Integrand size = 35, antiderivative size = 881 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^2 \left (e+f x^2\right )^2} \, dx=-\frac {b (d e-3 c f) x \sqrt {c+d x^2}}{2 e^2 f \sqrt {a+b x^2}}-\frac {\left (a^2 d f (5 d e-6 c f)-2 b^2 c e (d e-c f)-a b \left (5 d^2 e^2-8 c d e f+3 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 a e^2 (b e-a f) (d e-c f)}-\frac {d \left (a^2 d f (2 d e-3 c f)-4 b^2 c e (d e-c f)-a b \left (2 d^2 e^2-7 c d e f+6 c^2 f^2\right )\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 a c e^2 (b e-a f) (d e-c f)}-\frac {b d^2 (a f (2 d e-3 c f)-2 b e (d e-c f)) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 a c e^2 (b e-a f) (d e-c f)}-\frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}{a c e x \left (e+f x^2\right )}+\frac {f (a f (2 d e-3 c f)-2 b e (d e-c f)) x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}{2 a c e^2 (b e-a f) (d e-c f) \left (e+f x^2\right )}+\frac {\sqrt {a} \sqrt {b} (d e-3 c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{2 e^2 f \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} \sqrt {b} \left (2 b c^2 e f-a \left (d^2 e^2-2 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 c e^2 f (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (d e-c f) \left (3 a c f^2-b e (d e+2 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c e^3 f (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/2*b*(-3*c*f+d*e)*x*(d*x^2+c)^(1/2)/e^2/f/(b*x^2+a)^(1/2)-1/2*(a^2*d*f*( -6*c*f+5*d*e)-2*b^2*c*e*(-c*f+d*e)-a*b*(3*c^2*f^2-8*c*d*e*f+5*d^2*e^2))*x* (b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/e^2/(-a*f+b*e)/(-c*f+d*e)-1/2*d*(a^2*d*f *(-3*c*f+2*d*e)-4*b^2*c*e*(-c*f+d*e)-a*b*(6*c^2*f^2-7*c*d*e*f+2*d^2*e^2))* x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e^2/(-a*f+b*e)/(-c*f+d*e)-1/2*b*d^ 2*(a*f*(-3*c*f+2*d*e)-2*b*e*(-c*f+d*e))*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2 )/a/c/e^2/(-a*f+b*e)/(-c*f+d*e)-(b*x^2+a)^(3/2)*(d*x^2+c)^(5/2)/a/c/e/x/(f *x^2+e)+1/2*f*(a*f*(-3*c*f+2*d*e)-2*b*e*(-c*f+d*e))*x*(b*x^2+a)^(3/2)*(d*x ^2+c)^(5/2)/a/c/e^2/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)+1/2*a^(1/2)*b^(1/2)*(- 3*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))/e^2/f/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/ 2*a^(1/2)*b^(1/2)*(2*b*c^2*e*f-a*(3*c^2*f^2-2*c*d*e*f+d^2*e^2))*(d*x^2+c)^ (1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/c/e^2/f /(-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)*(- c*f+d*e)*(3*a*c*f^2-b*e*(2*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/e^3/f/(-a *f+b*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 = 4.91 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^2 \left (e+f x^2\right )^2} \, dx=\frac {\frac {e \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 c e+d e x^2-3 c f x^2\right )}{x \left (e+f x^2\right )}+\frac {i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (-b c e f (-d e+3 c f) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(d e-c f) \left (b e (d e+3 c f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-\left (-3 a c f^2+b e (d e+2 c f)\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{\sqrt {\frac {b}{a}} f^2}}{2 e^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(x^2*(e + f*x^2)^2),x]
Output:
((e*(a + b*x^2)*(c + d*x^2)*(-2*c*e + d*e*x^2 - 3*c*f*x^2))/(x*(e + f*x^2) ) + (I*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(-(b*c*e*f*(-(d*e) + 3*c*f) *EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]) - (d*e - c*f)*(b*e*(d*e + 3*c*f)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (-3*a*c*f^2 + b*e *(d*e + 2*c*f))*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c )])))/(Sqrt[b/a]*f^2))/(2*e^3*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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^2 \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^2 \left (e+f x^2\right )^2}dx\) |
Input:
Int[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(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]
Time = 19.74 (sec) , antiderivative size = 1102, normalized size of antiderivative = 1.25
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1102\) |
| risch | \(\text {Expression too large to display}\) | \(1423\) |
| default | \(\text {Expression too large to display}\) | \(1632\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^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*(c*f-d*e )/e^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+e)-c/e^2*(b*d*x^4+a*d*x ^2+b*c*x^2+a*c)^(1/2)/x-3/2*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)*b/e^2*EllipticF(x*(-b/a)^(1/2), (-1+(a*d+b*c)/c/b)^(1/2))+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^2/f^2+1/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))*b*c^2-1/2/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/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b*d^2+3/2*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)*b /e^2*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-3/2/e^3*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^2+3/ 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)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^( 1/2))*a*c*d+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) )/e/f*b*d*c-1/2*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d...
