Integrand size = 36, antiderivative size = 296 \[ \int \frac {x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d f \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {\sqrt {a} (d e+c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d f^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {a} e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f^2 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
a^(1/2)*(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)/d/f/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)-a^(1/2)*(c*f+ d*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)/d/f^2/(-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^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 3.18 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.61 \[ \int \frac {x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {i \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (c f E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-(d e+c f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )+d e \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{\sqrt {-\frac {b}{a}} d f^2 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[x^4/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
((-I)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(c*f*EllipticE[I*ArcSinh[Sqr t[-(b/a)]*x], -((a*d)/(b*c))] - (d*e + c*f)*EllipticF[I*ArcSinh[Sqrt[-(b/a )]*x], -((a*d)/(b*c))] + d*e*EllipticPi[-((a*f)/(b*e)), I*ArcSinh[Sqrt[-(b /a)]*x], -((a*d)/(b*c))]))/(Sqrt[-(b/a)]*d*f^2*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.
| \(\displaystyle \int \frac {x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )}dx\) |
Input:
Int[x^4/(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 = 5.67 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.69
| method | result | size |
| default | \(\frac {\left (-\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) c f -\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) d e +\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) c f +\operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) d e \right ) \sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{d \sqrt {\frac {b}{a}}\, f^{2} \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\) | \(204\) |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{f^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}+\frac {c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}+\frac {e \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right )}{f^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(426\) |
Input:
int(x^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOSE )
Output:
(-EllipticF(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*c*f-EllipticF(x*(b/a)^(1/2),(- a*d/b/c)^(1/2))*d*e+EllipticE(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*c*f+Elliptic Pi(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2))*d*e)*((d*x^2+c)/c)^( 1/2)*((-b*x^2+a)/a)^(1/2)*(d*x^2+c)^(1/2)*(-b*x^2+a)^(1/2)/d/(b/a)^(1/2)/f ^2/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)
Timed out. \[ \int \frac {x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(x^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="fri cas")
Output:
Timed out
\[ \int \frac {x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^{4}}{\sqrt {a - b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(x**4/(-b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral(x**4/(sqrt(a - b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)), x)
\[ \int \frac {x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {x^{4}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="max ima")
Output:
integrate(x^4/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
\[ \int \frac {x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {x^{4}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(x^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="gia c")
Output:
integrate(x^4/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^4}{\sqrt {a-b\,x^2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(x^4/((a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int(x^4/((a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{-b d f \,x^{6}+a d f \,x^{4}-b c f \,x^{4}-b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}-b c e \,x^{2}+a c e}d x \] Input:
int(x^4/(-b*x^2+a)^(1/2)/(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*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)