Integrand size = 34, antiderivative size = 532 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=-\frac {f x \sqrt {a+b x^2}}{3 e (d e-c f) \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}+\frac {d (3 d e+c f) x \sqrt {a+b x^2}}{3 c e (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {\left (a f \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )-b e \left (3 d^2 e^2+6 c d e f-c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} e^2 \sqrt {-b e+a f} (d e-c f)^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {\left (a d f (9 d e-c f)-b \left (3 d^2 e^2+6 c d e f-c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} e \sqrt {-b e+a f} (d e-c f)^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:
-1/3*f*x*(b*x^2+a)^(1/2)/e/(-c*f+d*e)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2)+1/3* d*(c*f+3*d*e)*x*(b*x^2+a)^(1/2)/c/e/(-c*f+d*e)^2/(d*x^2+c)^(1/2)/(f*x^2+e) ^(1/2)+1/3*(a*f*(-2*c^2*f^2+7*c*d*e*f+3*d^2*e^2)-b*e*(-c^2*f^2+6*c*d*e*f+3 *d^2*e^2))*(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)/e^2/(a*f-b*e)^(1/2)/(-c*f+d*e)^3/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x ^2+e))^(1/2)-1/3*(a*d*f*(-c*f+9*d*e)-b*(-c^2*f^2+6*c*d*e*f+3*d^2*e^2))*(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)/e/(a*f-b *e)^(1/2)/(-c*f+d*e)^3/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx \] Input:
Integrate[Sqrt[a + b*x^2]/((c + d*x^2)^(3/2)*(e + f*x^2)^(5/2)),x]
Output:
Integrate[Sqrt[a + b*x^2]/((c + d*x^2)^(3/2)*(e + f*x^2)^(5/2)), 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.
\(\displaystyle \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 434 |
\(\displaystyle \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}}dx\) |
Input:
Int[Sqrt[a + b*x^2]/((c + d*x^2)^(3/2)*(e + f*x^2)^(5/2)),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]
\[\int \frac {\sqrt {b \,x^{2}+a}}{\left (x^{2} d +c \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x\]
Input:
int((b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x)
Output:
int((b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x, algorithm="fr icas")
Output:
integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(d^2*f^3*x^10 + ( 3*d^2*e*f^2 + 2*c*d*f^3)*x^8 + (3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*x^6 + c^2*e^3 + (d^2*e^3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*x^4 + (2*c*d*e^3 + 3*c^2* e^2*f)*x^2), x)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {a + b x^{2}}}{\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)/(d*x**2+c)**(3/2)/(f*x**2+e)**(5/2),x)
Output:
Integral(sqrt(a + b*x**2)/((c + d*x**2)**(3/2)*(e + f*x**2)**(5/2)), x)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)^(5/2)), x)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)^(5/2)), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}}{{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{5/2}} \,d x \] Input:
int((a + b*x^2)^(1/2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(5/2)),x)
Output:
int((a + b*x^2)^(1/2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(5/2)), x)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {b \,x^{2}+a}}{\left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x \] Input:
int((b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x)
Output:
int((b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(5/2),x)