Integrand size = 35, antiderivative size = 542 \[ \int \frac {x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\frac {(3 b c e-a d e-2 a c f) x \sqrt {a+b x^2}}{2 (b e-a f) (d e-c f)^2 \sqrt {c+d x^2}}-\frac {b (d e+2 c f) x \sqrt {c+d x^2}}{2 d f (d e-c f)^2 \sqrt {a+b x^2}}+\frac {e x \left (a+b x^2\right )^{3/2}}{2 (b e-a f) (d e-c f) \sqrt {c+d x^2} \left (e+f x^2\right )}+\frac {\sqrt {a} \sqrt {b} (d e+2 c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{2 d f (d e-c f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \sqrt {b} e \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 c f (b e-a f) (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (3 a c f^2+b e (d e-4 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c f (b e-a f) (d e-c f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/2*(-2*a*c*f-a*d*e+3*b*c*e)*x*(b*x^2+a)^(1/2)/(-a*f+b*e)/(-c*f+d*e)^2/(d* x^2+c)^(1/2)-1/2*b*(2*c*f+d*e)*x*(d*x^2+c)^(1/2)/d/f/(-c*f+d*e)^2/(b*x^2+a )^(1/2)+1/2*e*x*(b*x^2+a)^(3/2)/(-a*f+b*e)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(f*x ^2+e)+1/2*a^(1/2)*b^(1/2)*(2*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))/d/f/(-c*f+d*e)^2/(b*x^2+a)^(1 /2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/2*a^(3/2)*b^(1/2)*e*(d*x^2+c)^(1/2)* InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/c/f/(-a*f+b*e )/(-c*f+d*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/2*a^(3/2)*( 3*a*c*f^2+b*e*(-4*c*f+d*e))*(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)/(-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 = 5.26 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.66 \[ \int \frac {x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\frac {\sqrt {\frac {b}{a}} d f^2 x \left (a+b x^2\right ) \left (3 c e+d e x^2+2 c f x^2\right )+i b c f (d e+2 c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b \left (d^2 e^2-3 c d e f+2 c^2 f^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+i d \left (3 a c f^2+b e (d e-4 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{2 \sqrt {\frac {b}{a}} d f^2 (d e-c f)^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[(x^4*Sqrt[a + b*x^2])/((c + d*x^2)^(3/2)*(e + f*x^2)^2),x]
Output:
(Sqrt[b/a]*d*f^2*x*(a + b*x^2)*(3*c*e + d*e*x^2 + 2*c*f*x^2) + I*b*c*f*(d* e + 2*c*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticE[I *ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*(d^2*e^2 - 3*c*d*e*f + 2*c^2*f^2 )*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticF[I*ArcSinh[ Sqrt[b/a]*x], (a*d)/(b*c)] + I*d*(3*a*c*f^2 + b*e*(d*e - 4*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticPi[(a*f)/(b*e), I*ArcSi nh[Sqrt[b/a]*x], (a*d)/(b*c)])/(2*Sqrt[b/a]*d*f^2*(d*e - c*f)^2*Sqrt[a + b *x^2]*Sqrt[c + d*x^2]*(e + f*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.
| \(\displaystyle \int \frac {x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}dx\) |
Input:
Int[(x^4*Sqrt[a + b*x^2])/((c + d*x^2)^(3/2)*(e + f*x^2)^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 = 8.82 (sec) , antiderivative size = 972, normalized size of antiderivative = 1.79
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {e x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{2 \left (c f -d e \right )^{2} \left (f \,x^{2}+e \right )}+\frac {\left (b d \,x^{2}+a d \right ) c x}{d \left (c f -d e \right )^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \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 ) b}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, f^{2} d}-\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 ) b d \,e^{2}}{2 \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, f^{2} \left (c f -d e \right )^{2}}+\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, b e \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{2 \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, \left (c f -d e \right )^{2} f}-\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, b e \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{2 \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, \left (c f -d e \right )^{2} f}-\frac {c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, b \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 \left (c f -d e \right )^{2}}-\frac {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 ) a c}{2 \left (c f -d e \right )^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {2 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 ) b c}{\left (c f -d e \right )^{2} f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\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 ) b d}{2 f^{2} \left (c f -d e \right )^{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}}\) | \(972\) |
| default | \(\text {Expression too large to display}\) | \(1222\) |
Input:
int(x^4*(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^2,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)*(1/2*e/(c*f-d* e)^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+e)+(b*d*x^2+a*d)*c/d*x/( c*f-d*e)^2/((x^2+c/d)*(b*d*x^2+a*d))^(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))*b/f^2/d-1/2/(-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))*b*d*e^2/f^2/(c*f-d*e)^2+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)* b*e/(c*f-d*e)^2/f*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-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)*b*e/(c*f-d*e)^2/f*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^ (1/2))-c^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)/d*b/(c*f-d*e)^2*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b* c)/c/b)^(1/2))-3/2/(c*f-d*e)^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*c+2*e/(c*f-d*e)^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))*b*c-1/2*e^2/f^2/(c*f- d*e)^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/...
