Integrand size = 32, antiderivative size = 1047 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=-\frac {b \left (3 a b c f^2-3 a^2 d f^2-2 b^2 e (d e-c f)\right ) x}{6 a (b c-a d) e (b e-a f)^2 (d e-c f) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}-\frac {b \left (6 a^3 b c d f^3-3 a^4 d^2 f^3-4 b^4 c e^2 (d e-c f)+4 a b^3 e \left (3 d^2 e^2+c d e f-4 c^2 f^2\right )-3 a^2 b^2 f \left (8 d^2 e^2-8 c d e f+c^2 f^2\right )\right ) x}{6 a^2 (b c-a d)^2 e (b e-a f)^3 (d e-c f) \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {f^2 x}{2 e (b e-a f) (d e-c f) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )}+\frac {\sqrt {d} \left (4 b^5 c^2 e^2 (d e-c f)^2+3 a^5 d^3 f^3 (2 d e+c f)-2 a b^4 c e (d e-c f)^2 (7 d e+8 c f)-9 a^4 b d^2 f^2 \left (2 d^2 e^2+c^2 f^2\right )+9 a^3 b^2 d f \left (2 d^3 e^3+c^3 f^3\right )-a^2 b^3 \left (6 d^4 e^4-26 c d^3 e^3 f+52 c^2 d^2 e^2 f^2-26 c^3 d e f^3+3 c^4 f^4\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 )}{6 a^2 \sqrt {c} (b c-a d)^3 e (b e-a f)^3 (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 (2 b^4 c e (d e-c f)^3-3 a^4 d^3 f^3 (4 d e-c f)+3 a^3 b d^2 f^2 \left (2 d^2 e^2+10 c d e f-3 c^2 f^2\right )+3 a^2 b^2 d f \left (8 d^3 e^3-20 c d^2 e^2 f+3 c^3 f^3\right )-3 a b^3 \left (6 d^4 e^4-10 c d^3 e^3 f+c^4 f^4\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 )}{6 a^2 (b c-a d)^3 e (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}}+\frac {c^{3/2} f^4 (3 b e (3 d e-2 c f)-a f (4 d e-c f)) \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 )}{2 a \sqrt {d} e^2 (b e-a f)^3 (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:
-1/6*b*(3*a*b*c*f^2-3*a^2*d*f^2-2*b^2*e*(-c*f+d*e))*x/a/(-a*d+b*c)/e/(-a*f +b*e)^2/(-c*f+d*e)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)-1/6*b*(6*a^3*b*c*d*f^3- 3*a^4*d^2*f^3-4*b^4*c*e^2*(-c*f+d*e)+4*a*b^3*e*(-4*c^2*f^2+c*d*e*f+3*d^2*e ^2)-3*a^2*b^2*f*(c^2*f^2-8*c*d*e*f+8*d^2*e^2))*x/a^2/(-a*d+b*c)^2/e/(-a*f+ b*e)^3/(-c*f+d*e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+1/2*f^2*x/e/(-a*f+b*e)/( -c*f+d*e)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)+1/6*d^(1/2)*(4*b^5*c^2 *e^2*(-c*f+d*e)^2+3*a^5*d^3*f^3*(c*f+2*d*e)-2*a*b^4*c*e*(-c*f+d*e)^2*(8*c* f+7*d*e)-9*a^4*b*d^2*f^2*(c^2*f^2+2*d^2*e^2)+9*a^3*b^2*d*f*(c^3*f^3+2*d^3* e^3)-a^2*b^3*(3*c^4*f^4-26*c^3*d*e*f^3+52*c^2*d^2*e^2*f^2-26*c*d^3*e^3*f+6 *d^4*e^4))*(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))/a^2/c^(1/2)/(-a*d+b*c)^3/e/(-a*f+b*e)^3/(-c*f+d*e)^2/(c* (b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/6*c^(1/2)*d^(1/2)*(2*b^4*c* e*(-c*f+d*e)^3-3*a^4*d^3*f^3*(-c*f+4*d*e)+3*a^3*b*d^2*f^2*(-3*c^2*f^2+10*c *d*e*f+2*d^2*e^2)+3*a^2*b^2*d*f*(3*c^3*f^3-20*c*d^2*e^2*f+8*d^3*e^3)-3*a*b ^3*(c^4*f^4-10*c*d^3*e^3*f+6*d^4*e^4))*(b*x^2+a)^(1/2)*InverseJacobiAM(arc tan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a^2/(-a*d+b*c)^3/e/(-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)+1/2*c^(3/2)*f ^4*(3*b*e*(-2*c*f+3*d*e)-a*f*(-c*f+4*d*e))*(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^2 /(-a*f+b*e)^3/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1...
Result contains complex when optimal does not.
