Integrand size = 44, antiderivative size = 967 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\frac {\left (C e^2-B e f+A f^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 e (b e-a f) (d e-c f) \left (e+f x^2\right )^2}-\frac {\left (b \left (C e^3 (d e+2 c f)+e f (3 A f (3 d e-2 c f)-B e (5 d e-2 c f))\right )+a f \left (C e^2 (2 d e-5 c f)-f (3 A f (2 d e-c f)-B e (2 d e+c f))\right )\right ) x \sqrt {a+b x^2}}{8 e^2 f (b e-a f)^2 (d e-c f) \sqrt {c+d x^2} \left (e+f x^2\right )}+\frac {\sqrt {c} \sqrt {d} \left (b \left (C e^3 (d e+2 c f)+e f (3 A f (3 d e-2 c f)-B e (5 d e-2 c f))\right )+a f \left (C e^2 (2 d e-5 c f)-f (3 A f (2 d e-c f)-B e (2 d e+c f))\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{8 e^2 f (b e-a f)^2 (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {d} \left (a f^2 \left (8 A d^2 e^2-4 c d e (B e+2 A f)+c^2 \left (3 C e^2+B e f+3 A f^2\right )\right )-b e \left (8 A d^2 e^2 f-c d e \left (C e^2+3 B e f+9 A f^2\right )+4 c^2 \left (C e^2 f+A f^3\right )\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{8 a e^2 f (b e-a f) (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} \left (a^2 f^4 \left (8 A d^2 e^2-4 c d e (B e+2 A f)+c^2 \left (3 C e^2+B e f+3 A f^2\right )\right )+2 a b e f \left (C d e^3 (2 d e-5 c f)+f^2 \left (B c e (5 d e-2 c f)-A \left (10 d^2 e^2-11 c d e f+4 c^2 f^2\right )\right )\right )-b^2 \left (C d e^5 (d e-4 c f)+e^2 f \left (3 B d^2 e^3-A f \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right )\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{8 a \sqrt {d} e^3 f (b e-a f)^2 (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/4*(A*f^2-B*e*f+C*e^2)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/(-a*f+b*e)/(-c *f+d*e)/(f*x^2+e)^2-1/8*(b*(C*e^3*(2*c*f+d*e)+e*f*(3*A*f*(-2*c*f+3*d*e)-B* e*(-2*c*f+5*d*e)))+a*f*(C*e^2*(-5*c*f+2*d*e)-f*(3*A*f*(-c*f+2*d*e)-B*e*(c* f+2*d*e))))*x*(b*x^2+a)^(1/2)/e^2/f/(-a*f+b*e)^2/(-c*f+d*e)/(d*x^2+c)^(1/2 )/(f*x^2+e)+1/8*c^(1/2)*d^(1/2)*(b*(C*e^3*(2*c*f+d*e)+e*f*(3*A*f*(-2*c*f+3 *d*e)-B*e*(-2*c*f+5*d*e)))+a*f*(C*e^2*(-5*c*f+2*d*e)-f*(3*A*f*(-c*f+2*d*e) -B*e*(c*f+2*d*e))))*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c )^(1/2),(1-b*c/a/d)^(1/2))/e^2/f/(-a*f+b*e)^2/(-c*f+d*e)^2/(c*(b*x^2+a)/a/ (d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/8*c^(1/2)*d^(1/2)*(a*f^2*(8*A*d^2*e^2-4 *c*d*e*(2*A*f+B*e)+c^2*(3*A*f^2+B*e*f+3*C*e^2))-b*e*(8*A*d^2*e^2*f-c*d*e*( 9*A*f^2+3*B*e*f+C*e^2)+4*c^2*(A*f^3+C*e^2*f)))*(b*x^2+a)^(1/2)*InverseJaco biAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/e^2/f/(-a*f+b*e)/(-c*f +d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/8*c^(3/2)*(a^2*f ^4*(8*A*d^2*e^2-4*c*d*e*(2*A*f+B*e)+c^2*(3*A*f^2+B*e*f+3*C*e^2))+2*a*b*e*f *(C*d*e^3*(-5*c*f+2*d*e)+f^2*(B*c*e*(-2*c*f+5*d*e)-A*(4*c^2*f^2-11*c*d*e*f +10*d^2*e^2)))-b^2*(C*d*e^5*(-4*c*f+d*e)+e^2*f*(3*B*d^2*e^3-A*f*(8*c^2*f^2 -20*c*d*e*f+15*d^2*e^2))))*(b*x^2+a)^(1/2)*EllipticPi(d^(1/2)*x/c^(1/2)/(1 +d*x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d^(1/2)/e^3/f/(-a*f+b*e)^2/ (-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 15.17 (sec) , antiderivative size = 732, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\frac {\sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (2 e (b e-a f) (d e-c f) \left (C e^2+f (-B e+A f)\right )+\left (a f \left (C e^2 (2 d e-5 c f)+f (3 A f (-2 d e+c f)+B e (2 d e+c f))\right )+b \left (C e^3 (d e+2 c f)+e f (3 A f (3 d e-2 c f)+B e (-5 d e+2 c f))\right )\right ) \left (e+f x^2\right )\right )-i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right )^2 \left (-b c e f \left (a f \left (C e^2 (2 d e-5 c f)+f (3 A f (-2 d e+c f)+B e (2 d e+c f))\right )+b \left (C e^3 (d e+2 c f)+e f (3 A f (3 d e-2 c f)+B e (-5 d e+2 c f))\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+b e (d e-c f) \left (-a f \left (C e^2 (4 d e-5 c f)+f^2 (B c e-4 A d e+3 A