Integrand size = 35, antiderivative size = 576 \[ \int \frac {x^8}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {(3 b d e+2 b c f+4 a d f) x \sqrt {c+d x^2}}{3 b^2 d^2 f^2 \sqrt {a+b x^2}}+\frac {x^3 \sqrt {c+d x^2}}{3 b d f \sqrt {a+b x^2}}+\frac {\sqrt {a} \left (8 a^3 d^2 f^2+b^3 c e (3 d e+2 c f)-a^2 b d f (2 d e+3 c f)-a b^2 \left (3 d^2 e^2+2 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 b^{5/2} d^2 (b c-a d) f^2 (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (4 a^3 c d f^3-b^3 c e^2 (3 d e+c f)-a^2 b c f^2 (5 d e+c f)+a b^2 e \left (3 d^2 e^2+c d e f+2 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{5/2} c d (b c-a d) f^2 (b e-a f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} e^3 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c f^2 (b e-a f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/3*(4*a*d*f+2*b*c*f+3*b*d*e)*x*(d*x^2+c)^(1/2)/b^2/d^2/f^2/(b*x^2+a)^(1/ 2)+1/3*x^3*(d*x^2+c)^(1/2)/b/d/f/(b*x^2+a)^(1/2)+1/3*a^(1/2)*(8*a^3*d^2*f^ 2+b^3*c*e*(2*c*f+3*d*e)-a^2*b*d*f*(3*c*f+2*d*e)-a*b^2*(2*c^2*f^2+3*d^2*e^2 ))*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/ c)^(1/2))/b^(5/2)/d^2/(-a*d+b*c)/f^2/(-a*f+b*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+ c)/c/(b*x^2+a))^(1/2)+1/3*a^(3/2)*(4*a^3*c*d*f^3-b^3*c*e^2*(c*f+3*d*e)-a^2 *b*c*f^2*(c*f+5*d*e)+a*b^2*e*(2*c^2*f^2+c*d*e*f+3*d^2*e^2))*(d*x^2+c)^(1/2 )*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/2)/c/d /(-a*d+b*c)/f^2/(-a*f+b*e)^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/ 2)+a^(3/2)*e^3*(d*x^2+c)^(1/2)*EllipticPi(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1 /2),1-a*f/b/e,(1-a*d/b/c)^(1/2))/b^(1/2)/c/f^2/(-a*f+b*e)^2/(b*x^2+a)^(1/2 )/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 6.05 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.83 \[ \int \frac {x^8}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {i c f \left (8 a^3 d^2 f^2+b^3 c e (3 d e+2 c f)-a^2 b d f (2 d e+3 c f)-a b^2 \left (3 d^2 e^2+2 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i (-b c+a d) \left (4 a^2 c d f^3+a b c f^2 (-d e+2 c f)-b^2 e \left (3 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d \left (\sqrt {\frac {b}{a}} f^2 x \left (c+d x^2\right ) \left (4 a^3 d f+b^3 c e x^2+a b^2 \left (-d e x^2+c \left (e-f x^2\right )\right )-a^2 b \left (c f+d \left (e-f x^2\right )\right )\right )-3 i b^2 d (-b c+a d) e^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 a^2 \left (\frac {b}{a}\right )^{5/2} d^2 (b c-a d) f^3 (b e-a f) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[x^8/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
(I*c*f*(8*a^3*d^2*f^2 + b^3*c*e*(3*d*e + 2*c*f) - a^2*b*d*f*(2*d*e + 3*c*f ) - a*b^2*(3*d^2*e^2 + 2*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c] *EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(-(b*c) + a*d)*(4*a^2* c*d*f^3 + a*b*c*f^2*(-(d*e) + 2*c*f) - b^2*e*(3*d^2*e^2 + 3*c*d*e*f + 2*c^ 2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b /a]*x], (a*d)/(b*c)] + d*(Sqrt[b/a]*f^2*x*(c + d*x^2)*(4*a^3*d*f + b^3*c*e *x^2 + a*b^2*(-(d*e*x^2) + c*(e - f*x^2)) - a^2*b*(c*f + d*(e - f*x^2))) - (3*I)*b^2*d*(-(b*c) + a*d)*e^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*El lipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*a^2*(b/a)^ (5/2)*d^2*(b*c - a*d)*f^3*(b*e - a*f)*Sqrt[a + b*x^2]*Sqrt[c + d*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.
