Integrand size = 36, antiderivative size = 771 \[ \int \frac {x^8}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=-\frac {x \sqrt {c+d x^2}}{b d f^2 \sqrt {a-b x^2}}+\frac {\left (b^3 c e^2 (3 d e-2 c f)+4 a^3 d f^2 (d e-c f)+a b^2 e \left (3 d^2 e^2+2 c d e f-4 c^2 f^2\right )+2 a^2 b f \left (2 d^2 e^2-c d e f-c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{2 b d (b c+a d) f^2 (b e+a f)^2 (d e-c f) \sqrt {a-b x^2}}-\frac {e^3 x \sqrt {c+d x^2}}{2 f^2 (b e+a f) (d e-c f) \sqrt {a-b x^2} \left (e+f x^2\right )}-\frac {\sqrt {a} \left (b^3 c e^2 (3 d e-2 c f)+4 a^3 d f^2 (d e-c f)+a b^2 e \left (3 d^2 e^2+2 c d e f-4 c^2 f^2\right )+2 a^2 b f \left (2 d^2 e^2-c d e f-c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{2 b^{3/2} d (b c+a d) f^2 (b e+a f)^2 (d e-c f) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} \left (2 a^2 c f^3+b^2 e^2 (3 d e+2 c f)+2 a b e f (3 d e+2 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{2 b^{3/2} d f^3 (b e+a f)^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} e^2 (a f (6 d e-7 c f)+b e (3 d e-4 c f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{2 \sqrt {b} f^3 (b e+a f)^2 (d e-c f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
-x*(d*x^2+c)^(1/2)/b/d/f^2/(-b*x^2+a)^(1/2)+1/2*(b^3*c*e^2*(-2*c*f+3*d*e)+ 4*a^3*d*f^2*(-c*f+d*e)+a*b^2*e*(-4*c^2*f^2+2*c*d*e*f+3*d^2*e^2)+2*a^2*b*f* (-c^2*f^2-c*d*e*f+2*d^2*e^2))*x*(d*x^2+c)^(1/2)/b/d/(a*d+b*c)/f^2/(a*f+b*e )^2/(-c*f+d*e)/(-b*x^2+a)^(1/2)-1/2*e^3*x*(d*x^2+c)^(1/2)/f^2/(a*f+b*e)/(- c*f+d*e)/(-b*x^2+a)^(1/2)/(f*x^2+e)-1/2*a^(1/2)*(b^3*c*e^2*(-2*c*f+3*d*e)+ 4*a^3*d*f^2*(-c*f+d*e)+a*b^2*e*(-4*c^2*f^2+2*c*d*e*f+3*d^2*e^2)+2*a^2*b*f* (-c^2*f^2-c*d*e*f+2*d^2*e^2))*(1-b*x^2/a)^(1/2)*(d*x^2+c)^(1/2)*EllipticE( b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(3/2)/d/(a*d+b*c)/f^2/(a*f+b*e)^2/(- c*f+d*e)/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)+1/2*a^(1/2)*(2*a^2*c*f^3+b^2*e ^2*(2*c*f+3*d*e)+2*a*b*e*f*(2*c*f+3*d*e))*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1 /2)*EllipticF(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(3/2)/d/f^3/(a*f+b*e)^ 2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)-1/2*a^(1/2)*e^2*(a*f*(-7*c*f+6*d*e)+b*e *(-4*c*f+3*d*e))*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticPi(b^(1/2)*x/ a^(1/2),-a*f/b/e,(-a*d/b/c)^(1/2))/b^(1/2)/f^3/(a*f+b*e)^2/(-c*f+d*e)/(-b* x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 10.23 (sec) , antiderivative size = 567, normalized size of antiderivative = 0.74 \[ \int \frac {x^8}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {\sqrt {-\frac {b}{a}} d f^2 x \left (c+d x^2\right ) \left (a^2 b d e^3-b^3 c e^3 x^2+a b^2 e^3 \left (c-d x^2\right )+2 a^3 f (-d e+c f) \left (e+f x^2\right )\right )+i c f \left (4 a^3 d f^2 (-d e+c f)+b^3 c e^2 (-3 d e+2 c f)+2 a^2 b f \left (-2 d^2 e^2+c d e f+c^2 f^2\right )+a b^2 e \left (-3 d^2 e^2-2 c d e f+4 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i (b c+a d) (-d e+c f) \left (2 a^2 c f^3+b^2 e^2 (3 d e+2 c f)+2 a b e f (3 d e+2 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+i b d (b c+a d) e^2 (b e (-3 d e+4 c f)+a f (-6 d e+7 c f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{2 b \sqrt {-\frac {b}{a}} d (b c+a d) f^3 (b e+a f)^2 (-d e+c f) \sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[x^8/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
(Sqrt[-(b/a)]*d*f^2*x*(c + d*x^2)*(a^2*b*d*e^3 - b^3*c*e^3*x^2 + a*b^2*e^3 *(c - d*x^2) + 2*a^3*f*(-(d*e) + c*f)*(e + f*x^2)) + I*c*f*(4*a^3*d*f^2*(- (d*e) + c*f) + b^3*c*e^2*(-3*d*e + 2*c*f) + 2*a^2*b*f*(-2*d^2*e^2 + c*d*e* f + c^2*f^2) + a*b^2*e*(-3*d^2*e^2 - 2*c*d*e*f + 4*c^2*f^2))*Sqrt[1 - (b*x ^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x] , -((a*d)/(b*c))] - I*(b*c + a*d)*(-(d*e) + c*f)*(2*a^2*c*f^3 + b^2*e^2*(3 *d*e + 2*c*f) + 2*a*b*e*f*(3*d*e + 2*c*f))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d *x^2)/c]*(e + f*x^2)*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] + I*b*d*(b*c + a*d)*e^2*(b*e*(-3*d*e + 4*c*f) + a*f*(-6*d*e + 7*c*f))*Sqrt [1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticPi[-((a*f)/(b*e)), I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(2*b*Sqrt[-(b/a)]*d*(b*c + a* d)*f^3*(b*e + a*f)^2*(-(d*e) + c*f)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f *x^2))
