Integrand size = 34, antiderivative size = 716 \[ \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 a b^2 \left (a+b x^2\right )^{3/2}}+\frac {d^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 b^2 \sqrt {a+b x^2}}+\frac {\sqrt {b c-a d} e \left (4 b^3 c^2 e+15 a^3 d^2 f+2 a b^2 c (3 d e-c f)-a^2 b d (13 d e+10 c f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{6 a^2 b^3 \sqrt {c} (b e-a f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {b c-a d} \left (33 a^2 b d^2 e f-15 a^3 d^2 f^2-2 b^3 c e (d e-c f)-2 a b^2 d e (8 d e+c f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{6 a b^4 \sqrt {c} (b e-a f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a d^2 (b d e+5 b c f-5 a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 b^4 \sqrt {c} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:
1/3*(-a*d+b*c)^2*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a/b^2/(b*x^2+a)^(3/2)+1 /2*d^2*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b^2/(b*x^2+a)^(1/2)+1/6*(-a*d+b*c )^(1/2)*e*(4*b^3*c^2*e+15*a^3*d^2*f+2*a*b^2*c*(-c*f+3*d*e)-a^2*b*d*(10*c*f +13*d*e))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticE((-a*d+ b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/a^ 2/b^3/c^(1/2)/(-a*f+b*e)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)-1 /6*(-a*d+b*c)^(1/2)*(33*a^2*b*d^2*e*f-15*a^3*d^2*f^2-2*b^3*c*e*(-c*f+d*e)- 2*a*b^2*d*e*(c*f+8*d*e))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*E llipticF((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f+b*e)/(-a*d+b* c)/e)^(1/2))/a/b^4/c^(1/2)/(-a*f+b*e)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x ^2+e)^(1/2)+1/2*a*d^2*(-5*a*d*f+5*b*c*f+b*d*e)*(d*x^2+c)^(1/2)*(a*(f*x^2+e )/e/(b*x^2+a))^(1/2)*EllipticPi((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2) ,b*c/(-a*d+b*c),(c*(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/b^4/c^(1/2)/(-a*d+b*c)^ (1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)
\[ \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx \] Input:
Integrate[((c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(a + b*x^2)^(5/2),x]
Output:
Integrate[((c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(a + b*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.
\(\displaystyle \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 434 |
\(\displaystyle \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}}dx\) |
Input:
Int[((c + d*x^2)^(5/2)*Sqrt[e + f*x^2])/(a + b*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]
\[\int \frac {\left (x^{2} d +c \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}}}d x\]
Input:
int((d*x^2+c)^(5/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x)
Output:
int((d*x^2+c)^(5/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x^2+c)^(5/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x, algorithm="fr icas")
Output:
Timed out
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x**2+c)**(5/2)*(f*x**2+e)**(1/2)/(b*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(5/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x, algorithm="ma xima")
Output:
integrate((d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)/(b*x^2 + a)^(5/2), x)
\[ \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(5/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x, algorithm="gi ac")
Output:
integrate((d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)/(b*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}\,\sqrt {f\,x^2+e}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(((c + d*x^2)^(5/2)*(e + f*x^2)^(1/2))/(a + b*x^2)^(5/2),x)
Output:
int(((c + d*x^2)^(5/2)*(e + f*x^2)^(1/2))/(a + b*x^2)^(5/2), x)
\[ \int \frac {\left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:
int((d*x^2+c)^(5/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x)
Output:
int((d*x^2+c)^(5/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^(5/2),x)