Integrand size = 35, antiderivative size = 1095 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Output:
-1/30*b*(4*b^2*c^2*e^2-4*a*b*c*e*(-5*c*f+d*e)+a^2*(-105*c^2*f^2+20*c*d*e*f +4*d^2*e^2))*x*(d*x^2+c)^(1/2)/a^2/c^2/e^4/(b*x^2+a)^(1/2)+1/30*d*(15*a^2* c*f^2*(-7*c*f+6*d*e)-4*b^2*c*e^2*(-c*f+d*e)+2*a*b*e*(10*c^2*f^2-11*c*d*e*f +d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^4/(-c*f+d*e)-1/10*b *(5*a^2*c*f^3*(-7*c*f+6*d*e)-2*b^2*d*e^3*(-c*f+d*e)+2*a*b*e*f*(15*c^2*f^2- 16*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/a^2/c^2/e^4/(-a*f+b *e)/(-c*f+d*e)-1/5*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)/a/c/e/x^5/(f*x^2+e)+1/1 5*(7*a*c*f+2*a*d*e+2*b*c*e)*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)/a^2/c^2/e^2/x^ 3/(f*x^2+e)+1/15*(a*f*(-35*c*f+2*d*e)-b*e*(-2*c*f+3*d*e))*(b*x^2+a)^(3/2)* (d*x^2+c)^(3/2)/a^2/c^2/e^3/x/(f*x^2+e)+1/10*f*(5*a^2*c*f^3*(-7*c*f+6*d*e) -2*b^2*d*e^3*(-c*f+d*e)+2*a*b*e*f*(15*c^2*f^2-16*c*d*e*f+d^2*e^2))*x*(b*x^ 2+a)^(3/2)*(d*x^2+c)^(3/2)/a^2/c^2/e^4/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)+1/3 0*b^(1/2)*(4*b^2*c^2*e^2-4*a*b*c*e*(-5*c*f+d*e)+a^2*(-105*c^2*f^2+20*c*d*e *f+4*d^2*e^2))*(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))/a^(3/2)/c^2/e^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^ 2+a))^(1/2)-1/30*b^(1/2)*(2*b^2*c*d*e^3-a^2*f*(-105*c^2*f^2+55*c*d*e*f+2*d ^2*e^2)+2*a*b*e*(-45*c^2*f^2+19*c*d*e*f+d^2*e^2))*(d*x^2+c)^(1/2)*InverseJ acobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/a^(1/2)/c^2/e^4/(-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)*f^2*(a*f *(-7*c*f+6*d*e)-b*e*(-6*c*f+5*d*e))*(d*x^2+c)^(1/2)*EllipticPi(b^(1/2)*...
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\frac {-\frac {e \left (a+b x^2\right ) \left (c+d x^2\right ) \left (15 a^2 c^2 f^3 x^6+6 a^2 c^2 e^2 \left (e+f x^2\right )+2 a c e (b c e+a d e-10 a c f) x^2 \left (e+f x^2\right )-2 \left (2 b^2 c^2 e^2+2 a b c e (-d e+5 c f)+a^2 \left (2 d^2 e^2+10 c d e f-45 c^2 f^2\right )\right ) x^4 \left (e+f x^2\right )\right )}{x^5 \left (e+f x^2\right )}+\frac {i c \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (b e \left (4 b^2 c^2 e^2+4 a b c e (-d e+5 c f)+a^2 \left (4 d^2 e^2+20 c d e f-105 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-b e \left (4 b^2 c^2 e^2+2 a b c e (-3 d e+10 c f)+a^2 \left (2 d^2 e^2+55 c d e f-105 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+15 a^2 c f (b e (5 d e-6 c f)+a f (-6 d e+7 c f)) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}}}}{30 a^2 c^2 e^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^6*(e + f*x^2)^2),x]
Output:
(-((e*(a + b*x^2)*(c + d*x^2)*(15*a^2*c^2*f^3*x^6 + 6*a^2*c^2*e^2*(e + f*x ^2) + 2*a*c*e*(b*c*e + a*d*e - 10*a*c*f)*x^2*(e + f*x^2) - 2*(2*b^2*c^2*e^ 2 + 2*a*b*c*e*(-(d*e) + 5*c*f) + a^2*(2*d^2*e^2 + 10*c*d*e*f - 45*c^2*f^2) )*x^4*(e + f*x^2)))/(x^5*(e + f*x^2))) + (I*c*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(b*e*(4*b^2*c^2*e^2 + 4*a*b*c*e*(-(d*e) + 5*c*f) + a^2*(4*d^2* e^2 + 20*c*d*e*f - 105*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/( b*c)] - b*e*(4*b^2*c^2*e^2 + 2*a*b*c*e*(-3*d*e + 10*c*f) + a^2*(2*d^2*e^2 + 55*c*d*e*f - 105*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c) ] + 15*a^2*c*f*(b*e*(5*d*e - 6*c*f) + a*f*(-6*d*e + 7*c*f))*EllipticPi[(a* f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/Sqrt[b/a])/(30*a^2*c^2*e^ 5*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.
\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 450 |
\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2}dx\) |
Input:
Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^6*(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 = 22.35 (sec) , antiderivative size = 1785, normalized size of antiderivative = 1.63
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1785\) |
elliptic | \(\text {Expression too large to display}\) | \(1829\) |
default | \(\text {Expression too large to display}\) | \(3178\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x,method=_RETURNVERBOS E)
Output:
-1/15*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(45*a^2*c^2*f^2*x^4-10*a^2*c*d*e*f*x ^4-2*a^2*d^2*e^2*x^4-10*a*b*c^2*e*f*x^4+2*a*b*c*d*e^2*x^4-2*b^2*c^2*e^2*x^ 4-10*a^2*c^2*e*f*x^2+a^2*c*d*e^2*x^2+a*b*c^2*e^2*x^2+3*a^2*c^2*e^2)/a^2/c^ 2/e^4/x^5+1/15/a^2/c^2/e^4*(-b*(45*a^2*c^2*f^2-10*a^2*c*d*e*f-2*a^2*d^2*e^ 2-10*a*b*c^2*e*f+2*a*b*c*d*e^2-2*b^2*c^2*e^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)*(EllipticF(x*(- b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c )/c/b)^(1/2)))-a*b^2*c^2*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)*EllipticF(x*(-b/a)^(1/2),(-1+(a* d+b*c)/c/b)^(1/2))-a^2*b*c*d^2*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)*EllipticF(x*(-b/a)^(1/2),(-1 +(a*d+b*c)/c/b)^(1/2))-15*a^2*c^2*f*(3*a*c*f^2-2*a*d*e*f-2*b*c*e*f+b*d*e^2 )/e/(-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))+10*a^2*b*c^2*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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c /b)^(1/2))-15*a^2*c^2*e*f*(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)*(1/2*f^2/(a*c* f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^ 2+e)-1/2*d*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*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)*EllipticF(x*(...
Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}{x^{6} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)/x**6/(f*x**2+e)**2,x)
Output:
Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)/(x**6*(e + f*x**2)**2), x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (f x^{2} + e\right )}^{2} x^{6}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/((f*x^2 + e)^2*x^6), x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (f x^{2} + e\right )}^{2} x^{6}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/((f*x^2 + e)^2*x^6), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}}{x^6\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2))/(x^6*(e + f*x^2)^2),x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2))/(x^6*(e + f*x^2)^2), x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{x^{6} \left (f \,x^{2}+e \right )^{2}}d x \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x)
Output:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/x^6/(f*x^2+e)^2,x)