Integrand size = 35, antiderivative size = 719 \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {\left (4 a^2 d f^2 (d e-c f)-b^2 e \left (15 d^2 e^2-8 c d e f-4 c^2 f^2\right )+4 a b f \left (2 d^2 e^2-c d e f-c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{6 b d^2 f^3 (b e-a f) (d e-c f) \sqrt {a+b x^2}}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d f^2}+\frac {e^3 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 f^2 (b e-a f) (d e-c f) \left (e+f x^2\right )}-\frac {\sqrt {a} \left (4 a^2 d f^2 (d e-c f)-b^2 e \left (15 d^2 e^2-8 c d e f-4 c^2 f^2\right )+4 a b f \left (2 d^2 e^2-c d e f-c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{6 b^{3/2} d^2 f^3 (b e-a f) (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (2 a^2 c f^3-2 a b e f (9 d e+2 c f)+b^2 e^2 (15 d e+2 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{6 b^{3/2} c d f^3 (b e-a f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} e^2 (a f (6 d e-7 c f)-b e (5 d e-6 c f)) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c f^3 (b e-a f)^2 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/6*(4*a^2*d*f^2*(-c*f+d*e)-b^2*e*(-4*c^2*f^2-8*c*d*e*f+15*d^2*e^2)+4*a*b* f*(-c^2*f^2-c*d*e*f+2*d^2*e^2))*x*(d*x^2+c)^(1/2)/b/d^2/f^3/(-a*f+b*e)/(-c *f+d*e)/(b*x^2+a)^(1/2)+1/3*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d/f^2+1/2* e^3*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^2/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)- 1/6*a^(1/2)*(4*a^2*d*f^2*(-c*f+d*e)-b^2*e*(-4*c^2*f^2-8*c*d*e*f+15*d^2*e^2 )+4*a*b*f*(-c^2*f^2-c*d*e*f+2*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^(3/2)/d^2/f^3/(-a*f+b*e)/ (-c*f+d*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/6*a^(3/2)*(2* a^2*c*f^3-2*a*b*e*f*(2*c*f+9*d*e)+b^2*e^2*(2*c*f+15*d*e))*(d*x^2+c)^(1/2)* InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(3/2)/c/d/f ^3/(-a*f+b*e)^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/2*a^(3/2 )*e^2*(a*f*(-7*c*f+6*d*e)-b*e*(-6*c*f+5*d*e))*(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^3/(-a*f+b*e)^2/(-c*f+d*e)/(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 = 5.73 (sec) , antiderivative size = 537, normalized size of antiderivative = 0.75 \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {i c f \left (4 a^2 d f^2 (-d e+c f)+b^2 e \left (15 d^2 e^2-8 c d e f-4 c^2 f^2\right )+4 a b f \left (-2 d^2 e^2+c d e f+c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i (-d e+c f) \left (2 a^2 c d f^3+2 a b f \left (9 d^2 e^2+5 c d e f+2 c^2 f^2\right )-b^2 e \left (15 d^2 e^2+12 c d e f+4 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d \left (\sqrt {\frac {b}{a}} f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (2 a f (-d e+c f) \left (e+f x^2\right )+b e \left (-2 c f \left (e+f x^2\right )+d e \left (5 e+2 f x^2\right )\right )\right )+3 i b d e^2 (b e (5 d e-6 c f)+a f (-6 d e+7 c f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{6 b \sqrt {\frac {b}{a}} d^2 f^4 (b e-a f) (d e-c f) \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[x^8/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
(I*c*f*(4*a^2*d*f^2*(-(d*e) + c*f) + b^2*e*(15*d^2*e^2 - 8*c*d*e*f - 4*c^2 *f^2) + 4*a*b*f*(-2*d^2*e^2 + c*d*e*f + 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*(-(d*e) + c*f)*(2*a^2*c*d*f^3 + 2*a*b*f*(9*d^2*e^2 + 5*c*d*e*f + 2*c^ 2*f^2) - b^2*e*(15*d^2*e^2 + 12*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)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b *c)] + d*(Sqrt[b/a]*f^2*x*(a + b*x^2)*(c + d*x^2)*(2*a*f*(-(d*e) + c*f)*(e + f*x^2) + b*e*(-2*c*f*(e + f*x^2) + d*e*(5*e + 2*f*x^2))) + (3*I)*b*d*e^ 2*(b*e*(5*d*e - 6*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)]))/(6*b*Sqrt[b/a]*d^2*f^4*(b*e - a*f)*(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.
| \(\displaystyle \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}dx\) |
Input:
Int[x^8/(Sqrt[a + b*x^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. \(1453\) vs. \(2(681)=1362\).
Time = 25.77 (sec) , antiderivative size = 1454, normalized size of antiderivative = 2.02
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1454\) |
| elliptic | \(\text {Expression too large to display}\) | \(1700\) |
| default | \(\text {Expression too large to display}\) | \(3438\) |
Input:
int(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x,method=_RETURNVERBOS E)
Output:
1/3*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d/f^2-1/3/f^2/b/d*(1/f^2*(-2*f*(a* d*f+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)))+a*c*f^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))-9*b*d*e^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)))+12*b*d*e^2/f^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)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))- 3*b*d*e^4/f^2*(1/2*f^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e*x*(b*d*x^4+a*d* x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+e)-1/2*d*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^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))+1/2*f*b/(a* c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/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)*EllipticF(x*(-b/a)^(1/2),(-1 +(a*d+b*c)/c/b)^(1/2))-1/2*f*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/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)*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+1/2/(a*c*f^2-a*d* e*f-b*c*e*f+b*d*e^2)/e^2*f^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)...
Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {x^{8}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(x**8/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**2,x)
Output:
Integral(x**8/(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**2), x)
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {x^{8}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="ma xima")
Output:
integrate(x^8/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {x^{8}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="gi ac")
Output:
integrate(x^8/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {x^8}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(x^8/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2),x)
Output:
int(x^8/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2), x)
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {too large to display} \] Input:
int(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)
Output:
( - 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 - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*e*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*x**3 + int((sqrt(c + d*x**2)*sqrt(a + b* x**2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2 *a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2* x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*a*b*c*d*e*f**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2 *x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*a*b*c*d*f**3*x**2 - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x **2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2* a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x **6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*a*b*d**2*e**2*f - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e* *2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*a*b*d**2*e*f**2*x**2 - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f **2*x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*b**2*c*d*...