Integrand size = 32, antiderivative size = 747 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=-\frac {b \left (3 a b c f^2-3 a^2 d f^2-2 b^2 e (d e-c f)\right ) x \sqrt {c+d x^2}}{6 a (b c-a d) e (b e-a f)^2 (d e-c f) \left (a+b x^2\right )^{3/2}}+\frac {f^2 x \sqrt {c+d x^2}}{2 e (b e-a f) (d e-c f) \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}-\frac {\sqrt {b} \left (6 a^3 b c d f^3-3 a^4 d^2 f^3-4 b^4 c e^2 (d e-c f)+8 a b^3 e \left (d^2 e^2+c d e f-2 c^2 f^2\right )-a^2 b^2 f \left (20 d^2 e^2-20 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{6 a^{3/2} (b c-a d)^2 e (b e-a f)^3 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {b} \left (2 b^4 c d e^3+3 a^4 d^2 f^3-6 a^3 b d f^2 (6 d e+c f)+a^2 b^2 f \left (24 d^2 e^2+50 c d e f+3 c^2 f^2\right )-2 a b^3 e \left (3 d^2 e^2+8 c d e f+9 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{6 \sqrt {a} c (b c-a d)^2 e (b e-a f)^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} f^3 (b e (7 d e-6 c f)-a f (2 d e-c f)) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c e^2 (b e-a f)^4 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] 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*(d*x^2+c)^(1/2)/a/(- a*d+b*c)/e/(-a*f+b*e)^2/(-c*f+d*e)/(b*x^2+a)^(3/2)+1/2*f^2*x*(d*x^2+c)^(1/ 2)/e/(-a*f+b*e)/(-c*f+d*e)/(b*x^2+a)^(3/2)/(f*x^2+e)-1/6*b^(1/2)*(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)+8*a*b^3*e*(-2*c^2*f^2+c*d*e*f +d^2*e^2)-a^2*b^2*f*(3*c^2*f^2-20*c*d*e*f+20*d^2*e^2))*(d*x^2+c)^(1/2)*Ell ipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(3/2)/(-a* d+b*c)^2/e/(-a*f+b*e)^3/(-c*f+d*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a ))^(1/2)-1/6*b^(1/2)*(2*b^4*c*d*e^3+3*a^4*d^2*f^3-6*a^3*b*d*f^2*(c*f+6*d*e )+a^2*b^2*f*(3*c^2*f^2+50*c*d*e*f+24*d^2*e^2)-2*a*b^3*e*(9*c^2*f^2+8*c*d*e *f+3*d^2*e^2))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),( 1-a*d/b/c)^(1/2))/a^(1/2)/c/(-a*d+b*c)^2/e/(-a*f+b*e)^4/(b*x^2+a)^(1/2)/(a *(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/2*a^(3/2)*f^3*(b*e*(-6*c*f+7*d*e)-a*f*(-c* f+2*d*e))*(d*x^2+c)^(1/2)*EllipticPi(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),1 -a*f/b/e,(1-a*d/b/c)^(1/2))/b^(1/2)/c/e^2/(-a*f+b*e)^4/(-c*f+d*e)/(b*x^2+a )^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 8.32 (sec) , antiderivative size = 569, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {\sqrt {\frac {b}{a}} e x \left (c+d x^2\right ) \left (3 a^2 (b c-a d)^2 f^4 \left (a+b x^2\right )^2+2 a b^3 (b c-a d) e (b e-a f) (d e-c f) \left (e+f x^2\right )+4 b^3 e (d e-c f) \left (b^2 c e+5 a^2 d f-2 a b (d e+2 c f)\right ) \left (a+b x^2\right ) \left (e+f x^2\right )\right )+i \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \left (-b c e \left (6 a^3 b c d f^3-3 a^4 d^2 f^3+4 b^4 c e^2 (-d e+c f)+a^2 b^2 f \left (-20 d^2 e^2+20 c d e f-3 c^2 f^2\right )+8 a b^3 e \left (d^2 e^2+c d e f-2 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+(b c-a d) \left (b e (-d e+c f) \left (4 b^3 c e^2+3 a^3 d f^2-3 a^2 b f (-6 d e+c f)-2 a b^2 e (3 d e+8 c f)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+3 a^2 (-b c+a d) f^2 (b e (7 d e-6 c f)+a f (-2 d e+c f)) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{6 a^2 \sqrt {\frac {b}{a}} (b c-a d)^2 e^2 (b e-a f)^3 (d e-c f) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[1/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
(Sqrt[b/a]*e*x*(c + d*x^2)*(3*a^2*(b*c - a*d)^2*f^4*(a + b*x^2)^2 + 2*a*b^ 3*(b*c - a*d)*e*(b*e - a*f)*(d*e - c*f)*(e + f*x^2) + 4*b^3*e*(d*e - c*f)* (b^2*c*e + 5*a^2*d*f - 2*a*b*(d*e + 2*c*f))*(a + b*x^2)*(e + f*x^2)) + I*( a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*(-(b*c*e*(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) + a^2*b^2*f*(- 20*d^2*e^2 + 20*c*d*e*f - 3*c^2*f^2) + 8*a*b^3*e*(d^2*e^2 + c*d*e*f - 2*c^ 2*f^2))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]) + (b*c - a*d)*(b*e *(-(d*e) + c*f)*(4*b^3*c*e^2 + 3*a^3*d*f^2 - 3*a^2*b*f*(-6*d*e + c*f) - 2* a*b^2*e*(3*d*e + 8*c*f))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + 3*a^2*(-(b*c) + a*d)*f^2*(b*e*(7*d*e - 6*c*f) + a*f*(-2*d*e + c*f))*Ellipt icPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])))/(6*a^2*Sqrt[b/a] *(b*c - a*d)^2*e^2*(b*e - a*f)^3*(d*e - c*f)*(a + b*x^2)^(3/2)*Sqrt[c + d* x^2]*(e + f*x^2))
Time = 1.95 (sec) , antiderivative size = 1283, normalized size of antiderivative = 1.72, number of steps used = 26, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {426, 421, 25, 402, 25, 400, 313, 320, 413, 413, 412, 426, 421, 25, 400, 313, 320, 414, 424, 406, 320, 388, 313, 413, 413, 412}
