Integrand size = 35, antiderivative size = 538 \[ \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {b (3 d e-2 c f) x \sqrt {c+d x^2}}{2 d f^2 (d e-c f) \sqrt {a+b x^2}}-\frac {b e x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 f (b e-a f) (d e-c f)}+\frac {e x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{2 (b e-a f) (d e-c f) \left (e+f x^2\right )}-\frac {\sqrt {a} \sqrt {b} (3 d e-2 c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{2 d f^2 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} (3 b e-2 a f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c f^2 (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (b e (3 d e-4 c f)-a f (2 d e-3 c f)) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c f^2 (b e-a f) (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/2*b*(-2*c*f+3*d*e)*x*(d*x^2+c)^(1/2)/d/f^2/(-c*f+d*e)/(b*x^2+a)^(1/2)-1/ 2*b*e*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f/(-a*f+b*e)/(-c*f+d*e)+1/2*e*x*(b *x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)-1/2*a^(1/2)* b^(1/2)*(-2*c*f+3*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^2/(-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)*(-2*a*f+3*b*e)*(d*x^2+c)^(1/2)*InverseJac obiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/f^2/(-a*f+b*e )/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/2*a^(3/2)*(b*e*(-4*c*f +3*d*e)-a*f*(-3*c*f+2*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^2/(-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)
Result contains complex when optimal does not.
Time = 3.97 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.67 \[ \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {i b c f (-3 d e+2 c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-d e+c f) (-3 b d e-2 b c f+2 a d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-d \left (\sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right )+i (b e (3 d e-4 c f)+a f (-2 d e+3 c f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{2 \sqrt {\frac {b}{a}} d f^3 (d e-c f) \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[(x^4*Sqrt[a + b*x^2])/(Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
(I*b*c*f*(-3*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*(-(d*e) + c*f)*(-3* b*d*e - 2*b*c*f + 2*a*d*f)*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)] - d*(Sqrt[b/a]*e*f^2*x *(a + b*x^2)*(c + d*x^2) + I*(b*e*(3*d*e - 4*c*f) + a*f*(-2*d*e + 3*c*f))* Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticPi[(a*f)/(b*e) , I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(2*Sqrt[b/a]*d*f^3*(d*e - c*f)*Sq rt[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.
| \(\displaystyle \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2}dx\) |
Input:
Int[(x^4*Sqrt[a + b*x^2])/(Sqrt[c + d*x^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]
Leaf count of result is larger than twice the leaf count of optimal. \(1213\) vs. \(2(500)=1000\).
Time = 6.74 (sec) , antiderivative size = 1214, normalized size of antiderivative = 2.26
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1214\) |
| default | \(\text {Expression too large to display}\) | \(1568\) |
Input:
int(x^4*(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(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)/f*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+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)*Elliptic F(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*a/f^2-2/(-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/f^3*e-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^3/(c*f-d*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*b/f^2*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* b/f^2*EllipticE(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/f^2/(c*f-d*e)*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/f^2/(c*f-d*e)*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^ (1/2))-3/2/(c*f-d*e)/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))*a*c+e/(c*f-d*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/...
Timed out. \[ \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^4*(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {x^{4} \sqrt {a + b x^{2}}}{\sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(x**4*(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**2,x)
Output:
Integral(x**4*sqrt(a + b*x**2)/(sqrt(c + d*x**2)*(e + f*x**2)**2), x)
\[ \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} x^{4}}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^4*(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*x^4/(sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
\[ \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} x^{4}}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^4*(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*x^4/(sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
Timed out. \[ \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {x^4\,\sqrt {b\,x^2+a}}{\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int((x^4*(a + b*x^2)^(1/2))/((c + d*x^2)^(1/2)*(e + f*x^2)^2),x)
Output:
int((x^4*(a + b*x^2)^(1/2))/((c + d*x^2)^(1/2)*(e + f*x^2)^2), x)
\[ \int \frac {x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {too large to display} \] Input:
int(x^4*(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)
Output:
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*x - int((sqrt(c + d*x**2)*sqrt(a + b* x**2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2 *a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2* x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*a*b*d*e*f - int( (sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c* f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f **2*x**8),x)*a*b*d*f**2*x**2 + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x* *6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f *x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b *d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*b**2*d*e**2 + 3*int((sqr t(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2 *x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2 *b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2* x**8),x)*b**2*d*e*f*x**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a *c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e** 2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*a**2*c*e*f + int((sqrt(c + d*x **2)*sqrt(a + b*x**2)*x**2)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a *d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e...