Integrand size = 34, antiderivative size = 616 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\frac {b^2 x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 d f \sqrt {c+d x^2}}-\frac {\sqrt {e} \left (4 a b c d f-2 a^2 d^2 f+b^2 c (d e-3 c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 c d^2 f \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {b \sqrt {e} \left (8 a b c d f-4 a^2 d^2 f-b^2 c (d e+3 c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a d^3 f \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {b^2 c \sqrt {e} (b d e+3 b c f-5 a d f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a d^3 f \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:
1/2*b^2*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/d/f/(d*x^2+c)^(1/2)-1/2*e^(1/2)* (4*a*b*c*d*f-2*a^2*d^2*f+b^2*c*(-3*c*f+d*e))*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/ e/(d*x^2+c))^(1/2)*EllipticE((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^(1/2),(- (-a*d+b*c)*e/a/(-c*f+d*e))^(1/2))/c/d^2/f/(-c*f+d*e)^(1/2)/(c*(b*x^2+a)/a/ (d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)-1/2*b*e^(1/2)*(8*a*b*c*d*f-4*a^2*d^2*f-b^ 2*c*(3*c*f+d*e))*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)*EllipticF ((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^(1/2),(-(-a*d+b*c)*e/a/(-c*f+d*e))^( 1/2))/a/d^3/f/(-c*f+d*e)^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^( 1/2)-1/2*b^2*c*e^(1/2)*(-5*a*d*f+3*b*c*f+b*d*e)*(b*x^2+a)^(1/2)*(c*(f*x^2+ e)/e/(d*x^2+c))^(1/2)*EllipticPi((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^(1/2 ),d*e/(-c*f+d*e),(-(-a*d+b*c)*e/a/(-c*f+d*e))^(1/2))/a/d^3/f/(-c*f+d*e)^(1 /2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx \] Input:
Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(3/2)*Sqrt[e + f*x^2]),x]
Output:
Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(3/2)*Sqrt[e + f*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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx\) |
| \(\Big \downarrow \) 434 |
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}dx\) |
Input:
Int[(a + b*x^2)^(5/2)/((c + d*x^2)^(3/2)*Sqrt[e + f*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]
\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{\left (x^{2} d +c \right )^{\frac {3}{2}} \sqrt {f \,x^{2}+e}}d x\]
Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x)
Output:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x, algorithm="fr icas")
Output:
integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt( f*x^2 + e)/(d^2*f*x^6 + (d^2*e + 2*c*d*f)*x^4 + c^2*e + (2*c*d*e + c^2*f)* x^2), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{\left (c + d x^{2}\right )^{\frac {3}{2}} \sqrt {e + f x^{2}}}\, dx \] Input:
integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(3/2)/(f*x**2+e)**(1/2),x)
Output:
Integral((a + b*x**2)**(5/2)/((c + d*x**2)**(3/2)*sqrt(e + f*x**2)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x, algorithm="ma xima")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(3/2)*sqrt(f*x^2 + e)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x, algorithm="gi ac")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(3/2)*sqrt(f*x^2 + e)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{3/2}\,\sqrt {f\,x^2+e}} \,d x \] Input:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(1/2)),x)
Output:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(1/2)), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x)
Output:
(3*sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*x - 5*int((sq rt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a**2*c**2*d*e*f + a**2*c**2*d*f**2*x**2 + 2*a**2*c*d**2*e*f*x**2 + 2*a**2*c*d**2*f**2*x**4 + a**2*d**3*e*f*x**4 + a**2*d**3*f**2*x**6 + 3*a*b*c**3*e*f + 3*a*b*c**3*f* *2*x**2 + a*b*c**2*d*e**2 + 8*a*b*c**2*d*e*f*x**2 + 7*a*b*c**2*d*f**2*x**4 + 2*a*b*c*d**2*e**2*x**2 + 7*a*b*c*d**2*e*f*x**4 + 5*a*b*c*d**2*f**2*x**6 + a*b*d**3*e**2*x**4 + 2*a*b*d**3*e*f*x**6 + a*b*d**3*f**2*x**8 + 3*b**2* c**3*e*f*x**2 + 3*b**2*c**3*f**2*x**4 + b**2*c**2*d*e**2*x**2 + 7*b**2*c** 2*d*e*f*x**4 + 6*b**2*c**2*d*f**2*x**6 + 2*b**2*c*d**2*e**2*x**4 + 5*b**2* c*d**2*e*f*x**6 + 3*b**2*c*d**2*f**2*x**8 + b**2*d**3*e**2*x**6 + b**2*d** 3*e*f*x**8),x)*a**2*b**3*c*d**2*f**2 - 5*int((sqrt(e + f*x**2)*sqrt(c + d* x**2)*sqrt(a + b*x**2)*x**6)/(a**2*c**2*d*e*f + a**2*c**2*d*f**2*x**2 + 2* a**2*c*d**2*e*f*x**2 + 2*a**2*c*d**2*f**2*x**4 + a**2*d**3*e*f*x**4 + a**2 *d**3*f**2*x**6 + 3*a*b*c**3*e*f + 3*a*b*c**3*f**2*x**2 + a*b*c**2*d*e**2 + 8*a*b*c**2*d*e*f*x**2 + 7*a*b*c**2*d*f**2*x**4 + 2*a*b*c*d**2*e**2*x**2 + 7*a*b*c*d**2*e*f*x**4 + 5*a*b*c*d**2*f**2*x**6 + a*b*d**3*e**2*x**4 + 2* a*b*d**3*e*f*x**6 + a*b*d**3*f**2*x**8 + 3*b**2*c**3*e*f*x**2 + 3*b**2*c** 3*f**2*x**4 + b**2*c**2*d*e**2*x**2 + 7*b**2*c**2*d*e*f*x**4 + 6*b**2*c**2 *d*f**2*x**6 + 2*b**2*c*d**2*e**2*x**4 + 5*b**2*c*d**2*e*f*x**6 + 3*b**2*c *d**2*f**2*x**8 + b**2*d**3*e**2*x**6 + b**2*d**3*e*f*x**8),x)*a**2*b**...