Integrand size = 34, antiderivative size = 565 \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {b x \sqrt {e+f x^2}}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {b \left (2 b^2 c e+5 a^2 d f-a b (6 d e+c f)\right ) x \sqrt {e+f x^2}}{3 a^2 (b c-a d)^2 (b e-a f) \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {e} \sqrt {d e-c f} \left (2 b^3 c^2 e+3 a^3 d^2 f-3 a^2 b d (d e-2 c f)-a b^2 c (7 d e+c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a^2 c (b c-a d)^3 (b e-a f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} \left (3 a^2 d^2 f-3 a b d (3 d e-2 c f)+b^2 c (d e-c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a^2 (b c-a d)^3 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:
1/3*b*x*(f*x^2+e)^(1/2)/a/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)+1/3*b *(2*b^2*c*e+5*a^2*d*f-a*b*(c*f+6*d*e))*x*(f*x^2+e)^(1/2)/a^2/(-a*d+b*c)^2/ (-a*f+b*e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+1/3*e^(1/2)*(-c*f+d*e)^(1/2)*(2 *b^3*c^2*e+3*a^3*d^2*f-3*a^2*b*d*(-2*c*f+d*e)-a*b^2*c*(c*f+7*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))/a^2/c/(-a*d+b*c)^3 /(-a*f+b*e)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)-1/3*e^(1/2)*(3 *a^2*d^2*f-3*a*b*d*(-2*c*f+3*d*e)+b^2*c*(-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^2/(-a*d+b*c)^3/(-c*f+d*e)^(1/2) /(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx \] Input:
Integrate[Sqrt[e + f*x^2]/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]
Output:
Integrate[Sqrt[e + f*x^2]/((a + b*x^2)^(5/2)*(c + d*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.
\(\displaystyle \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 434 |
\(\displaystyle \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}dx\) |
Input:
Int[Sqrt[e + f*x^2]/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),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]
\[\int \frac {\sqrt {f \,x^{2}+e}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (x^{2} d +c \right )^{\frac {3}{2}}}d x\]
Input:
int((f*x^2+e)^(1/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x)
Output:
int((f*x^2+e)^(1/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^(1/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x, algorithm="fr icas")
Output:
integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b^3*d^2*x^10 + ( 2*b^3*c*d + 3*a*b^2*d^2)*x^8 + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*x^6 + a^3*c^2 + (3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*x^4 + (3*a^2*b*c^2 + 2*a^ 3*c*d)*x^2), x)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e + f x^{2}}}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((f*x**2+e)**(1/2)/(b*x**2+a)**(5/2)/(d*x**2+c)**(3/2),x)
Output:
Integral(sqrt(e + f*x**2)/((a + b*x**2)**(5/2)*(c + d*x**2)**(3/2)), x)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^(1/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x, algorithm="ma xima")
Output:
integrate(sqrt(f*x^2 + e)/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)), x)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^(1/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x, algorithm="gi ac")
Output:
integrate(sqrt(f*x^2 + e)/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)), x)
Timed out. \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {f\,x^2+e}}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int((e + f*x^2)^(1/2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x)
Output:
int((e + f*x^2)^(1/2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)), x)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {f \,x^{2}+e}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}d x \] Input:
int((f*x^2+e)^(1/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x)
Output:
int((f*x^2+e)^(1/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x)