Integrand size = 35, antiderivative size = 597 \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\left (8 a^2 d f+a b (10 d e+7 c f)+b^2 \left (10 c e+\frac {15 d e^2}{f}+\frac {8 c^2 f}{d}\right )\right ) x \sqrt {c+d x^2}}{15 b^2 d^2 f^2 \sqrt {a+b x^2}}-\frac {(5 b d e+4 b c f+4 a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d^2 f^2}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d f}-\frac {\sqrt {a} \left (8 a^2 d^2 f^2+a b d f (10 d e+7 c f)+b^2 \left (15 d^2 e^2+10 c d e f+8 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{5/2} d^3 f^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (4 a^2 c d f^3+a b c f^2 (d e+4 c f)-b^2 e \left (15 d^2 e^2+5 c d e f+4 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{5/2} c d^2 f^3 (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} e^3 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c f^3 (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/15*(8*a^2*d*f+a*b*(7*c*f+10*d*e)+b^2*(10*c*e+15*d*e^2/f+8*c^2*f/d))*x*(d *x^2+c)^(1/2)/b^2/d^2/f^2/(b*x^2+a)^(1/2)-1/15*(4*a*d*f+4*b*c*f+5*b*d*e)*x *(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^2/f^2+1/5*x^3*(b*x^2+a)^(1/2)*(d*x^ 2+c)^(1/2)/b/d/f-1/15*a^(1/2)*(8*a^2*d^2*f^2+a*b*d*f*(7*c*f+10*d*e)+b^2*(8 *c^2*f^2+10*c*d*e*f+15*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^3/f^3/(b*x^2+a)^(1/2)/(a *(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*a^(3/2)*(4*a^2*c*d*f^3+a*b*c*f^2*(4*c*f +d*e)-b^2*e*(4*c^2*f^2+5*c*d*e*f+15*d^2*e^2))*(d*x^2+c)^(1/2)*InverseJacob iAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/2)/c/d^2/f^3/(-a*f+b *e)/(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^3/(-a*f+b*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 = 4.53 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.70 \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-i c f \left (8 a^2 d^2 f^2+a b d f (10 d e+7 c f)+b^2 \left (15 d^2 e^2+10 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i \left (4 a^2 c d^2 f^3+a b c d f^2 (5 d e+3 c f)+b^2 \left (15 d^3 e^3+15 c d^2 e^2 f+10 c^2 d e f^2+8 c^3 f^3\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-d \left (\sqrt {\frac {b}{a}} f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d f+b \left (5 d e+4 c f-3 d f x^2\right )\right )+15 i b^2 d^2 e^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 f^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[x^8/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
((-I)*c*f*(8*a^2*d^2*f^2 + a*b*d*f*(10*d*e + 7*c*f) + b^2*(15*d^2*e^2 + 10 *c*d*e*f + 8*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*(4*a^2*c*d^2*f^3 + a*b*c*d*f^2*(5* d*e + 3*c*f) + b^2*(15*d^3*e^3 + 15*c*d^2*e^2*f + 10*c^2*d*e*f^2 + 8*c^3*f ^3))*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*(a + b*x^2)*(c + d*x^2)*(4*a*d*f + b*(5*d*e + 4*c*f - 3*d*f*x^2)) + (15*I)*b^2*d^2*e^3*Sqrt[1 + (b*x^2)/a]*Sq rt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b *c)]))/(15*a^2*(b/a)^(5/2)*d^3*f^4*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.
| \(\displaystyle \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )}dx\) |
Input:
Int[x^8/(Sqrt[a + b*x^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 = 20.19 (sec) , antiderivative size = 528, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \(-\frac {x \left (-3 b d f \,x^{2}+4 a d f +4 b c f +5 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b^{2} d^{2} f^{2}}+\frac {\left (\frac {\left (4 a^{2} c d \,f^{3}+4 a b \,c^{2} f^{3}+5 a b c d e \,f^{2}-15 b^{2} d^{2} e^{3}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (8 a^{2} d^{2} f^{2}+7 a b c d \,f^{2}+10 a b \,d^{2} e f +8 b^{2} c^{2} f^{2}+10 b^{2} c d e f +15 b^{2} d^{2} e^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {15 b^{2} d^{2} e^{3} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 f^{2} b^{2} d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(528\) |
| default | \(\text {Expression too large to display}\) | \(1178\) |
| elliptic | \(\text {Expression too large to display}\) | \(1506\) |
Input:
int(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
-1/15*x*(-3*b*d*f*x^2+4*a*d*f+4*b*c*f+5*b*d*e)*(b*x^2+a)^(1/2)*(d*x^2+c)^( 1/2)/b^2/d^2/f^2+1/15/f^2/b^2/d^2*((4*a^2*c*d*f^3+4*a*b*c^2*f^3+5*a*b*c*d* e*f^2-15*b^2*d^2*e^3)/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))-1/f*(8*a^2*d^2*f^2+7*a*b*c*d*f^2+10*a*b*d^2*e*f+8*b^2*c^2*f^ 2+10*b^2*c*d*e*f+15*b^2*d^2*e^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)/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 )))+15*b^2*d^2*e^3/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)))*((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2 +c)^(1/2)
Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^{8}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(x**8/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral(x**8/(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)), x)
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {x^{8}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="maxi ma")
Output:
integrate(x^8/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {x^{8}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="giac ")
Output:
integrate(x^8/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^8}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(x^8/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int(x^8/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:
int(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x)
Output:
( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f*x - 4*sqrt(c + d*x**2)*sqrt( a + b*x**2)*b*c*f*x - 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*x + 3*sqrt (c + d*x**2)*sqrt(a + b*x**2)*b*d*f*x**3 + 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a**2*d**2*f**2 + 7*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f *x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*b*c*d*f**2 + 10*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a *d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6 ),x)*a*b*d**2*e*f + 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x **4 + b*d*f*x**6),x)*b**2*c**2*f**2 + 10*int((sqrt(c + d*x**2)*sqrt(a + b* x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b *c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b**2*c*d*e*f + 15*int((sqrt(c + d* x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b**2*d**2*e**2 + 4*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e *x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x) *a**2*c*d*f**2 + 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a *c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x...