Integrand size = 36, antiderivative size = 905 \[ \int \frac {1}{x^6 \left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {\sqrt {c+d x^2}}{5 a c e x^5 \sqrt {a-b x^2}}-\frac {(6 b c e-4 a d e-5 a c f) \sqrt {c+d x^2}}{15 a^2 c^2 e^2 x^3 \sqrt {a-b x^2}}-\frac {\left (\frac {24 b^2 c e}{a}-13 b d e+\frac {8 a d^2 e}{c}-20 b c f+10 a d f+\frac {15 a c f^2}{e}\right ) \sqrt {c+d x^2}}{15 a^2 c^2 e^2 x \sqrt {a-b x^2}}+\frac {b \left (48 b^4 c^3 e^3+8 a b^3 c^2 e^2 (2 d e+c f)-a^2 b^2 c e \left (9 d^2 e^2-c d e f+10 c^2 f^2\right )+a^4 d f \left (8 d^2 e^2+10 c d e f+15 c^2 f^2\right )+a^3 b \left (8 d^3 e^3+c d^2 e^2 f+15 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{15 a^4 c^3 (b c+a d) e^3 (b e+a f) \sqrt {a-b x^2}}-\frac {\sqrt {b} \left (48 b^4 c^3 e^3+8 a b^3 c^2 e^2 (2 d e+c f)-a^2 b^2 c e \left (9 d^2 e^2-c d e f+10 c^2 f^2\right )+a^4 d f \left (8 d^2 e^2+10 c d e f+15 c^2 f^2\right )+a^3 b \left (8 d^3 e^3+c d^2 e^2 f+15 c^3 f^3\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{15 a^{7/2} c^3 (b c+a d) e^3 (b e+a f) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {b} \left (48 b^3 c^2 e^3-8 a b^2 c e^2 (d e-c f)+a^2 b e \left (4 d^2 e^2-3 c d e f-10 c^2 f^2\right )+a^3 f \left (4 d^2 e^2+5 c d e f+15 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{15 a^{7/2} c^2 e^3 (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} f^4 \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e^4 (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
-1/5*(d*x^2+c)^(1/2)/a/c/e/x^5/(-b*x^2+a)^(1/2)-1/15*(-5*a*c*f-4*a*d*e+6*b *c*e)*(d*x^2+c)^(1/2)/a^2/c^2/e^2/x^3/(-b*x^2+a)^(1/2)-1/15*(24*b^2*c*e/a- 13*b*d*e+8*a*d^2*e/c-20*b*c*f+10*a*d*f+15*a*c*f^2/e)*(d*x^2+c)^(1/2)/a^2/c ^2/e^2/x/(-b*x^2+a)^(1/2)+1/15*b*(48*b^4*c^3*e^3+8*a*b^3*c^2*e^2*(c*f+2*d* e)-a^2*b^2*c*e*(10*c^2*f^2-c*d*e*f+9*d^2*e^2)+a^4*d*f*(15*c^2*f^2+10*c*d*e *f+8*d^2*e^2)+a^3*b*(15*c^3*f^3+c*d^2*e^2*f+8*d^3*e^3))*x*(d*x^2+c)^(1/2)/ a^4/c^3/(a*d+b*c)/e^3/(a*f+b*e)/(-b*x^2+a)^(1/2)-1/15*b^(1/2)*(48*b^4*c^3* e^3+8*a*b^3*c^2*e^2*(c*f+2*d*e)-a^2*b^2*c*e*(10*c^2*f^2-c*d*e*f+9*d^2*e^2) +a^4*d*f*(15*c^2*f^2+10*c*d*e*f+8*d^2*e^2)+a^3*b*(15*c^3*f^3+c*d^2*e^2*f+8 *d^3*e^3))*(1-b*x^2/a)^(1/2)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2),( -a*d/b/c)^(1/2))/a^(7/2)/c^3/(a*d+b*c)/e^3/(a*f+b*e)/(-b*x^2+a)^(1/2)/(1+d *x^2/c)^(1/2)+1/15*b^(1/2)*(48*b^3*c^2*e^3-8*a*b^2*c*e^2*(-c*f+d*e)+a^2*b* e*(-10*c^2*f^2-3*c*d*e*f+4*d^2*e^2)+a^3*f*(15*c^2*f^2+5*c*d*e*f+4*d^2*e^2) )*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),(-a*d/b/ c)^(1/2))/a^(7/2)/c^2/e^3/(a*f+b*e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)-a^(1/ 2)*f^4*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticPi(b^(1/2)*x/a^(1/2),-a *f/b/e,(-a*d/b/c)^(1/2))/b^(1/2)/e^4/(a*f+b*e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^ (1/2)
Result contains complex when optimal does not.