Time = 12.69 (sec) , antiderivative size = 765, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\frac {-\sqrt {\frac {b}{a}} e x \left (3 a^2 c (b c-a d)^3 f^5 \left (a+b x^2\right )^2 \left (c+d x^2\right )+6 a^2 d^5 e (b e-a f)^3 \left (a+b x^2\right )^2 \left (e+f x^2\right )-2 a b^4 c (-b c+a d) e (-b e+a f) (d e-c f)^2 \left (c+d x^2\right ) \left (e+f x^2\right )-2 b^4 c e (d e-c f)^2 \left (2 b^2 c e+13 a^2 d f-a b (7 d e+8 c f)\right ) \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )\right )-i c \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \left (-b e \left (4 b^5 c^2 e^2 (d e-c f)^2+3 a^5 d^3 f^3 (2 d e+c f)-2 a b^4 c e (d e-c f)^2 (7 d e+8 c f)-9 a^4 b d^2 f^2 \left (2 d^2 e^2+c^2 f^2\right )+9 a^3 b^2 d f \left (2 d^3 e^3+c^3 f^3\right )+a^2 b^3 \left (-6 d^4 e^4+26 c d^3 e^3 f-52 c^2 d^2 e^2 f^2+26 c^3 d e f^3-3 c^4 f^4\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+(b c-a d) \left (b e (d e-c f) \left (-6 a^3 b c d f^3+3 a^4 d^2 f^3+4 b^4 c e^2 (d e-c f)-4 a b^3 e \left (3 d^2 e^2+c d e f-4 c^2 f^2\right )+3 a^2 b^2 f \left (8 d^2 e^2-8 c d e f+c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 a^2 (b c-a d)^2 f^3 (3 b e (3 d e-2 c f)+a f (-4 d e+c f)) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{6 a^2 \sqrt {\frac {b}{a}} c (b c-a d)^3 e^2 (b e-a f)^3 (d e-c f)^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^2),x]
Output:
(-(Sqrt[b/a]*e*x*(3*a^2*c*(b*c - a*d)^3*f^5*(a + b*x^2)^2*(c + d*x^2) + 6* a^2*d^5*e*(b*e - a*f)^3*(a + b*x^2)^2*(e + f*x^2) - 2*a*b^4*c*(-(b*c) + a* d)*e*(-(b*e) + a*f)*(d*e - c*f)^2*(c + d*x^2)*(e + f*x^2) - 2*b^4*c*e*(d*e - c*f)^2*(2*b^2*c*e + 13*a^2*d*f - a*b*(7*d*e + 8*c*f))*(a + b*x^2)*(c + d*x^2)*(e + f*x^2))) - I*c*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2 )/c]*(e + f*x^2)*(-(b*e*(4*b^5*c^2*e^2*(d*e - c*f)^2 + 3*a^5*d^3*f^3*(2*d* e + c*f) - 2*a*b^4*c*e*(d*e - c*f)^2*(7*d*e + 8*c*f) - 9*a^4*b*d^2*f^2*(2* d^2*e^2 + c^2*f^2) + 9*a^3*b^2*d*f*(2*d^3*e^3 + c^3*f^3) + a^2*b^3*(-6*d^4 *e^4 + 26*c*d^3*e^3*f - 52*c^2*d^2*e^2*f^2 + 26*c^3*d*e*f^3 - 3*c^4*f^4))* EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]) + (b*c - a*d)*(b*e*(d*e - c*f)*(-6*a^3*b*c*d*f^3 + 3*a^4*d^2*f^3 + 4*b^4*c*e^2*(d*e - c*f) - 4*a*b^3 *e*(3*d^2*e^2 + c*d*e*f - 4*c^2*f^2) + 3*a^2*b^2*f*(8*d^2*e^2 - 8*c*d*e*f + c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - 3*a^2*(b*c - a*d)^2*f^3*(3*b*e*(3*d*e - 2*c*f) + a*f*(-4*d*e + c*f))*EllipticPi[(a*f)/( b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])))/(6*a^2*Sqrt[b/a]*c*(b*c - a* d)^3*e^2*(b*e - a*f)^3*(d*e - c*f)^2*(a + b*x^2)^(3/2)*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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \int \frac {1}{\left (b x^2+a\right )^{5/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {b \int -\frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{5/2} \left (d x^2+c\right )^{3/2}}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \int \frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{5/2} \left (d x^2+c\right )^{3/2}}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}-\frac {\int -\frac {6 d f a^2-b (3 d e+5 c f) a+3 b d (b e-a f) x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}dx}{3 a (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {6 d f a^2-3 b d e a-5 b c f a+3 b d (b e-a f) x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}dx}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}-\frac {\int -\frac {d \left (b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2+b (3 d e+2 c f) a+b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\int \frac {d \left (b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2+b (3 d e+2 c f) a+b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{a (b c-a d)}+\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {d \int \frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2+b (3 d e+2 c f) a+b^2 c e\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{a (b c-a d)}+\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {d \left (\frac {\left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}-\frac {a b \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{a (b c-a d)}+\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a b \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{a (b c-a d)}+\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {(a d f-2 b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {d (d e-c f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {f^2 \left (\frac {d \left (\frac {(a d f-2 b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(d e-c f)^2}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {b \int -\frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx f^2}{(b e-a f)^2}+\frac {b \int \frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\int \frac {a (b d e+b c f-2 a d f)-b d (b e-a f) x^2}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\frac {a b (2 b d e+b c f-3 a d f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {d \left (-2 d f a^2+b d e a+b^2 c e\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}}{a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\frac {a b (2 b d e+b c f-3 a d