Integrand size = 35, antiderivative size = 852 \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\frac {\left (8 a^2 f^2 (d e-c f)^2+b^2 e^2 \left (15 d^2 e^2-26 c d e f+8 c^2 f^2\right )-a b e f \left (26 d^2 e^2-45 c d e f+16 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{8 d f^3 (b e-a f)^2 (d e-c f)^2 \sqrt {a+b x^2}}+\frac {e^3 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 f^2 (b e-a f) (d e-c f) \left (e+f x^2\right )^2}+\frac {e^2 (a f (10 d e-13 c f)-b e (7 d e-10 c f)) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 f^2 (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )}-\frac {\sqrt {a} \left (8 a^2 f^2 (d e-c f)^2+b^2 e^2 \left (15 d^2 e^2-26 c d e f+8 c^2 f^2\right )-a b e f \left (26 d^2 e^2-45 c d e f+16 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{8 \sqrt {b} d f^3 (b e-a f)^2 (d e-c f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} e \left (a b e f (36 d e-37 c f)-b^2 e^2 (15 d e-16 c f)-24 a^2 f^2 (d e-c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{8 \sqrt {b} c f^3 (b e-a f)^3 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} e \left (3 b^2 e^2 \left (5 d^2 e^2-12 c d e f+8 c^2 f^2\right )-2 a b e f \left (18 d^2 e^2-43 c d e f+28 c^2 f^2\right )+a^2 f^2 \left (24 d^2 e^2-56 c d e f+35 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{8 \sqrt {b} c f^3 (b e-a f)^3 (d e-c f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/8*(8*a^2*f^2*(-c*f+d*e)^2+b^2*e^2*(8*c^2*f^2-26*c*d*e*f+15*d^2*e^2)-a*b* e*f*(16*c^2*f^2-45*c*d*e*f+26*d^2*e^2))*x*(d*x^2+c)^(1/2)/d/f^3/(-a*f+b*e) ^2/(-c*f+d*e)^2/(b*x^2+a)^(1/2)+1/4*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)^2+1/8*e^2*(a*f*(-13*c*f+10*d*e)-b*e*(- 10*c*f+7*d*e))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^2/(-a*f+b*e)^2/(-c*f+d* e)^2/(f*x^2+e)-1/8*a^(1/2)*(8*a^2*f^2*(-c*f+d*e)^2+b^2*e^2*(8*c^2*f^2-26*c *d*e*f+15*d^2*e^2)-a*b*e*f*(16*c^2*f^2-45*c*d*e*f+26*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^(1 /2)/d/f^3/(-a*f+b*e)^2/(-c*f+d*e)^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+ a))^(1/2)-1/8*a^(3/2)*e*(a*b*e*f*(-37*c*f+36*d*e)-b^2*e^2*(-16*c*f+15*d*e) -24*a^2*f^2*(-c*f+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^(1/2)/c/f^3/(-a*f+b*e)^3/(-c*f+d*e)/(b*x^2+a) ^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/8*a^(3/2)*e*(3*b^2*e^2*(8*c^2*f^2 -12*c*d*e*f+5*d^2*e^2)-2*a*b*e*f*(28*c^2*f^2-43*c*d*e*f+18*d^2*e^2)+a^2*f^ 2*(35*c^2*f^2-56*c*d*e*f+24*d^2*e^2))*(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)^3/(-c*f+d*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 = 5.09 (sec) , antiderivative size = 559, normalized size of antiderivative = 0.66 \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\frac {\sqrt {\frac {b}{a}} d e^2 f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (2 e (b e-a f) (d e-c f)-(b e (7 d e-10 c f)+a f (-10 d e+13 c f)) \left (e+f x^2\right )\right )-i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right )^2 \left (c f \left (8 a^2 f^2 (d e-c f)^2+a b e f \left (-26 d^2 e^2+45 c d e f-16 c^2 f^2\right )+b^2 e^2 \left (15 d^2 e^2-26 c d e f+8 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(d e-c f) \left (b^2 e^2 \left (15 d^2 e^2-6 c d e f-8 c^2 f^2\right )-8 a^2 f^2 \left (-3 d^2 e^2+2 c d e f+c^2 f^2\right )+a b e f \left (-36 d^2 e^2+19 c d e f+16 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d e \left (3 b^2 e^2 \left (5 d^2 e^2-12 c d e f+8 c^2 f^2\right )-2 a b e f \left (18 d^2 e^2-43 c d e f+28 c^2 f^2\right )+a^2 f^2 \left (24 d^2 e^2-56 c d e f+35 c^2 f^2\right )\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{8 \sqrt {\frac {b}{a}} d f^4 (b e-a f)^2 (d e-c f)^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \] Input:
Integrate[x^8/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^3),x]
Output:
(Sqrt[b/a]*d*e^2*f^2*x*(a + b*x^2)*(c + d*x^2)*(2*e*(b*e - a*f)*(d*e - c*f ) - (b*e*(7*d*e - 10*c*f) + a*f*(-10*d*e + 13*c*f))*(e + f*x^2)) - I*Sqrt[ 1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)^2*(c*f*(8*a^2*f^2*(d*e - c* f)^2 + a*b*e*f*(-26*d^2*e^2 + 45*c*d*e*f - 16*c^2*f^2) + b^2*e^2*(15*d^2*e ^2 - 26*c*d*e*f + 8*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c )] - (d*e - c*f)*(b^2*e^2*(15*d^2*e^2 - 6*c*d*e*f - 8*c^2*f^2) - 8*a^2*f^2 *(-3*d^2*e^2 + 2*c*d*e*f + c^2*f^2) + a*b*e*f*(-36*d^2*e^2 + 19*c*d*e*f + 16*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + d*e*(3*b^2*e ^2*(5*d^2*e^2 - 12*c*d*e*f + 8*c^2*f^2) - 2*a*b*e*f*(18*d^2*e^2 - 43*c*d*e *f + 28*c^2*f^2) + a^2*f^2*(24*d^2*e^2 - 56*c*d*e*f + 35*c^2*f^2))*Ellipti cPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(8*Sqrt[b/a]*d*f^4 *(b*e - a*f)^2*(d*e - c*f)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^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.
