Integrand size = 35, antiderivative size = 427 \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {x \sqrt {c+d x^2}}{b d f \sqrt {a+b x^2}}-\frac {\sqrt {a} \left (b^2 c e+2 a^2 d f-a b (d e+c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{b^{3/2} d (b c-a d) f (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (b^2 c e^2-a^2 c f^2-a b e (d e-c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{b^{3/2} c (b c-a d) f (b e-a f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} e^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c f (b e-a f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
x*(d*x^2+c)^(1/2)/b/d/f/(b*x^2+a)^(1/2)-a^(1/2)*(b^2*c*e+2*a^2*d*f-a*b*(c* f+d*e))*(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/(-a*d+b*c)/f/(-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)*(b^2*c*e^2-a^2*c*f^2-a*b*e*(-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 ^(3/2)/c/(-a*d+b*c)/f/(-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^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/(-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 = 5.26 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.79 \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-a^2 \sqrt {\frac {b}{a}} d f^2 x \left (c+d x^2\right )-i c f \left (b^2 c e+2 a^2 d f-a b (d e+c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-b c+a d) \left (a c f^2-b e (d e+c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+i b d (-b c+a d) e^2 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{b \sqrt {\frac {b}{a}} d (b c-a d) f^2 (b e-a f) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[x^6/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
(-(a^2*Sqrt[b/a]*d*f^2*x*(c + d*x^2)) - I*c*f*(b^2*c*e + 2*a^2*d*f - a*b*( d*e + c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sq rt[b/a]*x], (a*d)/(b*c)] + I*(-(b*c) + a*d)*(a*c*f^2 - b*e*(d*e + c*f))*Sq rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a *d)/(b*c)] + I*b*d*(-(b*c) + a*d)*e^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2) /c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(b*Sqrt[ b/a]*d*(b*c - a*d)*f^2*(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.
| \(\displaystyle \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )}dx\) |
Input:
Int[x^6/((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 = 8.68 (sec) , antiderivative size = 806, normalized size of antiderivative = 1.89
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+b c \right ) a^{2} x}{b^{2} \left (a d -b c \right ) \left (a f -b e \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}-\frac {\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 ) a}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, f \,b^{2}}-\frac {\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 ) e}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, f^{2} b}+\frac {\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 ) a^{2}}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, b^{2} \left (a f -b e \right )}-\frac {c \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}\, d f b}+\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\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}\, d f b}+\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, a^{2} \operatorname {EllipticE}\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}\, b \left (a d -b c \right ) \left (a f -b e \right )}-\frac {e^{2} \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 )}{\left (a f -b e \right ) f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(806\) |
| default | \(\frac {\left (-\sqrt {-\frac {b}{a}}\, a^{2} d^{2} f^{2} x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d \,f^{2}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f^{2}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e f +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,d^{2} e^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} e f -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c d \,e^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d \,f^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e f +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} e f -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a b \,d^{2} e^{2}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b^{2} c d \,e^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c d \,f^{2} x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{b \,f^{2} \left (a f -b e \right ) d \sqrt {-\frac {b}{a}}\, \left (a d -b c \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(840\) |
Input:
int(x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
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^2/(a*d-b*c)*x/(a*f-b*e)/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)-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/b^2*a-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*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)*E llipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*a^2/b^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)/d/f/b*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(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)/d/f/b*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(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)*a ^2/b/(a*d-b*c)/(a*f-b*e)*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2) )-e^2/(a*f-b*e)/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)))
Timed out. \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^{6}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(x**6/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral(x**6/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)), x)
\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="maxi ma")
Output:
integrate(x^6/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="giac ")
Output:
integrate(x^6/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^6}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(x^6/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int(x^6/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{6}}{b^{2} d f \,x^{8}+2 a b d f \,x^{6}+b^{2} c f \,x^{6}+b^{2} d e \,x^{6}+a^{2} d f \,x^{4}+2 a b c f \,x^{4}+2 a b d e \,x^{4}+b^{2} c e \,x^{4}+a^{2} c f \,x^{2}+a^{2} d e \,x^{2}+2 a b c e \,x^{2}+a^{2} c e}d x \] Input:
int(x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a**2*c*e + a**2*c*f*x**2 + a **2*d*e*x**2 + a**2*d*f*x**4 + 2*a*b*c*e*x**2 + 2*a*b*c*f*x**4 + 2*a*b*d*e *x**4 + 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b**2*d*e*x**6 + b **2*d*f*x**8),x)