Integrand size = 35, antiderivative size = 887 \[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\left (b^2 c^2 d e^2 (d e-c f)-a^2 d \left (48 d^3 e^3-8 c d^2 e^2 f-10 c^2 d e f^2-15 c^3 f^3\right )+a b c \left (32 d^3 e^3-7 c d^2 e^2 f-10 c^2 d e f^2-15 c^3 f^3\right )\right ) x \sqrt {a+b x^2}}{15 a^2 c^4 e^3 (d e-c f) \sqrt {c+d x^2}}-\frac {\left (a+b x^2\right )^{3/2}}{5 a c e x^5 \sqrt {c+d x^2}}+\frac {(2 b c e+6 a d e+5 a c f) \left (a+b x^2\right )^{3/2}}{15 a^2 c^2 e^2 x^3 \sqrt {c+d x^2}}+\frac {\left (b c d e^2-a \left (24 d^2 e^2+20 c d e f+15 c^2 f^2\right )\right ) \left (a+b x^2\right )^{3/2}}{15 a^2 c^3 e^3 x \sqrt {c+d x^2}}-\frac {b \left (2 b^2 c^2 e^2 (d e-c f)+a b c e \left (8 d^2 e^2-3 c d e f-5 c^2 f^2\right )-a^2 \left (48 d^3 e^3-8 c d^2 e^2 f-10 c^2 d e f^2-15 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{15 a^2 c^4 e^3 (d e-c f) \sqrt {a+b x^2}}+\frac {\sqrt {b} \left (2 b^2 c^2 e^2 (d e-c f)+a b c e \left (8 d^2 e^2-3 c d e f-5 c^2 f^2\right )-a^2 \left (48 d^3 e^3-8 c d^2 e^2 f-10 c^2 d e f^2-15 c^3 f^3\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{3/2} c^4 e^3 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {b} \left (b c d e^2-a \left (24 d^2 e^2+20 c d e f+15 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 \sqrt {a} c^4 e^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} f^4 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e^4 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/15*(b^2*c^2*d*e^2*(-c*f+d*e)-a^2*d*(-15*c^3*f^3-10*c^2*d*e*f^2-8*c*d^2*e ^2*f+48*d^3*e^3)+a*b*c*(-15*c^3*f^3-10*c^2*d*e*f^2-7*c*d^2*e^2*f+32*d^3*e^ 3))*x*(b*x^2+a)^(1/2)/a^2/c^4/e^3/(-c*f+d*e)/(d*x^2+c)^(1/2)-1/5*(b*x^2+a) ^(3/2)/a/c/e/x^5/(d*x^2+c)^(1/2)+1/15*(5*a*c*f+6*a*d*e+2*b*c*e)*(b*x^2+a)^ (3/2)/a^2/c^2/e^2/x^3/(d*x^2+c)^(1/2)+1/15*(b*c*d*e^2-a*(15*c^2*f^2+20*c*d *e*f+24*d^2*e^2))*(b*x^2+a)^(3/2)/a^2/c^3/e^3/x/(d*x^2+c)^(1/2)-1/15*b*(2* b^2*c^2*e^2*(-c*f+d*e)+a*b*c*e*(-5*c^2*f^2-3*c*d*e*f+8*d^2*e^2)-a^2*(-15*c ^3*f^3-10*c^2*d*e*f^2-8*c*d^2*e^2*f+48*d^3*e^3))*x*(d*x^2+c)^(1/2)/a^2/c^4 /e^3/(-c*f+d*e)/(b*x^2+a)^(1/2)+1/15*b^(1/2)*(2*b^2*c^2*e^2*(-c*f+d*e)+a*b *c*e*(-5*c^2*f^2-3*c*d*e*f+8*d^2*e^2)-a^2*(-15*c^3*f^3-10*c^2*d*e*f^2-8*c* d^2*e^2*f+48*d^3*e^3))*(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^4/e^3/(-c*f+d*e)/(b*x^2+a)^(1/2)/( a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*b^(1/2)*(b*c*d*e^2-a*(15*c^2*f^2+20*c* d*e*f+24*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))/a^(1/2)/c^4/e^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^ 2+a))^(1/2)+a^(3/2)*f^4*(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^4/(-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 = 6.00 (sec) , antiderivative size = 1430, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:
Integrate[Sqrt[a + b*x^2]/(x^6*(c + d*x^2)^(3/2)*(e + f*x^2)),x]
Output:
(3*a^3*Sqrt[b/a]*c^3*d*e^4 - 3*a^3*Sqrt[b/a]*c^4*e^3*f + 4*a^3*(b/a)^(3/2) *c^3*d*e^4*x^2 - 6*a^3*Sqrt[b/a]*c^2*d^2*e^4*x^2 - 4*a^3*(b/a)^(3/2)*c^4*e ^3*f*x^2 + a^3*Sqrt[b/a]*c^3*d*e^3*f*x^2 + 5*a^3*Sqrt[b/a]*c^4*e^2*f^2*x^2 - a*b^2*Sqrt[b/a]*c^3*d*e^4*x^4 - 13*a^3*(b/a)^(3/2)*c^2*d^2*e^4*x^4 + 24 *a^3*Sqrt[b/a]*c*d^3*e^4*x^4 + a*b^2*Sqrt[b/a]*c^4*e^3*f*x^4 + 3*a^3*(b/a) ^(3/2)*c^3*d*e^3*f*x^4 - 4*a^3*Sqrt[b/a]*c^2*d^2*e^3*f*x^4 + 10*a^3*(b/a)^ (3/2)*c^4*e^2*f^2*x^4 - 5*a^3*Sqrt[b/a]*c^3*d*e^2*f^2*x^4 - 15*a^3*Sqrt[b/ a]*c^4*e*f^3*x^4 - 2*b^3*Sqrt[b/a]*c^3*d*e^4*x^6 - 9*a*b^2*Sqrt[b/a]*c^2*d ^2*e^4*x^6 + 16*a^3*(b/a)^(3/2)*c*d^3*e^4*x^6 + 48*a^3*Sqrt[b/a]*d^4*e^4*x ^6 + 2*b^3*Sqrt[b/a]*c^4*e^3*f*x^6 + 4*a*b^2*Sqrt[b/a]*c^3*d*e^3*f*x^6 - a ^3*(b/a)^(3/2)*c^2*d^2*e^3*f*x^6 - 8*a^3*Sqrt[b/a]*c*d^3*e^3*f*x^6 + 5*a*b ^2*Sqrt[b/a]*c^4*e^2*f^2*x^6 - 10*a^3*Sqrt[b/a]*c^2*d^2*e^2*f^2*x^6 - 15*a ^3*(b/a)^(3/2)*c^4*e*f^3*x^6 - 15*a^3*Sqrt[b/a]*c^3*d*e*f^3*x^6 - 2*b^3*Sq rt[b/a]*c^2*d^2*e^4*x^8 - 8*a*b^2*Sqrt[b/a]*c*d^3*e^4*x^8 + 48*a^3*(b/a)^( 3/2)*d^4*e^4*x^8 + 2*b^3*Sqrt[b/a]*c^3*d*e^3*f*x^8 + 3*a*b^2*Sqrt[b/a]*c^2 *d^2*e^3*f*x^8 - 8*a^3*(b/a)^(3/2)*c*d^3*e^3*f*x^8 + 5*a*b^2*Sqrt[b/a]*c^3 *d*e^2*f^2*x^8 - 10*a^3*(b/a)^(3/2)*c^2*d^2*e^2*f^2*x^8 - 15*a^3*(b/a)^(3/ 2)*c^3*d*e*f^3*x^8 - I*b*c*e*(2*b^2*c^2*e^2*(d*e - c*f) + a*b*c*e*(8*d^2*e ^2 - 3*c*d*e*f - 5*c^2*f^2) + a^2*(-48*d^3*e^3 + 8*c*d^2*e^2*f + 10*c^2*d* e*f^2 + 15*c^3*f^3))*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipt...
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}dx\) |
Input:
Int[Sqrt[a + b*x^2]/(x^6*(c + d*x^2)^(3/2)*(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 = 23.89 (sec) , antiderivative size = 1123, normalized size of antiderivative = 1.27
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1123\) |
| elliptic | \(\text {Expression too large to display}\) | \(1780\) |
| default | \(\text {Expression too large to display}\) | \(2310\) |
Input:
int((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
-1/15*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(15*a^2*c^2*f^2*x^4+25*a^2*c*d*e*f*x ^4+33*a^2*d^2*e^2*x^4-5*a*b*c^2*e*f*x^4-8*a*b*c*d*e^2*x^4-2*b^2*c^2*e^2*x^ 4-5*a^2*c^2*e*f*x^2-9*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 ^4/e^3/x^5+1/15/e^3/c^4/a^2*(-b*(15*a^2*c^2*f^2+25*a^2*c*d*e*f+33*a^2*d^2* e^2-5*a*b*c^2*e*f-8*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))+9*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))+5*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^4*f^3*(a*f-b*e)/(c*f-d*e)/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))+15*a^ 2*c*d^3*e^3*(a*d-b*c)/(c*f-d*e)*((b*d*x^2+a*d)/c/(a*d-b*c)*x/((x^2+c/d)*(b *d*x^2+a*d))^(1/2)+(1/c-1/(a*d-b*c)/c*a*d)/(-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/(a*d-b*c)/(-b/a)^(1/2)*(1+b*x^2/a)^(1/...
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {a + b x^{2}}}{x^{6} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate((b*x**2+a)**(1/2)/x**6/(d*x**2+c)**(3/2)/(f*x**2+e),x)
Output:
Integral(sqrt(a + b*x**2)/(x**6*(c + d*x**2)**(3/2)*(e + f*x**2)), x)
\[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )} x^{6}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="maxi ma")
Output:
integrate(sqrt(b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)*x^6), x)
\[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )} x^{6}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="giac ")
Output:
integrate(sqrt(b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)*x^6), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {b\,x^2+a}}{x^6\,{\left (d\,x^2+c\right )}^{3/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int((a + b*x^2)^(1/2)/(x^6*(c + d*x^2)^(3/2)*(e + f*x^2)),x)
Output:
int((a + b*x^2)^(1/2)/(x^6*(c + d*x^2)^(3/2)*(e + f*x^2)), x)
\[ \int \frac {\sqrt {a+b x^2}}{x^6 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d^{2} f \,x^{12}+2 c d f \,x^{10}+d^{2} e \,x^{10}+c^{2} f \,x^{8}+2 c d e \,x^{8}+c^{2} e \,x^{6}}d x \] Input:
int((b*x^2+a)^(1/2)/x^6/(d*x^2+c)^(3/2)/(f*x^2+e),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c**2*e*x**6 + c**2*f*x**8 + 2*c*d *e*x**8 + 2*c*d*f*x**10 + d**2*e*x**10 + d**2*f*x**12),x)