| \(\displaystyle \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3}dx\) |
Input:
Int[x^8/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^3),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. \(2926\) vs. \(2(814)=1628\).
Time = 11.05 (sec) , antiderivative size = 2927, normalized size of antiderivative = 3.44
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2927\) |
| default | \(\text {Expression too large to display}\) | \(6862\) |
Input:
int(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x,method=_RETURNVERBOS E)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(7/8*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*e^4*b^2/f^3/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2*EllipticE(x*(-b/a)^( 1/2),(-1+(a*d+b*c)/c/b)^(1/2))-7*e^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/f /(-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)) *a^2*c*d-7*e^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/f/(-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))*a*b*c^2-3/(-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))*e/f^4-1/8*e^2*(13*a*c*f ^2-10*a*d*e*f-10*b*c*e*f+7*b*d*e^2)/f^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^ 2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+e)+9/8/(-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*d^2*e^5/f^4/(a*c*f^2-a*d*e*f -b*c*e*f+b*d*e^2)^2+43/4*e^3/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^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))*a*b*c *d-9/2*e^4/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/f^3/(-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)*EllipticP...
Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm="fr icas")
Output:
Timed out
Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**8/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int { \frac {x^{8}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm="ma xima")
Output:
integrate(x^8/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^3), x)
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int { \frac {x^{8}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm="gi ac")
Output:
integrate(x^8/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^3), x)
Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int \frac {x^8}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int(x^8/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^3),x)
Output:
int(x^8/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^3), x)
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {too large to display} \] Input:
int(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,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 + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f*x**3 + 4*sqr t(c + d*x**2)*sqrt(a + b*x**2)*b*c*e*x + 2*sqrt(c + d*x**2)*sqrt(a + b*x** 2)*b*c*f*x**3 - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*x**3 - 2*int((sq rt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(2*a**2*c*d*e**3*f + 6*a**2*c*d*e**2 *f**2*x**2 + 6*a**2*c*d*e*f**3*x**4 + 2*a**2*c*d*f**4*x**6 + 2*a**2*d**2*e **3*f*x**2 + 6*a**2*d**2*e**2*f**2*x**4 + 6*a**2*d**2*e*f**3*x**6 + 2*a**2 *d**2*f**4*x**8 + 2*a*b*c**2*e**3*f + 6*a*b*c**2*e**2*f**2*x**2 + 6*a*b*c* *2*e*f**3*x**4 + 2*a*b*c**2*f**4*x**6 - 3*a*b*c*d*e**4 - 5*a*b*c*d*e**3*f* x**2 + 3*a*b*c*d*e**2*f**2*x**4 + 9*a*b*c*d*e*f**3*x**6 + 4*a*b*c*d*f**4*x **8 - 3*a*b*d**2*e**4*x**2 - 7*a*b*d**2*e**3*f*x**4 - 3*a*b*d**2*e**2*f**2 *x**6 + 3*a*b*d**2*e*f**3*x**8 + 2*a*b*d**2*f**4*x**10 + 2*b**2*c**2*e**3* f*x**2 + 6*b**2*c**2*e**2*f**2*x**4 + 6*b**2*c**2*e*f**3*x**6 + 2*b**2*c** 2*f**4*x**8 - 3*b**2*c*d*e**4*x**2 - 7*b**2*c*d*e**3*f*x**4 - 3*b**2*c*d*e **2*f**2*x**6 + 3*b**2*c*d*e*f**3*x**8 + 2*b**2*c*d*f**4*x**10 - 3*b**2*d* *2*e**4*x**4 - 9*b**2*d**2*e**3*f*x**6 - 9*b**2*d**2*e**2*f**2*x**8 - 3*b* *2*d**2*e*f**3*x**10),x)*a**2*b*c*d**2*e**2*f**3 - 4*int((sqrt(c + d*x**2) *sqrt(a + b*x**2)*x**6)/(2*a**2*c*d*e**3*f + 6*a**2*c*d*e**2*f**2*x**2 + 6 *a**2*c*d*e*f**3*x**4 + 2*a**2*c*d*f**4*x**6 + 2*a**2*d**2*e**3*f*x**2 + 6 *a**2*d**2*e**2*f**2*x**4 + 6*a**2*d**2*e*f**3*x**6 + 2*a**2*d**2*f**4*...