Time = 8.92 (sec) , antiderivative size = 1531, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x^6 \left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:
Integrate[1/(x^6*(a - b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
-1/15*(3*a^3*b^3*c^4*e^4 + 3*a^4*b^2*c^3*d*e^4 + 3*a^4*b^2*c^4*e^3*f + 3*a ^5*b*c^3*d*e^3*f + 6*a^2*b^4*c^4*e^4*x^2 + 5*a^3*b^3*c^3*d*e^4*x^2 - a^4*b ^2*c^2*d^2*e^4*x^2 + a^3*b^3*c^4*e^3*f*x^2 - a^5*b*c^2*d^2*e^3*f*x^2 - 5*a ^4*b^2*c^4*e^2*f^2*x^2 - 5*a^5*b*c^3*d*e^2*f^2*x^2 + 24*a*b^5*c^4*e^4*x^4 + 17*a^2*b^4*c^3*d*e^4*x^4 - 3*a^3*b^3*c^2*d^2*e^4*x^4 + 4*a^4*b^2*c*d^3*e ^4*x^4 + 4*a^2*b^4*c^4*e^3*f*x^4 + 2*a^3*b^3*c^3*d*e^3*f*x^4 + 2*a^4*b^2*c ^2*d^2*e^3*f*x^4 + 4*a^5*b*c*d^3*e^3*f*x^4 - 5*a^3*b^3*c^4*e^2*f^2*x^4 + 5 *a^5*b*c^2*d^2*e^2*f^2*x^4 + 15*a^4*b^2*c^4*e*f^3*x^4 + 15*a^5*b*c^3*d*e*f ^3*x^4 - 48*b^6*c^4*e^4*x^6 + 8*a*b^5*c^3*d*e^4*x^6 + 20*a^2*b^4*c^2*d^2*e ^4*x^6 - 13*a^3*b^3*c*d^3*e^4*x^6 + 8*a^4*b^2*d^4*e^4*x^6 - 8*a*b^5*c^4*e^ 3*f*x^6 + 3*a^2*b^4*c^3*d*e^3*f*x^6 - 3*a^4*b^2*c*d^3*e^3*f*x^6 + 8*a^5*b* d^4*e^3*f*x^6 + 10*a^2*b^4*c^4*e^2*f^2*x^6 - 5*a^3*b^3*c^3*d*e^2*f^2*x^6 - 5*a^4*b^2*c^2*d^2*e^2*f^2*x^6 + 10*a^5*b*c*d^3*e^2*f^2*x^6 - 15*a^3*b^3*c ^4*e*f^3*x^6 + 15*a^5*b*c^2*d^2*e*f^3*x^6 - 48*b^6*c^3*d*e^4*x^8 - 16*a*b^ 5*c^2*d^2*e^4*x^8 + 9*a^2*b^4*c*d^3*e^4*x^8 - 8*a^3*b^3*d^4*e^4*x^8 - 8*a* b^5*c^3*d*e^3*f*x^8 - a^2*b^4*c^2*d^2*e^3*f*x^8 - a^3*b^3*c*d^3*e^3*f*x^8 - 8*a^4*b^2*d^4*e^3*f*x^8 + 10*a^2*b^4*c^3*d*e^2*f^2*x^8 - 10*a^4*b^2*c*d^ 3*e^2*f^2*x^8 - 15*a^3*b^3*c^3*d*e*f^3*x^8 - 15*a^4*b^2*c^2*d^2*e*f^3*x^8 + I*a*b*Sqrt[-(b/a)]*c*e*(48*b^4*c^3*e^3 + 8*a*b^3*c^2*e^2*(2*d*e + c*f) + a^2*b^2*c*e*(-9*d^2*e^2 + c*d*e*f - 10*c^2*f^2) + a^4*d*f*(8*d^2*e^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.