Timed out. \[ \int \frac {x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^4*(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int \frac {x^{4} \sqrt {a + b x^{2}}}{\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(x**4*(b*x**2+a)**(1/2)/(d*x**2+c)**(3/2)/(f*x**2+e)**2,x)
Output:
Integral(x**4*sqrt(a + b*x**2)/((c + d*x**2)**(3/2)*(e + f*x**2)**2), x)
\[ \int \frac {x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} x^{4}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^4*(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*x^4/((d*x^2 + c)^(3/2)*(f*x^2 + e)^2), x)
\[ \int \frac {x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} x^{4}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^4*(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*x^4/((d*x^2 + c)^(3/2)*(f*x^2 + e)^2), x)
Timed out. \[ \int \frac {x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int \frac {x^4\,\sqrt {b\,x^2+a}}{{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int((x^4*(a + b*x^2)^(1/2))/((c + d*x^2)^(3/2)*(e + f*x^2)^2),x)
Output:
int((x^4*(a + b*x^2)^(1/2))/((c + d*x^2)^(3/2)*(e + f*x^2)^2), x)
\[ \int \frac {x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\text {too large to display} \] Input:
int(x^4*(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x)
Output:
( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*x + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(2*a**2*c**2*e**2*f + 4*a**2*c**2*e*f**2*x**2 + 2*a**2*c* *2*f**3*x**4 + 4*a**2*c*d*e**2*f*x**2 + 8*a**2*c*d*e*f**2*x**4 + 4*a**2*c* d*f**3*x**6 + 2*a**2*d**2*e**2*f*x**4 + 4*a**2*d**2*e*f**2*x**6 + 2*a**2*d **2*f**3*x**8 - a*b*c**2*e**3 + 3*a*b*c**2*e*f**2*x**4 + 2*a*b*c**2*f**3*x **6 - 2*a*b*c*d*e**3*x**2 + 6*a*b*c*d*e*f**2*x**6 + 4*a*b*c*d*f**3*x**8 - a*b*d**2*e**3*x**4 + 3*a*b*d**2*e*f**2*x**8 + 2*a*b*d**2*f**3*x**10 - b**2 *c**2*e**3*x**2 - 2*b**2*c**2*e**2*f*x**4 - b**2*c**2*e*f**2*x**6 - 2*b**2 *c*d*e**3*x**4 - 4*b**2*c*d*e**2*f*x**6 - 2*b**2*c*d*e*f**2*x**8 - b**2*d* *2*e**3*x**6 - 2*b**2*d**2*e**2*f*x**8 - b**2*d**2*e*f**2*x**10),x)*a**2*b *c*d*e*f**2 + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(2*a**2*c**2* e**2*f + 4*a**2*c**2*e*f**2*x**2 + 2*a**2*c**2*f**3*x**4 + 4*a**2*c*d*e**2 *f*x**2 + 8*a**2*c*d*e*f**2*x**4 + 4*a**2*c*d*f**3*x**6 + 2*a**2*d**2*e**2 *f*x**4 + 4*a**2*d**2*e*f**2*x**6 + 2*a**2*d**2*f**3*x**8 - a*b*c**2*e**3 + 3*a*b*c**2*e*f**2*x**4 + 2*a*b*c**2*f**3*x**6 - 2*a*b*c*d*e**3*x**2 + 6* a*b*c*d*e*f**2*x**6 + 4*a*b*c*d*f**3*x**8 - a*b*d**2*e**3*x**4 + 3*a*b*d** 2*e*f**2*x**8 + 2*a*b*d**2*f**3*x**10 - b**2*c**2*e**3*x**2 - 2*b**2*c**2* e**2*f*x**4 - b**2*c**2*e*f**2*x**6 - 2*b**2*c*d*e**3*x**4 - 4*b**2*c*d*e* *2*f*x**6 - 2*b**2*c*d*e*f**2*x**8 - b**2*d**2*e**3*x**6 - 2*b**2*d**2*e** 2*f*x**8 - b**2*d**2*e*f**2*x**10),x)*a**2*b*c*d*f**3*x**2 + 2*int((sqr...