Integrand size = 34, antiderivative size = 711 \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {(b e-a f) x \sqrt {e+f x^2}}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}+\frac {2 \left (b^2 c e+2 a^2 d f-a b (4 d e-c f)\right ) x \sqrt {e+f x^2}}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {d \left (2 b^2 c^2 e-a^2 d (d e-6 c f)-a b c (9 d e-2 c f)\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{3 a^2 c (b c-a d)^3 \left (c+d x^2\right )^{3/2}}+\frac {2 \sqrt {e} \sqrt {d e-c f} \left (b^3 c^3 e-a^2 b c d (5 d e-6 c f)-a b^2 c^2 (5 d e-c f)+a^3 d^2 (d e+c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a^2 c^2 (b c-a d)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} \left (b^3 c^2 e (d e-c f)-a^3 d^2 f (d e+2 c f)+a^2 b d \left (d^2 e^2+17 c d e f-12 c^2 f^2\right )-a b^2 c \left (18 d^2 e^2-17 c d e f+2 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a^2 c (b c-a d)^4 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:
1/3*(-a*f+b*e)*x*(f*x^2+e)^(1/2)/a/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3 /2)+2/3*(b^2*c*e+2*a^2*d*f-a*b*(-c*f+4*d*e))*x*(f*x^2+e)^(1/2)/a^2/(-a*d+b *c)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)+1/3*d*(2*b^2*c^2*e-a^2*d*(-6*c*f+d*e )-a*b*c*(-2*c*f+9*d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/a^2/c/(-a*d+b*c) ^3/(d*x^2+c)^(3/2)+2/3*e^(1/2)*(-c*f+d*e)^(1/2)*(b^3*c^3*e-a^2*b*c*d*(-6*c *f+5*d*e)-a*b^2*c^2*(-c*f+5*d*e)+a^3*d^2*(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))/a^2/c^2/(-a*d+b*c)^4/(c*(b*x^2+a) /a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)-1/3*e^(1/2)*(b^3*c^2*e*(-c*f+d*e)-a^3* d^2*f*(2*c*f+d*e)+a^2*b*d*(-12*c^2*f^2+17*c*d*e*f+d^2*e^2)-a*b^2*c*(2*c^2* f^2-17*c*d*e*f+18*d^2*e^2))*(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/c/(-a*d+b*c)^4/(-c*f+d*e)^(1/2)/(c*(b*x^2+a)/a/(d*x^2 +c))^(1/2)/(f*x^2+e)^(1/2)
\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx \] Input:
Integrate[(e + f*x^2)^(3/2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)),x]
Output:
Integrate[(e + f*x^2)^(3/2)/((a + b*x^2)^(5/2)*(c + d*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.
\(\displaystyle \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 434 |
\(\displaystyle \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}}dx\) |
Input:
Int[(e + f*x^2)^(3/2)/((a + b*x^2)^(5/2)*(c + d*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]
\[\int \frac {\left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (x^{2} d +c \right )^{\frac {5}{2}}}d x\]
Input:
int((f*x^2+e)^(3/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2),x)
Output:
int((f*x^2+e)^(3/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2),x)
\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^(3/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2),x, algorithm="fr icas")
Output:
integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)/(b^3*d^3*x^12 + 3*(b^3*c*d^2 + a*b^2*d^3)*x^10 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3 )*x^8 + (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*x^6 + a^3*c^3 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*x^4 + 3*(a^2*b*c^3 + a^3*c^2*d )*x^2), x)
Timed out. \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((f*x**2+e)**(3/2)/(b*x**2+a)**(5/2)/(d*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^(3/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2),x, algorithm="ma xima")
Output:
integrate((f*x^2 + e)^(3/2)/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(5/2)), x)
\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^(3/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2),x, algorithm="gi ac")
Output:
integrate((f*x^2 + e)^(3/2)/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(5/2)), x)
Timed out. \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^{3/2}}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:
int((e + f*x^2)^(3/2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)),x)
Output:
int((e + f*x^2)^(3/2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)), x)
\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}d x \] Input:
int((f*x^2+e)^(3/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2),x)
Output:
int((f*x^2+e)^(3/2)/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2),x)