| \(\displaystyle \int \frac {1}{x^6 \left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {1}{x^6 \left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )}dx\) |
Input:
Int[1/(x^6*(a - b*x^2)^(3/2)*Sqrt[c + d*x^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 = 26.17 (sec) , antiderivative size = 1138, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1138\) |
| elliptic | \(\text {Expression too large to display}\) | \(1844\) |
| default | \(\text {Expression too large to display}\) | \(3145\) |
Input:
int(1/x^6/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBO SE)
Output:
-1/15*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(15*a^2*c^2*f^2*x^4+10*a^2*c*d*e*f* x^4+8*a^2*d^2*e^2*x^4-25*a*b*c^2*e*f*x^4-17*a*b*c*d*e^2*x^4+33*b^2*c^2*e^2 *x^4-5*a^2*c^2*e*f*x^2-4*a^2*c*d*e^2*x^2+9*a*b*c^2*e^2*x^2+3*a^2*c^2*e^2)/ a^4/c^3/e^3/x^5-1/15/a^4/c^3/e^3*(-b*(15*a^2*c^2*f^2+10*a^2*c*d*e*f+8*a^2* d^2*e^2-25*a*b*c^2*e*f-17*a*b*c*d*e^2+33*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)*(Ellipt icF(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)))-9*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))+4*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^4*c^3*f^4/(a*f+b*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)*Ellip ticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2))+15*a*b^4*c^3*e^3/ (a*f+b*e)*((-b*d*x^2-b*c)/a/(a*d+b*c)*x/((x^2-a/b)*(-b*d*x^2-b*c))^(1/2)+( -1/a+b*c/a/(a*d+b*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))-b/a/(a*d+b*c)*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))))+5*a^2*...
Timed out. \[ \int \frac {1}{x^6 \left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x^6/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="f ricas")
Output:
Timed out
\[ \int \frac {1}{x^6 \left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {1}{x^{6} \left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(1/x**6/(-b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral(1/(x**6*(a - b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)), x)
\[ \int \frac {1}{x^6 \left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )} x^{6}} \,d x } \] Input:
integrate(1/x^6/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="m axima")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)*x^6), x)
\[ \int \frac {1}{x^6 \left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )} x^{6}} \,d x } \] Input:
integrate(1/x^6/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="g iac")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)*x^6), x)
Timed out. \[ \int \frac {1}{x^6 \left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {1}{x^6\,{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(1/(x^6*(a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int(1/(x^6*(a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {1}{x^6 \left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {too large to display} \] Input:
int(1/x^6/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a*c*e + 5*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a*c*f*x**2 + 4*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a*d*e*x**2 + 12 *sqrt(c + d*x**2)*sqrt(a - b*x**2)*a*d*f*x**4 - 6*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*c*e*x**2 - 18*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*c*f*x**4 - 1 5*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*d*e*x**4 - 12*int((sqrt(c + d*x**2)* sqrt(a - b*x**2)*x**4)/(a**2*c*e + a**2*c*f*x**2 + a**2*d*e*x**2 + a**2*d* f*x**4 - 2*a*b*c*e*x**2 - 2*a*b*c*f*x**4 - 2*a*b*d*e*x**4 - 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b**2*d*e*x**6 + b**2*d*f*x**8),x)*a**2* b*d**2*f**2*x**5 + 18*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c *e + a**2*c*f*x**2 + a**2*d*e*x**2 + a**2*d*f*x**4 - 2*a*b*c*e*x**2 - 2*a* b*c*f*x**4 - 2*a*b*d*e*x**4 - 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x* *6 + b**2*d*e*x**6 + b**2*d*f*x**8),x)*a*b**2*c*d*f**2*x**5 + 15*int((sqrt (c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c*e + a**2*c*f*x**2 + a**2*d*e*x **2 + a**2*d*f*x**4 - 2*a*b*c*e*x**2 - 2*a*b*c*f*x**4 - 2*a*b*d*e*x**4 - 2 *a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b**2*d*e*x**6 + b**2*d*f*x **8),x)*a*b**2*d**2*e*f*x**5 + 12*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x **4)/(a**2*c*e + a**2*c*f*x**2 + a**2*d*e*x**2 + a**2*d*f*x**4 - 2*a*b*c*e *x**2 - 2*a*b*c*f*x**4 - 2*a*b*d*e*x**4 - 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b**2*d*e*x**6 + b**2*d*f*x**8),x)*a*b**2*d**2*f**2*x**7 - 18*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c*e + a**2*c*f*x...