx\) |
Input:
Int[(Sqrt[a + b*x^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]
\[\int \frac {\sqrt {b \,x^{2}+a}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\sqrt {x^{2} d +c}}d x\]
Input:
int((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x)
Output:
int((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(f*x**2+e)**(3/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(sqrt(a + b*x**2)*(e + f*x**2)**(3/2)/sqrt(c + d*x**2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^(3/2)/sqrt(d*x^2 + c), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^(3/2)/sqrt(d*x^2 + c), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (f\,x^2+e\right )}^{3/2}}{\sqrt {d\,x^2+c}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(1/2),x)
Output:
int(((a + b*x^2)^(1/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(1/2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, f x +\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a d \,f^{2}-3 \left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b c \,f^{2}+5 \left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b d e f -2 \left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a c \,f^{2}+6 \left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a d e f -2 \left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b c e f +4 \left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b d \,e^{2}-\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a c e f +4 \left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a d \,e^{2}}{4 d} \] Input:
int((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x)
Output:
(sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*f*x + 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*d*f**2 - 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*c*f**2 + 5*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*d*e*f - 2*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 + b*c*f*x**4 + b*d*e*x**4 + b*d *f*x**6),x)*a*c*f**2 + 6*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 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*d*e*f - 2*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 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b*c*e *f + 4*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 + b*c*f*x**4 + b*d*e *x**4 + b*d*f*x**6),x)*b*d*e**2 - int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*s qrt(a + b*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 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*c*e*f + 4*int((sqrt(e + ...