Integrand size = 34, antiderivative size = 564 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=-\frac {(b c-a d) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{5 c (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac {2 (2 a d (d e-2 c f)+b c (d e+c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{15 c^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (2 b^2 c^2 e (d e-5 c f)+a b c \left (3 d^2 e^2-7 c d e f+20 c^2 f^2\right )-a^2 d \left (8 d^2 e^2-23 c d e f+23 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 c^3 (b c-a d) (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (b e-a f) (b c (d e-5 c f)-4 a d (d e-2 c f)) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{15 c^2 (b c-a d) (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:
-1/5*(-a*d+b*c)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/(-c*f+d*e)/(d*x^2+c)^( 5/2)+2/15*(2*a*d*(-2*c*f+d*e)+b*c*(c*f+d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^( 1/2)/c^2/(-c*f+d*e)^2/(d*x^2+c)^(3/2)+1/15*e^(1/2)*(2*b^2*c^2*e*(-5*c*f+d* e)+a*b*c*(20*c^2*f^2-7*c*d*e*f+3*d^2*e^2)-a^2*d*(23*c^2*f^2-23*c*d*e*f+8*d ^2*e^2))*(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^ 3/(-a*d+b*c)/(-c*f+d*e)^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1 /2)-1/15*e^(1/2)*(-a*f+b*e)*(b*c*(-5*c*f+d*e)-4*a*d*(-2*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))/c^2/(-a*d+b*c)/(-c* f+d*e)^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx \] Input:
Integrate[(a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*Sqrt[e + f*x^2]),x]
Output:
Integrate[(a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx\) |
\(\Big \downarrow \) 434 |
\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}}dx\) |
Input:
Int[(a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*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]
\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (x^{2} d +c \right )^{\frac {7}{2}} \sqrt {f \,x^{2}+e}}d x\]
Input:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x)
Output:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x, algorithm="fr icas")
Output:
integral((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(d^4*f*x^10 + ( d^4*e + 4*c*d^3*f)*x^8 + 2*(2*c*d^3*e + 3*c^2*d^2*f)*x^6 + c^4*e + 2*(3*c^ 2*d^2*e + 2*c^3*d*f)*x^4 + (4*c^3*d*e + c^4*f)*x^2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(3/2)/(d*x**2+c)**(7/2)/(f*x**2+e)**(1/2),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x, algorithm="ma xima")
Output:
integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^(7/2)*sqrt(f*x^2 + e)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x, algorithm="gi ac")
Output:
integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^(7/2)*sqrt(f*x^2 + e)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{7/2}\,\sqrt {f\,x^2+e}} \,d x \] Input:
int((a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(1/2)),x)
Output:
int((a + b*x^2)^(3/2)/((c + d*x^2)^(7/2)*(e + f*x^2)^(1/2)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{\frac {7}{2}} \sqrt {f \,x^{2}+e}}d x \] Input:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x)
Output:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(7/2)/(f*x^2+e)^(1/2),x)