Integrand size = 33, antiderivative size = 774 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {d^2 x}{3 c (b c+a d) (d e-c f) \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}}-\frac {b \left (a b d^2 e+a^2 d^2 f-3 b^2 c (d e-c f)\right ) x}{3 a c (b c+a d)^2 (b e+a f) (d e-c f) \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {d \left (a b^2 c d^2 e (7 d e-10 c f)+a^3 d^3 f (2 d e-5 c f)-3 b^3 c^2 (d e-c f)^2+2 a^2 b d^2 \left (d^2 e^2+c d e f-5 c^2 f^2\right )\right ) x \sqrt {a-b x^2}}{3 a c^2 (b c+a d)^3 (b e+a f) (d e-c f)^2 \sqrt {c+d x^2}}+\frac {\sqrt {b} \left (a b^2 c d^2 e (7 d e-10 c f)+a^3 d^3 f (2 d e-5 c f)-3 b^3 c^2 (d e-c f)^2+2 a^2 b d^2 \left (d^2 e^2+c d e f-5 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{3 \sqrt {a} c^2 (b c+a d)^3 (b e+a f) (d e-c f)^2 \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {\sqrt {b} \left (a b d^2 e+a^2 d^2 f-3 b^2 c (d e-c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{3 \sqrt {a} c (b c+a d)^2 (b e+a f) (d e-c f) \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {a} f^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e (b e+a f) (d e-c f)^2 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
1/3*d^2*x/c/(a*d+b*c)/(-c*f+d*e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)-1/3*b*(a *b*d^2*e+a^2*d^2*f-3*b^2*c*(-c*f+d*e))*x/a/c/(a*d+b*c)^2/(a*f+b*e)/(-c*f+d *e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+1/3*d*(a*b^2*c*d^2*e*(-10*c*f+7*d*e)+ a^3*d^3*f*(-5*c*f+2*d*e)-3*b^3*c^2*(-c*f+d*e)^2+2*a^2*b*d^2*(-5*c^2*f^2+c* d*e*f+d^2*e^2))*x*(-b*x^2+a)^(1/2)/a/c^2/(a*d+b*c)^3/(a*f+b*e)/(-c*f+d*e)^ 2/(d*x^2+c)^(1/2)+1/3*b^(1/2)*(a*b^2*c*d^2*e*(-10*c*f+7*d*e)+a^3*d^3*f*(-5 *c*f+2*d*e)-3*b^3*c^2*(-c*f+d*e)^2+2*a^2*b*d^2*(-5*c^2*f^2+c*d*e*f+d^2*e^2 ))*(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^(1/2)/c^2/(a*d+b*c)^3/(a*f+b*e)/(-c*f+d*e)^2/(-b*x^2+a)^(1/2)/( 1+d*x^2/c)^(1/2)-1/3*b^(1/2)*(a*b*d^2*e+a^2*d^2*f-3*b^2*c*(-c*f+d*e))*(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^(1/2)/c/(a*d+b*c)^2/(a*f+b*e)/(-c*f+d*e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^ (1/2)+a^(1/2)*f^3*(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/(a*f+b*e)/(-c*f+d*e)^2/(-b*x ^2+a)^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 12.98 (sec) , antiderivative size = 1710, normalized size of antiderivative = 2.21 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:
Integrate[1/((a - b*x^2)^(3/2)*(c + d*x^2)^(5/2)*(e + f*x^2)),x]
Output:
(I*b*c*e*(3*b^3*c^2*(d*e - c*f)^2 + a^3*d^3*f*(-2*d*e + 5*c*f) + a*b^2*c*d ^2*e*(-7*d*e + 10*c*f) - 2*a^2*b*d^2*(d^2*e^2 + c*d*e*f - 5*c^2*f^2))*Sqrt [1 - (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[- (b/a)]*x], -((a*d)/(b*c))] + (-3*b^5*c^4*d^2*e^3*x - 8*a^2*b^3*c^2*d^4*e^3 *x - 3*a^3*b^2*c*d^5*e^3*x + 6*b^5*c^5*d*e^2*f*x + 11*a^2*b^3*c^3*d^3*e^2* f*x - 2*a^3*b^2*c^2*d^4*e^2*f*x - 3*a^4*b*c*d^5*e^2*f*x - 3*b^5*c^6*e*f^2* x + 11*a^3*b^2*c^3*d^3*e*f^2*x + 6*a^4*b*c^2*d^4*e*f^2*x - 6*b^5*c^3*d^3*e ^3*x^3 + 8*a*b^4*c^2*d^4*e^3*x^3 - 4*a^2*b^3*c*d^5*e^3*x^3 - 2*a^3*b^2*d^6 *e^3*x^3 + 12*b^5*c^4*d^2*e^2*f*x^3 - 11*a*b^4*c^3*d^3*e^2*f*x^3 + 12*a^2* b^3*c^2*d^4*e^2*f*x^3 + a^3*b^2*c*d^5*e^2*f*x^3 - 2*a^4*b*d^6*e^2*f*x^3 - 6*b^5*c^5*d*e*f^2*x^3 - 11*a^2*b^3*c^3*d^3*e*f^2*x^3 + 4*a^3*b^2*c^2*d^4*e *f^2*x^3 + 5*a^4*b*c*d^5*e*f^2*x^3 - 3*b^5*c^2*d^4*e^3*x^5 + 7*a*b^4*c*d^5 *e^3*x^5 + 2*a^2*b^3*d^6*e^3*x^5 + 6*b^5*c^3*d^3*e^2*f*x^5 - 10*a*b^4*c^2* d^4*e^2*f*x^5 + 2*a^2*b^3*c*d^5*e^2*f*x^5 + 2*a^3*b^2*d^6*e^2*f*x^5 - 3*b^ 5*c^4*d^2*e*f^2*x^5 - 10*a^2*b^3*c^2*d^4*e*f^2*x^5 - 5*a^3*b^2*c*d^5*e*f^2 *x^5 - I*a*b*Sqrt[-(b/a)]*c*(b*c + a*d)*e*(-(d*e) + c*f)*(a*b*d^2*e + a^2* d^2*f + 3*b^2*c*(-(d*e) + c*f))*Sqrt[1 - (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + ( d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] + ((3*I)*a* b^4*c^6*f^3*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[-((a*f)/(b* e)), I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/Sqrt[-(b/a)] + (9*I)*a...
