Integrand size = 35, antiderivative size = 771 \[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=-\frac {\left (4 a^2 d^2 f^2+4 a b d f (5 d e-c f)-b^2 \left (105 d^2 e^2-20 c d e f-4 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{30 b d^2 f^4 \sqrt {a+b x^2}}+\frac {\left (\frac {2 a f}{b}-\frac {35 d^2 e^2-16 c d e f-4 c^2 f^2}{d (d e-c f)}\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{30 f^3}+\frac {(b e (7 d e-2 c f)-2 a f (d e-c f)) x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{10 d f^2 (b e-a f) (d e-c f)}-\frac {e^2 x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{2 f (b e-a f) (d e-c f) \left (e+f x^2\right )}+\frac {\sqrt {a} \left (4 a^2 d^2 f^2+4 a b d f (5 d e-c f)-b^2 \left (105 d^2 e^2-20 c d e f-4 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{30 b^{3/2} d^2 f^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (2 a^2 c d f^3+b^2 e \left (105 d^2 e^2-55 c d e f-2 c^2 f^2\right )-2 a b f \left (45 d^2 e^2-19 c d e f-c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{30 b^{3/2} c d f^4 (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} e (b e (7 d e-6 c f)-a f (6 d e-5 c f)) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c f^4 (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/30*(4*a^2*d^2*f^2+4*a*b*d*f*(-c*f+5*d*e)-b^2*(-4*c^2*f^2-20*c*d*e*f+105 *d^2*e^2))*x*(d*x^2+c)^(1/2)/b/d^2/f^4/(b*x^2+a)^(1/2)+1/30*(2*a*f/b-(-4*c ^2*f^2-16*c*d*e*f+35*d^2*e^2)/d/(-c*f+d*e))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1 /2)/f^3+1/10*(b*e*(-2*c*f+7*d*e)-2*a*f*(-c*f+d*e))*x*(b*x^2+a)^(1/2)*(d*x^ 2+c)^(3/2)/d/f^2/(-a*f+b*e)/(-c*f+d*e)-1/2*e^2*x*(b*x^2+a)^(3/2)*(d*x^2+c) ^(3/2)/f/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)+1/30*a^(1/2)*(4*a^2*d^2*f^2+4*a*b *d*f*(-c*f+5*d*e)-b^2*(-4*c^2*f^2-20*c*d*e*f+105*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))/b^(3/2)/ d^2/f^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/30*a^(3/2)*(2*a^ 2*c*d*f^3+b^2*e*(-2*c^2*f^2-55*c*d*e*f+105*d^2*e^2)-2*a*b*f*(-c^2*f^2-19*c *d*e*f+45*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))/b^(3/2)/c/d/f^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)*e*(b*e*(-6*c*f+7*d*e)-a*f*(-5*c*f+6*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^4/(-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 = 4.79 (sec) , antiderivative size = 505, normalized size of antiderivative = 0.65 \[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\frac {i c f \left (4 a^2 d^2 f^2+4 a b d f (5 d e-c f)+b^2 \left (-105 d^2 e^2+20 c d e f+4 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i \left (2 a^2 c d^2 f^3+2 a b d f \left (45 d^2 e^2-10 c d e f-3 c^2 f^2\right )+b^2 \left (-105 d^3 e^3-15 c d^2 e^2 f+20 c^2 d e f^2+4 c^3 f^3\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d \left (\sqrt {\frac {b}{a}} f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (2 b c f \left (e+f x^2\right )+2 a d f \left (e+f x^2\right )+b d \left (-35 e^2-14 e f x^2+6 f^2 x^4\right )\right )-15 i b d e (b e (7 d e-6 c f)+a f (-6 d e+5 c f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{30 b \sqrt {\frac {b}{a}} d^2 f^5 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[(x^6*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(e + f*x^2)^2,x]
Output:
(I*c*f*(4*a^2*d^2*f^2 + 4*a*b*d*f*(5*d*e - c*f) + b^2*(-105*d^2*e^2 + 20*c *d*e*f + 4*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*E llipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(2*a^2*c*d^2*f^3 + 2*a*b *d*f*(45*d^2*e^2 - 10*c*d*e*f - 3*c^2*f^2) + b^2*(-105*d^3*e^3 - 15*c*d^2* e^2*f + 20*c^2*d*e*f^2 + 4*c^3*f^3))*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]*f^2*x*(a + b*x^2)*(c + d*x^2)*(2*b*c*f*(e + f*x^2) + 2*a*d*f*(e + f*x^2 ) + b*d*(-35*e^2 - 14*e*f*x^2 + 6*f^2*x^4)) - (15*I)*b*d*e*(b*e*(7*d*e - 6 *c*f) + a*f*(-6*d*e + 5*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)]))/(30 *b*Sqrt[b/a]*d^2*f^5*Sqrt[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.
| \(\displaystyle \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2}dx\) |
Input:
Int[(x^6*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. \(1537\) vs. \(2(727)=1454\).
Time = 20.29 (sec) , antiderivative size = 1538, normalized size of antiderivative = 1.99
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1538\) |
| elliptic | \(\text {Expression too large to display}\) | \(1932\) |
| default | \(\text {Expression too large to display}\) | \(2886\) |
Input:
int(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^2,x,method=_RETURNVERBOS E)
Output:
1/15*x*(3*b*d*f*x^2+a*d*f+b*c*f-10*b*d*e)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/ b/d/f^3-1/15/b/d/f^3*((a^2*c*d*f^3+a*b*c^2*f^3+20*a*b*c*d*e*f^2-45*a*b*d^2 *e^2*f-45*b^2*c*d*e^2*f+60*b^2*d^2*e^3)/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)*EllipticF(x*(-b/a)^ (1/2),(-1+(a*d+b*c)/c/b)^(1/2))-1/f*(2*a^2*d^2*f^2-2*a*b*c*d*f^2+10*a*b*d^ 2*e*f+2*b^2*c^2*f^2+10*b^2*c*d*e*f-45*b^2*d^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)/d*(Ellipti cF(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)))-15*b*d*e/f^2*(3*a*c*f^2-4*a*d*e*f-4*b*c*e*f+5*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)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2 ))+15*b*d*e^3*(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/f^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*(-b/a)^(1/2) ,(-1+(a*d+b*c)/c/b)^(1/2))+1/2*f*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-1/2*f*b/(a*c* f^2-a*d*e*f-b*c*e*f+b*d*e^2)/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)*EllipticE(x*(-b/a)^(1/2),(-...
Timed out. \[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\int \frac {x^{6} \sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}{\left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(x**6*(b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)/(f*x**2+e)**2,x)
Output:
Integral(x**6*sqrt(a + b*x**2)*sqrt(c + d*x**2)/(e + f*x**2)**2, x)
\[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{6}}{{\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^6*(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)*sqrt(d*x^2 + c)*x^6/(f*x^2 + e)^2, x)
\[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{6}}{{\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(x^6*(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)*sqrt(d*x^2 + c)*x^6/(f*x^2 + e)^2, x)
Timed out. \[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\int \frac {x^6\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}}{{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int((x^6*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2))/(e + f*x^2)^2,x)
Output:
int((x^6*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2))/(e + f*x^2)^2, x)
\[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\int \frac {x^{6} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{\left (f \,x^{2}+e \right )^{2}}d x \] Input:
int(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)
Output:
int(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)