c f)\right )+b \left (C e^3 (d e-2 c f)+e f (B e (3 d e-2 c f)+A f (-7 d e+6 c f))\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+\left (a^2 f^4 \left (8 A d^2 e^2-4 c d e (B e+2 A f)+c^2 \left (3 C e^2+B e f+3 A f^2\right )\right )+2 a b e f \left (C d e^3 (2 d e-5 c f)+f^2 \left (B c e (5 d e-2 c f)+A \left (-10 d^2 e^2+11 c d e f-4 c^2 f^2\right )\right )\right )+b^2 \left (C d e^5 (-d e+4 c f)+e^2 f \left (-3 B d^2 e^3+A f \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right )\right )\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{8 \sqrt {\frac {b}{a}} e^3 f^2 (b e-a f)^2 (d e-c f)^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2) ^3),x]
Output:
(Sqrt[b/a]*e*f^2*x*(a + b*x^2)*(c + d*x^2)*(2*e*(b*e - a*f)*(d*e - c*f)*(C *e^2 + f*(-(B*e) + A*f)) + (a*f*(C*e^2*(2*d*e - 5*c*f) + f*(3*A*f*(-2*d*e + c*f) + B*e*(2*d*e + c*f))) + b*(C*e^3*(d*e + 2*c*f) + e*f*(3*A*f*(3*d*e - 2*c*f) + B*e*(-5*d*e + 2*c*f))))*(e + f*x^2)) - I*Sqrt[1 + (b*x^2)/a]*Sq rt[1 + (d*x^2)/c]*(e + f*x^2)^2*(-(b*c*e*f*(a*f*(C*e^2*(2*d*e - 5*c*f) + f *(3*A*f*(-2*d*e + c*f) + B*e*(2*d*e + c*f))) + b*(C*e^3*(d*e + 2*c*f) + e* f*(3*A*f*(3*d*e - 2*c*f) + B*e*(-5*d*e + 2*c*f))))*EllipticE[I*ArcSinh[Sqr t[b/a]*x], (a*d)/(b*c)]) + b*e*(d*e - c*f)*(-(a*f*(C*e^2*(4*d*e - 5*c*f) + f^2*(B*c*e - 4*A*d*e + 3*A*c*f))) + b*(C*e^3*(d*e - 2*c*f) + e*f*(B*e*(3* d*e - 2*c*f) + A*f*(-7*d*e + 6*c*f))))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], ( a*d)/(b*c)] + (a^2*f^4*(8*A*d^2*e^2 - 4*c*d*e*(B*e + 2*A*f) + c^2*(3*C*e^2 + B*e*f + 3*A*f^2)) + 2*a*b*e*f*(C*d*e^3*(2*d*e - 5*c*f) + f^2*(B*c*e*(5* d*e - 2*c*f) + A*(-10*d^2*e^2 + 11*c*d*e*f - 4*c^2*f^2))) + b^2*(C*d*e^5*( -(d*e) + 4*c*f) + e^2*f*(-3*B*d^2*e^3 + A*f*(15*d^2*e^2 - 20*c*d*e*f + 8*c ^2*f^2))))*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/ (8*Sqrt[b/a]*e^3*f^2*(b*e - a*f)^2*(d*e - c*f)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {A f^2-B e f+C e^2}{f^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3}+\frac {B f-2 C e}{f^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}+\frac {C}{f^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d (2 C e-B f) \sqrt {b x^2+a} x}{2 e f (b e-a f) (d e-c f) \sqrt {d x^2+c}}-\frac {(2 C e-B f) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {\sqrt {c} \sqrt {d} (2 C e-B f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{2 e f (b e-a f) (d e-c f) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} \sqrt {d} (2 C e-B f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{2 a f^2 (b e-a f) (d e-c f) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {3 \sqrt {-a} \left (C e^2-B f e+A f^2\right ) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{8 \sqrt {b} e^3 f^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\sqrt {-a} (2 C e-B f) (b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{2 \sqrt {b} e^2 f^2 (b e-a f) (d e-c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {\sqrt {-a} C \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \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 f^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\left (C e^2-B f e+A f^2\right ) \int \frac {1}{\left (\sqrt {-e} \sqrt {f}-f x\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{8 (-e)^{3/2} \sqrt {f}}-\frac {3}{16} \left (\frac {C}{f}-\frac {B e-A f}{e^2}\right ) \int \frac {1}{\left (\sqrt {-e} \sqrt {f}-f x\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\frac {\left (C e^2-B f e+A f^2\right ) \int \frac {1}{\left (f x+\sqrt {-e} \sqrt {f}\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{8 (-e)^{3/2} \sqrt {f}}-\frac {3}{16} \left (\frac {C}{f}-\frac {B e-A f}{e^2}\right ) \int \frac {1}{\left (f x+\sqrt {-e} \sqrt {f}\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx\) |
Input:
Int[(A + B*x^2 + C*x^4)/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(6399\) vs. \(2(935)=1870\).