| \(\displaystyle \int \frac {x^8}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^8}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )}dx\) |
Input:
Int[x^8/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^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]
Time = 23.23 (sec) , antiderivative size = 772, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(\frac {x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 f \,b^{2} d}-\frac {\left (-\frac {\left (5 a d f +2 b c f +3 b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}-\frac {\left (3 a^{2} d \,f^{2}-a b c \,f^{2}+3 a b d e f +3 b^{2} d \,e^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{f^{2} b \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {3 f \,a^{4} d \left (-\frac {\left (b d \,x^{2}+b c \right ) x}{a \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {1}{a}+\frac {b c}{\left (a d -b c \right ) a}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{b \left (a f -b e \right )}-\frac {3 b^{2} d \,e^{3} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{f^{2} \left (a f -b e \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 d f \,b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(772\) |
| elliptic | \(\text {Expression too large to display}\) | \(1334\) |
| default | \(\text {Expression too large to display}\) | \(1541\) |
Input:
int(x^8/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
1/3/f/b^2/d*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)-1/3/d/f/b^2*(-1/f*(5*a*d*f+2 *b*c*f+3*b*d*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)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b )^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))-(3*a^2*d*f^2- a*b*c*f^2+3*a*b*d*e*f+3*b^2*d*e^2)/f^2/b/(-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))+3*f/b*a^4*d/(a*f-b*e)*(-(b*d*x^2+b*c)/a/(a*d- b*c)*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+(1/a+b*c/(a*d-b*c)/a)/(-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)*El lipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-b/(a*d-b*c)/a*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 )*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^( 1/2),(-1+(a*d+b*c)/c/b)^(1/2))))-3/f^2*b^2*d*e^3/(a*f-b*e)/(-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)*Elli pticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2)))*((b*x^2+a)*(d* x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Timed out. \[ \int \frac {x^8}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(x^8/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {x^8}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^{8}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(x**8/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral(x**8/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)), x)
\[ \int \frac {x^8}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {x^{8}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^8/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="maxi ma")
Output:
integrate(x^8/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
\[ \int \frac {x^8}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {x^{8}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^8/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="giac ")
Output:
integrate(x^8/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {x^8}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^8}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(x^8/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int(x^8/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {x^8}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {too large to display} \] Input:
int(x^8/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*f*x - 4*sqrt(c + d*x**2)*sqrt( a + b*x**2)*a*d*e*x + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f*x**3 - 2*s qrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*e*x + sqrt(c + d*x**2)*sqrt(a + b*x** 2)*b*d*e*x**3 - 16*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(2*a**3*c* e*f + 2*a**3*c*f**2*x**2 + 2*a**3*d*e*f*x**2 + 2*a**3*d*f**2*x**4 + a**2*b *c*e**2 + 5*a**2*b*c*e*f*x**2 + 4*a**2*b*c*f**2*x**4 + a**2*b*d*e**2*x**2 + 5*a**2*b*d*e*f*x**4 + 4*a**2*b*d*f**2*x**6 + 2*a*b**2*c*e**2*x**2 + 4*a* b**2*c*e*f*x**4 + 2*a*b**2*c*f**2*x**6 + 2*a*b**2*d*e**2*x**4 + 4*a*b**2*d *e*f*x**6 + 2*a*b**2*d*f**2*x**8 + b**3*c*e**2*x**4 + b**3*c*e*f*x**6 + b* *3*d*e**2*x**6 + b**3*d*e*f*x**8),x)*a**4*d**2*f**3 - 2*int((sqrt(c + d*x* *2)*sqrt(a + b*x**2)*x**6)/(2*a**3*c*e*f + 2*a**3*c*f**2*x**2 + 2*a**3*d*e *f*x**2 + 2*a**3*d*f**2*x**4 + a**2*b*c*e**2 + 5*a**2*b*c*e*f*x**2 + 4*a** 2*b*c*f**2*x**4 + a**2*b*d*e**2*x**2 + 5*a**2*b*d*e*f*x**4 + 4*a**2*b*d*f* *2*x**6 + 2*a*b**2*c*e**2*x**2 + 4*a*b**2*c*e*f*x**4 + 2*a*b**2*c*f**2*x** 6 + 2*a*b**2*d*e**2*x**4 + 4*a*b**2*d*e*f*x**6 + 2*a*b**2*d*f**2*x**8 + b* *3*c*e**2*x**4 + b**3*c*e*f*x**6 + b**3*d*e**2*x**6 + b**3*d*e*f*x**8),x)* a**3*b*c*d*f**3 - 20*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(2*a**3* c*e*f + 2*a**3*c*f**2*x**2 + 2*a**3*d*e*f*x**2 + 2*a**3*d*f**2*x**4 + a**2 *b*c*e**2 + 5*a**2*b*c*e*f*x**2 + 4*a**2*b*c*f**2*x**4 + a**2*b*d*e**2*x** 2 + 5*a**2*b*d*e*f*x**4 + 4*a**2*b*d*f**2*x**6 + 2*a*b**2*c*e**2*x**2 +...