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^2 \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^2/(f*x^2+e)^2,x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^2 \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}{x^{2} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)/x**2/(f*x**2+e)**2,x)
Output:
Integral(sqrt(a + b*x**2)*(c + d*x**2)**(3/2)/(x**2*(e + f*x**2)**2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^2 \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e\right )}^{2} x^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^2/(f*x^2+e)^2,x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)/((f*x^2 + e)^2*x^2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^2 \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e\right )}^{2} x^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^2/(f*x^2+e)^2,x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)/((f*x^2 + e)^2*x^2), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^2 \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}}{x^2\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(x^2*(e + f*x^2)^2),x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(x^2*(e + f*x^2)^2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^2 \left (e+f x^2\right )^2} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^2/(f*x^2+e)^2,x)
Output:
( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*c - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(2*a**2*c*d*e**2*f + 4*a**2*c*d*e*f**2*x**2 + 2*a**2*c*d*f* *3*x**4 + 2*a**2*d**2*e**2*f*x**2 + 4*a**2*d**2*e*f**2*x**4 + 2*a**2*d**2* f**3*x**6 + 2*a*b*c**2*e**2*f + 4*a*b*c**2*e*f**2*x**2 + 2*a*b*c**2*f**3*x **4 - a*b*c*d*e**3 + 2*a*b*c*d*e**2*f*x**2 + 7*a*b*c*d*e*f**2*x**4 + 4*a*b *c*d*f**3*x**6 - a*b*d**2*e**3*x**2 + 3*a*b*d**2*e*f**2*x**6 + 2*a*b*d**2* f**3*x**8 + 2*b**2*c**2*e**2*f*x**2 + 4*b**2*c**2*e*f**2*x**4 + 2*b**2*c** 2*f**3*x**6 - b**2*c*d*e**3*x**2 + 3*b**2*c*d*e*f**2*x**6 + 2*b**2*c*d*f** 3*x**8 - b**2*d**2*e**3*x**4 - 2*b**2*d**2*e**2*f*x**6 - b**2*d**2*e*f**2* x**8),x)*a*b*c*d**2*e*f**2*x - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x* *4)/(2*a**2*c*d*e**2*f + 4*a**2*c*d*e*f**2*x**2 + 2*a**2*c*d*f**3*x**4 + 2 *a**2*d**2*e**2*f*x**2 + 4*a**2*d**2*e*f**2*x**4 + 2*a**2*d**2*f**3*x**6 + 2*a*b*c**2*e**2*f + 4*a*b*c**2*e*f**2*x**2 + 2*a*b*c**2*f**3*x**4 - a*b*c *d*e**3 + 2*a*b*c*d*e**2*f*x**2 + 7*a*b*c*d*e*f**2*x**4 + 4*a*b*c*d*f**3*x **6 - a*b*d**2*e**3*x**2 + 3*a*b*d**2*e*f**2*x**6 + 2*a*b*d**2*f**3*x**8 + 2*b**2*c**2*e**2*f*x**2 + 4*b**2*c**2*e*f**2*x**4 + 2*b**2*c**2*f**3*x**6 - b**2*c*d*e**3*x**2 + 3*b**2*c*d*e*f**2*x**6 + 2*b**2*c*d*f**3*x**8 - b* *2*d**2*e**3*x**4 - 2*b**2*d**2*e**2*f*x**6 - b**2*d**2*e*f**2*x**8),x)*a* b*c*d**2*f**3*x**3 + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(2*a** 2*c*d*e**2*f + 4*a**2*c*d*e*f**2*x**2 + 2*a**2*c*d*f**3*x**4 + 2*a**2*d...