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.
| \(\displaystyle \int \frac {1}{\left (a-b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}+\frac {b \int \frac {-b f x^2+b e+2 a f}{\left (a-b x^2\right )^{3/2} \left (d x^2+c\right )^{5/2}}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}+\frac {b \left (\frac {\int \frac {3 b d (b e+a f) x^2+a (b d e+b c f+2 a d f)}{\sqrt {a-b x^2} \left (d x^2+c\right )^{5/2}}dx}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {-\frac {\int -\frac {b d \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) x^2+a \left (3 c (2 d e+c f) b^2+a d (2 d e+11 c f) b+4 a^2 d^2 f\right )}{\sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}dx}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {\int \frac {b d \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) x^2+a \left (3 c (2 d e+c f) b^2+a d (2 d e+11 c f) b+4 a^2 d^2 f\right )}{\sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}dx}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\frac {-\frac {\int -\frac {b \left (a c \left (3 c (3 d e+c f) b^2+a d (d e+13 c f) b+2 a^2 d^2 f\right )-d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x^2\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\int \frac {b \left (a c \left (3 c (3 d e+c f) b^2+a d (d e+13 c f) b+2 a^2 d^2 f\right )-d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x^2\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {b \int \frac {a c \left (3 c (3 d e+c f) b^2+a d (d e+13 c f) b+2 a^2 d^2 f\right )-d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x^2}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {b \left (c (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx-\left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {b \left (\frac {c \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{\sqrt {c+d x^2}}-\left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {b \left (\frac {c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{\sqrt {a-b x^2} \sqrt {c+d x^2}}-\left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {a-b x^2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {\sqrt {a-b x^2} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {\sqrt {a-b x^2} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {x \sqrt {a-b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}-\frac {\int -\frac {d \left (a (2 d e-5 c f)-b (d e-4 c f) x^2\right )}{\sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}dx}{3 c d}\right )}{(d e-c f)^2}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int \frac {d \left (a (2 d e-5 c f)-b (d e-4 c f) x^2\right )}{\sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}dx}{3 c d}+\frac {x \sqrt {a-b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int \frac {a (2 d e-5 c f)-b (d e-4 c f) x^2}{\sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \sqrt {a-b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\frac {x \sqrt {a-b x^2} (a d (2 d e-5 c f)+b c (d e-4 c f))}{c \sqrt {c+d x^2} (a d+b c)}-\frac {\int -\frac {b \left ((a d (2 d e-5 c f)+b c (d e-4 c f)) x^2+a c (d e-c f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{c (a d+b c)}}{3 c}+\frac {x \sqrt {a-b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\frac {\int \frac {b \left ((a d (2 d e-5 c f)+b c (d e-4 c f)) x^2+a c (d e-c