f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {d} \left (-2 d f a^2+b d e a+b^2 c e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\frac {b \sqrt {c} (2 b d e+b c f-3 a d 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 )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} \left (-2 d f a^2+b d e a+b^2 c e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 416 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\left (\frac {d \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{d e-c f}-\frac {f \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{d e-c f}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\frac {b \sqrt {c} (2 b d e+b c f-3 a d 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 )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} \left (-2 d f a^2+b d e a+b^2 c e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\left (\frac {\sqrt {d} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (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 {f \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{d e-c f}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\frac {b \sqrt {c} (2 b d e+b c f-3 a d 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 )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} \left (-2 d f a^2+b d e a+b^2 c e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\left (\frac {\sqrt {d} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (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 {a^{3/2} f \sqrt {d x^2+c} \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 (d e-c f) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\frac {b \sqrt {c} (2 b d e+b c f-3 a d 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 )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} \left (-2 d f a^2+b d e a+b^2 c e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\left (\frac {\sqrt {d} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (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 {a^{3/2} f \sqrt {d x^2+c} \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 (d e-c f) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\frac {b \sqrt {c} (2 b d e+b c f-3 a d 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 )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} \left (-2 d f a^2+b d e a+b^2 c e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\left (\frac {\sqrt {d} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (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 {a^{3/2} f \sqrt {d x^2+c} \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 (d e-c f) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\frac {b \sqrt {c} (2 b d e+b c f-3 a d 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 )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} \left (-2 d f a^2+b d e a+b^2 c e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {d \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\left (\frac {c^{3/2} \sqrt {b x^2+a} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d 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 )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+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}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}+\frac {\frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (\frac {\left (6 d^2 f a^3-b d (3 d e-7 c f) a^2-b^2 c (7 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \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} \left (15 d f a^2-9 b d e a-7 b c f a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\left (\frac {\sqrt {d} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (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 {a^{3/2} f \sqrt {d x^2+c} \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 (d e-c f) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{a (b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {\frac {b \sqrt {c} (2 b d e+b c f-3 a d 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 )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} \left (-2 d f a^2+b d e a+b^2 c e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {d \left (\frac {\int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{b e-a f}\right )}{b e-a f}\) |
Input:
Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^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[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[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 + 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[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ [d/c]
Int[Sqrt[(e_) + (f_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[b/(b*c - a*d) Int[Sqrt[e + f*x^2]/((a + b*x^2)* Sqrt[c + d*x^2]), x], x] - Simp[d/(b*c - a*d) Int[Sqrt[e + f*x^2]/(c + d* x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c] && PosQ[f/e ]
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]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[b/(b*c - a*d) Int[(a + b*x^2)^p*(c + d*x^2)^ (q + 1)*(e + f*x^2)^r, x], x] - Simp[d/(b*c - a*d) Int[(a + b*x^2)^(p + 1 )*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && ILtQ[p, 0] && LeQ[q, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(3093\) vs. \(2(1009)=2018\).
