Integrand size = 36, antiderivative size = 362 \[ \int \frac {x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {a x \sqrt {c+d x^2}}{(b c+a d) (b e+a f) \sqrt {a-b x^2}}-\frac {a^{3/2} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} (b c+a d) (b e+a f) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {\sqrt {a} e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {a} e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
a*x*(d*x^2+c)^(1/2)/(a*d+b*c)/(a*f+b*e)/(-b*x^2+a)^(1/2)-a^(3/2)*(1-b*x^2/ a)^(1/2)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^( 1/2)/(a*d+b*c)/(a*f+b*e)/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)-a^(1/2)*e*(1-b *x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/ 2))/b^(1/2)/f/(a*f+b*e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+a^(1/2)*e*(1-b*x^ 2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticPi(b^(1/2)*x/a^(1/2),-a*f/b/e,(-a*d/b /c)^(1/2))/b^(1/2)/f/(a*f+b*e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 4.81 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.02 \[ \int \frac {x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {a \sqrt {-\frac {b}{a}} c f x+a \sqrt {-\frac {b}{a}} d f x^3+i a c f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i (b c+a d) e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )-i b c e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )-i a d e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} (b c+a d) f (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[x^4/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
(a*Sqrt[-(b/a)]*c*f*x + a*Sqrt[-(b/a)]*d*f*x^3 + I*a*c*f*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)*e*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*Ar cSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] - I*b*c*e*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))] - I*a*d*e*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))])/(Sqrt[-(b/a)]*(b *c + a*d)*f*(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.
| \(\displaystyle \int \frac {x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )}dx\) |
Input:
Int[x^4/((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 = 8.41 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.10
| method | result | size |
| default | \(\frac {\left (\sqrt {\frac {b}{a}}\, a d f \,x^{3}-\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a d e -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c e -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a c f +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a d e +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) b c e +\sqrt {\frac {b}{a}}\, a c f x \right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{f \sqrt {\frac {b}{a}}\, \left (a d +b c \right ) \left (a f +b e \right ) \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\) | \(398\) |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}-b c \right ) a x}{b \left (a d +b c \right ) \left (a f +b e \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}-\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, f b}+\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right ) a}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, b \left (a f +b e \right )}-\frac {a c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\left (a d +b c \right ) \left (a f +b e \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {e \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right )}{\left (a f +b e \right ) f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(518\) |
Input:
int(x^4/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOSE )
Output:
((b/a)^(1/2)*a*d*f*x^3-((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF( x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*a*d*e-((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^( 1/2)*EllipticF(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b*c*e-((-b*x^2+a)/a)^(1/2)* ((d*x^2+c)/c)^(1/2)*EllipticE(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*a*c*f+((-b*x ^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c *d)^(1/2)/(b/a)^(1/2))*a*d*e+((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Elli pticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2))*b*c*e+(b/a)^(1/2 )*a*c*f*x)*(d*x^2+c)^(1/2)*(-b*x^2+a)^(1/2)/f/(b/a)^(1/2)/(a*d+b*c)/(a*f+b *e)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)
Timed out. \[ \int \frac {x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(x^4/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="fri cas")
Output:
Timed out
\[ \int \frac {x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^{4}}{\left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(x**4/(-b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral(x**4/((a - b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)), x)
\[ \int \frac {x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^4/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="max ima")
Output:
integrate(x^4/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
\[ \int \frac {x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^4/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="gia c")
Output:
integrate(x^4/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^4}{{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(x^4/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int(x^4/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {x^4}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b^{2} d f \,x^{8}-2 a b d f \,x^{6}+b^{2} c f \,x^{6}+b^{2} d e \,x^{6}+a^{2} d f \,x^{4}-2 a b c f \,x^{4}-2 a b d e \,x^{4}+b^{2} c e \,x^{4}+a^{2} c f \,x^{2}+a^{2} d e \,x^{2}-2 a b c e \,x^{2}+a^{2} c e}d x \] Input:
int(x^4/(-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**4)/(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)