Integrand size = 34, antiderivative size = 666 \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx=\frac {b (b d e-3 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{8 d^2 f \sqrt {a+b x^2}}+\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{4 d}-\frac {\sqrt {b c-a d} e (b d e-3 b c f+5 a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 \sqrt {c} d^2 f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {3 a \sqrt {b c-a d} (b d e-b c f+a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b \sqrt {c} d^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a \left (3 a^2 d^2 f^2+6 a b d f (d e-c f)-b^2 \left (d^2 e^2+2 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b \sqrt {c} d^2 \sqrt {b c-a d} f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:
1/8*b*(5*a*d*f-3*b*c*f+b*d*e)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d^2/f/(b*x ^2+a)^(1/2)+1/4*b*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*(5*a*d*f-3*b*c*f+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))/c^(1/2)/d^2/f/(a*(d*x^2+c)/c/(b*x^2+a))^(1/ 2)/(f*x^2+e)^(1/2)+3/8*a*(-a*d+b*c)^(1/2)*(a*d*f-b*c*f+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/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*(3*a^2*d^2*f^2+6*a*b*d*f*(-c*f +d*e)-b^2*(-3*c^2*f^2+2*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)*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/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)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx \] Input:
Integrate[((a + b*x^2)^(3/2)*Sqrt[e + f*x^2])/Sqrt[c + d*x^2],x]
Output:
Integrate[((a + b*x^2)^(3/2)*Sqrt[e + f*x^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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 434 |
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}}dx\) |
Input:
Int[((a + b*x^2)^(3/2)*Sqrt[e + f*x^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]
\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {f \,x^{2}+e}}{\sqrt {x^{2} d +c}}d x\]
Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(1/2),x)
Output:
int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(1/2),x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {e + f x^{2}}}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(f*x**2+e)**(1/2)/(d*x**2+c)**(1/2),x)
Output:
Integral((a + b*x**2)**(3/2)*sqrt(e + f*x**2)/sqrt(c + d*x**2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="ma xima")
Output:
integrate((b*x^2 + a)^(3/2)*sqrt(f*x^2 + e)/sqrt(d*x^2 + c), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="gi ac")
Output:
integrate((b*x^2 + a)^(3/2)*sqrt(f*x^2 + e)/sqrt(d*x^2 + c), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {f\,x^2+e}}{\sqrt {d\,x^2+c}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(1/2),x)
Output:
int(((a + b*x^2)^(3/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(1/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b x +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 ) a b d f -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^{2} c f +\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^{2} d e +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 ) a^{2} d 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 b c f +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 b d e -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^{2} c e +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^{2} d e -\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 b c e}{4 d} \] Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(1/2),x)
Output:
(sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*x + 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 )*a*b*d*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*f + 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*e + 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)*a**2*d*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*b*c*f + 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*b*d *e - 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**2*c*e + 4*int((sqrt(e + f*x**2)*sqrt(c + d*x**2) *sqrt(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**2*d*e - int((sqrt(e +...