Time = 21.74 (sec) , antiderivative size = 3094, normalized size of antiderivative = 2.96
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(3094\) |
| default | \(\text {Expression too large to display}\) | \(14457\) |
Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^2,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)*(7/3*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*b^5/(a*d-b*c)^3/a/(a^2*f^2-2*a*b*e*f+b^2*e^2)/(a*f-b*e)*e*EllipticE( x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-2/3*c^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)*b^6/(a*d-b*c)^ 3/a^2/(a^2*f^2-2*a*b*e*f+b^2*e^2)/(a*f-b*e)*e*EllipticE(x*(-b/a)^(1/2),(-1 +(a*d+b*c)/c/b)^(1/2))-1/2/(-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))*b*f^3*d/(c*f-d*e)^2/(a*f-b*e)/(a^2*f^2-2*a*b*e*f+b^2*e^2)-1 3/3/(-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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*b^3/(a*d -b*c)^2/(a^2*f^2-2*a*b*e*f+b^2*e^2)/(a*f-b*e)*d*f+1/(-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))*d^4/c/(a*c*d*f-a*d^2*e-b*c^2*f+b*c *d*e)^2+1/3/(-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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))* b^3*d/(a*d-b*c)^2/(a*f-b*e)^2/a-1/(-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^5/c/(a*d-b*c)/(a*c*d*f-a*d^2*e-b*c^2*f+b*c*d*e)^ 2+1/(-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...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**(3/2)/(f*x**2+e)**2,x)
Output:
Integral(1/((a + b*x**2)**(5/2)*(c + d*x**2)**(3/2)*(e + f*x**2)**2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="maxi ma")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)*(f*x^2 + e)^2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="giac ")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)*(f*x^2 + e)^2), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^2),x)
Output:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{3} d^{2} f^{2} x^{14}+3 a \,b^{2} d^{2} f^{2} x^{12}+2 b^{3} c d \,f^{2} x^{12}+2 b^{3} d^{2} e f \,x^{12}+3 a^{2} b \,d^{2} f^{2} x^{10}+6 a \,b^{2} c d \,f^{2} x^{10}+6 a \,b^{2} d^{2} e f \,x^{10}+b^{3} c^{2} f^{2} x^{10}+4 b^{3} c d e f \,x^{10}+b^{3} d^{2} e^{2} x^{10}+a^{3} d^{2} f^{2} x^{8}+6 a^{2} b c d \,f^{2} x^{8}+6 a^{2} b \,d^{2} e f \,x^{8}+3 a \,b^{2} c^{2} f^{2} x^{8}+12 a \,b^{2} c d e f \,x^{8}+3 a \,b^{2} d^{2} e^{2} x^{8}+2 b^{3} c^{2} e f \,x^{8}+2 b^{3} c d \,e^{2} x^{8}+2 a^{3} c d \,f^{2} x^{6}+2 a^{3} d^{2} e f \,x^{6}+3 a^{2} b \,c^{2} f^{2} x^{6}+12 a^{2} b c d e f \,x^{6}+3 a^{2} b \,d^{2} e^{2} x^{6}+6 a \,b^{2} c^{2} e f \,x^{6}+6 a \,b^{2} c d \,e^{2} x^{6}+b^{3} c^{2} e^{2} x^{6}+a^{3} c^{2} f^{2} x^{4}+4 a^{3} c d e f \,x^{4}+a^{3} d^{2} e^{2} x^{4}+6 a^{2} b \,c^{2} e f \,x^{4}+6 a^{2} b c d \,e^{2} x^{4}+3 a \,b^{2} c^{2} e^{2} x^{4}+2 a^{3} c^{2} e f \,x^{2}+2 a^{3} c d \,e^{2} x^{2}+3 a^{2} b \,c^{2} e^{2} x^{2}+a^{3} c^{2} e^{2}}d x \] Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^2,x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**3*c**2*e**2 + 2*a**3*c**2*e*f* x**2 + a**3*c**2*f**2*x**4 + 2*a**3*c*d*e**2*x**2 + 4*a**3*c*d*e*f*x**4 + 2*a**3*c*d*f**2*x**6 + a**3*d**2*e**2*x**4 + 2*a**3*d**2*e*f*x**6 + a**3*d **2*f**2*x**8 + 3*a**2*b*c**2*e**2*x**2 + 6*a**2*b*c**2*e*f*x**4 + 3*a**2* b*c**2*f**2*x**6 + 6*a**2*b*c*d*e**2*x**4 + 12*a**2*b*c*d*e*f*x**6 + 6*a** 2*b*c*d*f**2*x**8 + 3*a**2*b*d**2*e**2*x**6 + 6*a**2*b*d**2*e*f*x**8 + 3*a **2*b*d**2*f**2*x**10 + 3*a*b**2*c**2*e**2*x**4 + 6*a*b**2*c**2*e*f*x**6 + 3*a*b**2*c**2*f**2*x**8 + 6*a*b**2*c*d*e**2*x**6 + 12*a*b**2*c*d*e*f*x**8 + 6*a*b**2*c*d*f**2*x**10 + 3*a*b**2*d**2*e**2*x**8 + 6*a*b**2*d**2*e*f*x **10 + 3*a*b**2*d**2*f**2*x**12 + b**3*c**2*e**2*x**6 + 2*b**3*c**2*e*f*x* *8 + b**3*c**2*f**2*x**10 + 2*b**3*c*d*e**2*x**8 + 4*b**3*c*d*e*f*x**10 + 2*b**3*c*d*f**2*x**12 + b**3*d**2*e**2*x**10 + 2*b**3*d**2*e*f*x**12 + b** 3*d**2*f**2*x**14),x)