f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{c (a d+b c)}+\frac {x \sqrt {a-b x^2} (a d (2 d e-5 c f)+b c (d e-4 c f))}{c \sqrt {c+d x^2} (a d+b c)}}{3 c}+\frac {x \sqrt {a-b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\frac {b \int \frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) x^2+a c (d e-c f)}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{c (a d+b c)}+\frac {x \sqrt {a-b x^2} (a d (2 d e-5 c f)+b c (d e-4 c f))}{c \sqrt {c+d x^2} (a d+b c)}}{3 c}+\frac {x \sqrt {a-b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\frac {b \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-4 a^3 d^2 f-a^2 b d (13 c f+2 d e)-a b^2 c (c f+7 d e)+3 b^3 c^2 e\right )}{c \sqrt {c+d x^2} (a d+b c)}}{3 c (a d+b c)}-\frac {d x \sqrt {a-b x^2} \left (-2 a^2 d f-a b (d e-2 c f)+3 b^2 c e\right )}{3 c \left (c+d x^2\right )^{3/2} (a d+b c)}}{a (a d+b c)}+\frac {b x (a f+b e)}{a \sqrt {a-b x^2} \left (c+d x^2\right )^{3/2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\left (\frac {\int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) \sqrt {a-b x^2} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {a-b x^2} x}{c (b c+a d) \sqrt {d x^2+c}}+\frac {b \left (\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c (b c+a d) (d e-4 c f) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d}\right )}{c (b c+a d)}}{3 c}\right )}{(d e-c f)^2}\right ) f^2}{(b e+a f)^2}+\frac {b \left (\frac {b (b e+a f) x}{a (b c+a d) \sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {\frac {b \left (\frac {\sqrt {a} c (b c+a d) \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (b c+a d)}-\frac {d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {d x^2+c}}}{3 c (b c+a d)}-\frac {d \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) x \sqrt {a-b x^2}}{3 c (b c+a d) \left (d x^2+c\right )^{3/2}}}{a (b c+a d)}\right )}{(b e+a f)^2}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\left (\frac {\int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) \sqrt {a-b x^2} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {a-b x^2} x}{c (b c+a d) \sqrt {d x^2+c}}+\frac {b \left (\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c (b c+a d) (d e-4 c f) \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {d x^2+c}}\right )}{c (b c+a d)}}{3 c}\right )}{(d e-c f)^2}\right ) f^2}{(b e+a f)^2}+\frac {b \left (\frac {b (b e+a f) x}{a (b c+a d) \sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {\frac {b \left (\frac {\sqrt {a} c (b c+a d) \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (b c+a d)}-\frac {d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {d x^2+c}}}{3 c (b c+a d)}-\frac {d \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) x \sqrt {a-b x^2}}{3 c (b c+a d) \left (d x^2+c\right )^{3/2}}}{a (b c+a d)}\right )}{(b e+a f)^2}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\left (\frac {\int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) \sqrt {a-b x^2} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {a-b x^2} x}{c (b c+a d) \sqrt {d x^2+c}}+\frac {b \left (\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c (b c+a d) (d e-4 c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {a-b x^2} \sqrt {d x^2+c}}\right )}{c (b c+a d)}}{3 c}\right )}{(d e-c f)^2}\right ) f^2}{(b e+a f)^2}+\frac {b \left (\frac {b (b e+a f) x}{a (b c+a d) \sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {\frac {b \left (\frac {\sqrt {a} c (b c+a d) \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (b c+a d)}-\frac {d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {d x^2+c}}}{3 c (b c+a d)}-\frac {d \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) x \sqrt {a-b x^2}}{3 c (b c+a d) \left (d x^2+c\right )^{3/2}}}{a (b c+a d)}\right )}{(b e+a f)^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\left (\frac {\int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) \sqrt {a-b x^2} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {a-b x^2} x}{c (b c+a d) \sqrt {d x^2+c}}+\frac {b \left (\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {a} c (b c+a d) (d e-4 c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}\right )}{c (b c+a d)}}{3 c}\right )}{(d e-c f)^2}\right ) f^2}{(b e+a f)^2}+\frac {b \left (\frac {b (b e+a f) x}{a (b c+a d) \sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {\frac {b \left (\frac {\sqrt {a} c (b c+a d) \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (b c+a d)}-\frac {d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {d x^2+c}}}{3 c (b c+a d)}-\frac {d \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) x \sqrt {a-b x^2}}{3 c (b c+a d) \left (d x^2+c\right )^{3/2}}}{a (b c+a d)}\right )}{(b e+a f)^2}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle \frac {\left (\frac {\int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) \sqrt {a-b x^2} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {a-b x^2} x}{c (b c+a d) \sqrt {d x^2+c}}+\frac {b \left (\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\sqrt {a} c (b c+a d) (d e-4 c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}\right )}{c (b c+a d)}}{3 c}\right )}{(d e-c f)^2}\right ) f^2}{(b e+a f)^2}+\frac {b \left (\frac {b (b e+a f) x}{a (b c+a d) \sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {\frac {b \left (\frac {\sqrt {a} c (b c+a d) \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (b c+a d)}-\frac {d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {d x^2+c}}}{3 c (b c+a d)}-\frac {d \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) x \sqrt {a-b x^2}}{3 c (b c+a d) \left (d x^2+c\right )^{3/2}}}{a (b c+a d)}\right )}{(b e+a f)^2}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\left (\frac {\int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) \sqrt {a-b x^2} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {a-b x^2} x}{c (b c+a d) \sqrt {d x^2+c}}+\frac {b \left (\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (b c+a d) (d e-4 c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}\right )}{c (b c+a d)}}{3 c}\right )}{(d e-c f)^2}\right ) f^2}{(b e+a f)^2}+\frac {b \left (\frac {b (b e+a f) x}{a (b c+a d) \sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {\frac {b \left (\frac {\sqrt {a} c (b c+a d) \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (b c+a d)}-\frac {d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {d x^2+c}}}{3 c (b c+a d)}-\frac {d \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) x \sqrt {a-b x^2}}{3 c (b c+a d) \left (d x^2+c\right )^{3/2}}}{a (b c+a d)}\right )}{(b e+a f)^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\left (\frac {\int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) \sqrt {a-b x^2} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {a-b x^2} x}{c (b c+a d) \sqrt {d x^2+c}}+\frac {b \left (\frac {\sqrt {a} (a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (b c+a d) (d e-4 c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}\right )}{c (b c+a d)}}{3 c}\right )}{(d e-c f)^2}\right ) f^2}{(b e+a f)^2}+\frac {b \left (\frac {b (b e+a f) x}{a (b c+a d) \sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {\frac {b \left (\frac {\sqrt {a} c (b c+a d) \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (b c+a d)}-\frac {d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {d x^2+c}}}{3 c (b c+a d)}-\frac {d \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) x \sqrt {a-b x^2}}{3 c (b c+a d) \left (d x^2+c\right )^{3/2}}}{a (b c+a d)}\right )}{(b e+a f)^2}\) |