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.
| \(\displaystyle \int \frac {x^8}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^8}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}dx\) |
Input:
Int[x^8/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
$Aborted
Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ (q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q, r}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(1757\) vs. \(2(704)=1408\).
Time = 11.16 (sec) , antiderivative size = 1758, normalized size of antiderivative = 2.28
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1758\) |
| default | \(\text {Expression too large to display}\) | \(4457\) |
Input:
int(x^8/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x,method=_RETURNVERBO SE)
Output:
((-b*x^2+a)*(d*x^2+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-(-b*d*x^2- b*c)/b^2*a^3/(a*d+b*c)*x/(a*f+b*e)^2/((x^2-a/b)*(-b*d*x^2-b*c))^(1/2)+1/2/ f/(a*c*f^2-a*d*e*f+b*c*e*f-b*d*e^2)*e^3*x/(a*f+b*e)*(-b*d*x^4+a*d*x^2-b*c* x^2+a*c)^(1/2)/(f*x^2+e)-1/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2) /(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c )/c/b)^(1/2))/f^2/b^2*a+2/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/ (-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c) /c/b)^(1/2))/f^3/b*e+1/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b *d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/ b)^(1/2))/b^2*a^3/(a*f+b*e)^2-1/2*c/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2 /c)^(1/2)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)*b/f^2*e^3/(a*c*f^2-a*d*e*f+ b*c*e*f-b*d*e^2)/(a*f+b*e)*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2 ))+1/2*c/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a*d*x^2 -b*c*x^2+a*c)^(1/2)*b/f^2*e^3/(a*c*f^2-a*d*e*f+b*c*e*f-b*d*e^2)/(a*f+b*e)* EllipticE(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))+1/2/(b/a)^(1/2)*(1-b*x^2 /a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)*EllipticF (x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))*d*b*e^4/f^3/(a*c*f^2-a*d*e*f+b*c* e*f-b*d*e^2)/(a*f+b*e)-c/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/( -b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)*a^3/b/(a*d+b*c)/(a*f+b*e)^2*EllipticE( x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))-7/2*e^2/(a*c*f^2-a*d*e*f+b*c*e*...
Timed out. \[ \int \frac {x^8}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^8/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="f ricas")
Output:
Timed out
Timed out. \[ \int \frac {x^8}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x**8/(-b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**2,x)
Output:
Timed out
\[ \int \frac {x^8}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {x^{8}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^8/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="m axima")
Output:
integrate(x^8/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
\[ \int \frac {x^8}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {x^{8}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^8/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="g iac")
Output:
integrate(x^8/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
Timed out. \[ \int \frac {x^8}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {x^8}{{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(x^8/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2),x)
Output:
int(x^8/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2), x)
\[ \int \frac {x^8}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {x^{8}}{\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{2}}d x \] Input:
int(x^8/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)
Output:
int(x^8/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)