Time = 10.91 (sec) , antiderivative size = 6400, normalized size of antiderivative = 6.62
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(6400\) |
| default | \(\text {Expression too large to display}\) | \(11835\) |
Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x,method=_ RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, a lgorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((C*x**4+B*x**2+A)/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e) **3,x)
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, a lgorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e) ^3), x)
\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, a lgorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e) ^3), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int \frac {C\,x^4+B\,x^2+A}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int((A + B*x^2 + C*x^4)/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^3 ),x)
Output:
int((A + B*x^2 + C*x^4)/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^3 ), x)
\[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d \,f^{3} x^{10}+a d \,f^{3} x^{8}+b c \,f^{3} x^{8}+3 b d e \,f^{2} x^{8}+a c \,f^{3} x^{6}+3 a d e \,f^{2} x^{6}+3 b c e \,f^{2} x^{6}+3 b d \,e^{2} f \,x^{6}+3 a c e \,f^{2} x^{4}+3 a d \,e^{2} f \,x^{4}+3 b c \,e^{2} f \,x^{4}+b d \,e^{3} x^{4}+3 a c \,e^{2} f \,x^{2}+a d \,e^{3} x^{2}+b c \,e^{3} x^{2}+a c \,e^{3}}d x \right ) c +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,f^{3} x^{10}+a d \,f^{3} x^{8}+b c \,f^{3} x^{8}+3 b d e \,f^{2} x^{8}+a c \,f^{3} x^{6}+3 a d e \,f^{2} x^{6}+3 b c e \,f^{2} x^{6}+3 b d \,e^{2} f \,x^{6}+3 a c e \,f^{2} x^{4}+3 a d \,e^{2} f \,x^{4}+3 b c \,e^{2} f \,x^{4}+b d \,e^{3} x^{4}+3 a c \,e^{2} f \,x^{2}+a d \,e^{3} x^{2}+b c \,e^{3} x^{2}+a c \,e^{3}}d x \right ) b +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,f^{3} x^{10}+a d \,f^{3} x^{8}+b c \,f^{3} x^{8}+3 b d e \,f^{2} x^{8}+a c \,f^{3} x^{6}+3 a d e \,f^{2} x^{6}+3 b c e \,f^{2} x^{6}+3 b d \,e^{2} f \,x^{6}+3 a c e \,f^{2} x^{4}+3 a d \,e^{2} f \,x^{4}+3 b c \,e^{2} f \,x^{4}+b d \,e^{3} x^{4}+3 a c \,e^{2} f \,x^{2}+a d \,e^{3} x^{2}+b c \,e^{3} x^{2}+a c \,e^{3}}d x \right ) a \] Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e**3 + 3*a*c*e**2*f*x**2 + 3*a*c*e*f**2*x**4 + a*c*f**3*x**6 + a*d*e**3*x**2 + 3*a*d*e**2*f*x**4 + 3*a*d*e*f**2*x**6 + a*d*f**3*x**8 + b*c*e**3*x**2 + 3*b*c*e**2*f*x**4 + 3 *b*c*e*f**2*x**6 + b*c*f**3*x**8 + b*d*e**3*x**4 + 3*b*d*e**2*f*x**6 + 3*b *d*e*f**2*x**8 + b*d*f**3*x**10),x)*c + int((sqrt(c + d*x**2)*sqrt(a + b*x **2)*x**2)/(a*c*e**3 + 3*a*c*e**2*f*x**2 + 3*a*c*e*f**2*x**4 + a*c*f**3*x* *6 + a*d*e**3*x**2 + 3*a*d*e**2*f*x**4 + 3*a*d*e*f**2*x**6 + a*d*f**3*x**8 + b*c*e**3*x**2 + 3*b*c*e**2*f*x**4 + 3*b*c*e*f**2*x**6 + b*c*f**3*x**8 + b*d*e**3*x**4 + 3*b*d*e**2*f*x**6 + 3*b*d*e*f**2*x**8 + b*d*f**3*x**10),x )*b + int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c*e**3 + 3*a*c*e**2*f*x** 2 + 3*a*c*e*f**2*x**4 + a*c*f**3*x**6 + a*d*e**3*x**2 + 3*a*d*e**2*f*x**4 + 3*a*d*e*f**2*x**6 + a*d*f**3*x**8 + b*c*e**3*x**2 + 3*b*c*e**2*f*x**4 + 3*b*c*e*f**2*x**6 + b*c*f**3*x**8 + b*d*e**3*x**4 + 3*b*d*e**2*f*x**6 + 3* b*d*e*f**2*x**8 + b*d*f**3*x**10),x)*a