| \(\Big \downarrow \) 415 |
| \(\displaystyle \frac {\left (\frac {\left (\frac {(b e+a f) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}-\frac {b \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{f}\right ) f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) \sqrt {a-b x^2} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {(a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {a-b x^2} x}{c (b c+a d) \sqrt {d x^2+c}}+\frac {b \left (\frac {\sqrt {a} (a d (2 d e-5 c f)+b c (d e-4 c f)) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (b c+a d) (d e-4 c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}\right )}{c (b c+a d)}}{3 c}\right )}{(d e-c f)^2}\right ) f^2}{(b e+a f)^2}+\frac {b \left (\frac {b (b e+a f) x}{a (b c+a d) \sqrt {a-b x^2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {\frac {b \left (\frac {\sqrt {a} c (b c+a d) \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{c (b c+a d)}-\frac {d \left (-4 d^2 f a^3-b d (2 d e+13 c f) a^2-b^2 c (7 d e+c f) a+3 b^3 c^2 e\right ) x \sqrt {a-b x^2}}{c (b c+a d) \sqrt {d x^2+c}}}{3 c (b c+a d)}-\frac {d \left (-2 d f a^2-b (d e-2 c f) a+3 b^2 c e\right ) x \sqrt {a-b x^2}}{3 c (b c+a d) \left (d x^2+c\right )^{3/2}}}{a (b c+a d)}\right )}{(b e+a f)^2}\) |
Input:
Int[1/((a - b*x^2)^(3/2)*(c + d*x^2)^(5/2)*(e + f*x^2)),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[d/b Int[1/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Simp[(b*c - a*d)/b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2 ]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2 Int[(c + d*x^2)^(q + 2)*((e + f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2 Int[(c + d*x^2)^q*( e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} , x] && LtQ[q, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(2793\) vs. \(2(707)=1414\).
Time = 20.19 (sec) , antiderivative size = 2794, normalized size of antiderivative = 3.61
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2794\) |
| default | \(\text {Expression too large to display}\) | \(4122\) |
Input:
int(1/(-b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
((-b*x^2+a)*(d*x^2+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-1/3/c*d/(a *d+b*c)*x/(a*c*d*f-a*d^2*e+b*c^2*f-b*c*d*e)*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c) ^(1/2)/(x^2+c/d)^2-1/3*(-b*d*x^2+a*d)/c^2*d^2/(a*d+b*c)^2*x*(5*a*c*d*f-2*a *d^2*e+10*b*c^2*f-7*b*c*d*e)/(c*f-d*e)/(a*c*d*f-a*d^2*e+b*c^2*f-b*c*d*e)/( (x^2+c/d)*(-b*d*x^2+a*d))^(1/2)-(-b*d*x^2-b*c)*b^3/a/(a*d+b*c)^3*x/(a*f+b* e)/((x^2-a/b)*(-b*d*x^2-b*c))^(1/2)+5/(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))*a*d^3/(a*d+b*c)^2/(c*f-d*e)/(a*c*d*f-a*d^2*e+b*c^ 2*f-b*c*d*e)*b*f+10/3*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^2*b^2/(a*d+b*c)^2/(c*f-d*e)/(a*c*d*f- a*d^2*e+b*c^2*f-b*c*d*e)*f*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2 ))-10/3*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^2*b^2/(a*d+b*c)^2/(c*f-d*e)/(a*c*d*f-a*d^2*e+b*c^2* f-b*c*d*e)*f*EllipticE(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))-7/3/(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^3*b^2/(a*d+b*c)^2/(c*f-d*e)/(a*c*d*f-a*d^2*e+b*c^2*f-b*c*d*e)*e*Elli pticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))+7/3/(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^3*b^2/(a*d+ b*c)^2/(c*f-d*e)/(a*c*d*f-a*d^2*e+b*c^2*f-b*c*d*e)*e*EllipticE(x*(b/a)^(1/ 2),(-1-(a*d-b*c)/c/b)^(1/2))-5/3/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2...
Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(-b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="frica s")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(-b*x**2+a)**(3/2)/(d*x**2+c)**(5/2)/(f*x**2+e),x)
Output:
Timed out
\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(-b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="maxim a")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*(d*x^2 + c)^(5/2)*(f*x^2 + e)), x)
\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(-b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="giac" )
Output:
integrate(1/((-b*x^2 + a)^(3/2)*(d*x^2 + c)^(5/2)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{{\left (a-b\,x^2\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{5/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(1/((a - b*x^2)^(3/2)*(c + d*x^2)^(5/2)*(e + f*x^2)),x)
Output:
int(1/((a - b*x^2)^(3/2)*(c + d*x^2)^(5/2)*(e + f*x^2)), x)
\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d^{3} f \,x^{12}-2 a b \,d^{3} f \,x^{10}+3 b^{2} c \,d^{2} f \,x^{10}+b^{2} d^{3} e \,x^{10}+a^{2} d^{3} f \,x^{8}-6 a b c \,d^{2} f \,x^{8}-2 a b \,d^{3} e \,x^{8}+3 b^{2} c^{2} d f \,x^{8}+3 b^{2} c \,d^{2} e \,x^{8}+3 a^{2} c \,d^{2} f \,x^{6}+a^{2} d^{3} e \,x^{6}-6 a b \,c^{2} d f \,x^{6}-6 a b c \,d^{2} e \,x^{6}+b^{2} c^{3} f \,x^{6}+3 b^{2} c^{2} d e \,x^{6}+3 a^{2} c^{2} d f \,x^{4}+3 a^{2} c \,d^{2} e \,x^{4}-2 a b \,c^{3} f \,x^{4}-6 a b \,c^{2} d e \,x^{4}+b^{2} c^{3} e \,x^{4}+a^{2} c^{3} f \,x^{2}+3 a^{2} c^{2} d e \,x^{2}-2 a b \,c^{3} e \,x^{2}+a^{2} c^{3} e}d x \] Input:
int(1/(-b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*c**3*e + a**2*c**3*f*x**2 + 3*a**2*c**2*d*e*x**2 + 3*a**2*c**2*d*f*x**4 + 3*a**2*c*d**2*e*x**4 + 3*a** 2*c*d**2*f*x**6 + a**2*d**3*e*x**6 + a**2*d**3*f*x**8 - 2*a*b*c**3*e*x**2 - 2*a*b*c**3*f*x**4 - 6*a*b*c**2*d*e*x**4 - 6*a*b*c**2*d*f*x**6 - 6*a*b*c* d**2*e*x**6 - 6*a*b*c*d**2*f*x**8 - 2*a*b*d**3*e*x**8 - 2*a*b*d**3*f*x**10 + b**2*c**3*e*x**4 + b**2*c**3*f*x**6 + 3*b**2*c**2*d*e*x**6 + 3*b**2*c** 2*d*f*x**8 + 3*b**2*c*d**2*e*x**8 + 3*b**2*c*d**2*f*x**10 + b**2*d**3*e*x* *10 + b**2*d**3*f*x**12),x)