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \int \frac {1}{\left (b x^2+a\right )^{5/2} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {b \int -\frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{5/2} \sqrt {d x^2+c}}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \int \frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{5/2} \sqrt {d x^2+c}}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}-\frac {\int -\frac {6 d f a^2-b (3 d e+5 c f) a+b d (b e-a f) x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{3 a (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {6 d f a^2-3 b d e a-5 b c f a+b d (b e-a f) x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {b \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right )^{3/2}}dx}{b c-a d}-\frac {d \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {f^2 \sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{\sqrt {a+b x^2} (b e-a f)^2}+\frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx}{\sqrt {a+b x^2} \sqrt {c+d x^2} (b e-a f)^2}+\frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {b \int -\frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \int \frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right )^{3/2}}dx}{b c-a d}-\frac {(-2 a d f+b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {(-2 a d f+b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {b \left (\frac {b (b e-a f) \sqrt {d x^2+c} x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {b} \left (7 d f a^2-4 b d e a-5 b c f a+2 b^2 c e\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (6 d f a^2-b (3 d e+4 c f) a+b^2 c e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(b e-a f)^2}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 a d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {b \left (\frac {b (b e-a f) \sqrt {d x^2+c} x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {b} \left (7 d f a^2-4 b d e a-5 b c f a+2 b^2 c e\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (6 d f a^2-b (3 d e+4 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 )}{a (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}}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {a^{3/2} \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 ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 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}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 424 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {b \left (\frac {b (b e-a f) \sqrt {d x^2+c} x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {b} \left (7 d f a^2-4 b d e a-5 b c f a+2 b^2 c e\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (6 d f a^2-b (3 d e+4 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 )}{a (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}}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {a^{3/2} \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 ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 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}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}-\frac {b d \int \frac {f x^2+e}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{2 e (b e-a f) (d e-c f)}\right )}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {b \left (\frac {b (b e-a f) \sqrt {d x^2+c} x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {b} \left (7 d f a^2-4 b d e a-5 b c f a+2 b^2 c e\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (6 d f a^2-b (3 d e+4 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 )}{a (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}}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {a^{3/2} \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 ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 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}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (e \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}\right )}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {b \left (\frac {b (b e-a f) \sqrt {d x^2+c} x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {b} \left (7 d f a^2-4 b d e a-5 b c f a+2 b^2 c e\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (6 d f a^2-b (3 d e+4 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 )}{a (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}}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {a^{3/2} \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 ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 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}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (\frac {\sqrt {c} e \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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}\right )}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {b \left (\frac {b (b e-a f) \sqrt {d x^2+c} x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {b} \left (7 d f a^2-4 b d e a-5 b c f a+2 b^2 c e\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (6 d f a^2-b (3 d e+4 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 )}{a (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}}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {a^{3/2} \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 ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 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}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (\frac {\sqrt {c} e \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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}\right )}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {b \left (\frac {b (b e-a f) \sqrt {d x^2+c} x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {b} \left (7 d f a^2-4 b d e a-5 b c f a+2 b^2 c e\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (6 d f a^2-b (3 d e+4 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 )}{a (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}}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {a^{3/2} \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 ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 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}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}\right )}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {b \left (\frac {b (b e-a f) \sqrt {d x^2+c} x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {b} \left (7 d f a^2-4 b d e a-5 b c f a+2 b^2 c e\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (6 d f a^2-b (3 d e+4 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 )}{a (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}}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {a^{3/2} \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 ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 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}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f) \sqrt {b x^2+a}}\right )}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {b \left (\frac {b (b e-a f) \sqrt {d x^2+c} x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {b} \left (7 d f a^2-4 b d e a-5 b c f a+2 b^2 c e\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (6 d f a^2-b (3 d e+4 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 )}{a (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}}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {a^{3/2} \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 ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 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}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}\right )}{b e-a f}\right )}{b e-a f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {b \left (\frac {b (b e-a f) \sqrt {d x^2+c} x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {b} \left (7 d f a^2-4 b d e a-5 b c f a+2 b^2 c e\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (6 d f a^2-b (3 d e+4 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 )}{a (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}}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {b \left (\frac {a^{3/2} \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 ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 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}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {\sqrt {-a} (b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{2 \sqrt {b} e^2 (b e-a f) (d e-c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}\right )}{b e-a f}\right )}{b e-a f}\) |
Input:
Int[1/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
(b*((b*((b*(b*e - a*f)*x*Sqrt[c + d*x^2])/(3*a*(b*c - a*d)*(a + b*x^2)^(3/ 2)) + ((Sqrt[b]*(2*b^2*c*e - 4*a*b*d*e - 5*a*b*c*f + 7*a^2*d*f)*Sqrt[c + d *x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[a]*(b *c - a*d)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (Sqrt[c ]*Sqrt[d]*(b^2*c*e + 6*a^2*d*f - a*b*(3*d*e + 4*c*f))*Sqrt[a + b*x^2]*Elli pticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*(b*c - a*d)*Sqrt[( c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*a*(b*c - a*d))))/(b*e - a*f)^2 + (Sqrt[-a]*f^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elliptic Pi[(a*f)/(b*e), ArcSin[(Sqrt[b]*x)/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*e*(b* e - a*f)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])))/(b*e - a*f) - (f*(-((f*((f^2 *x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(2*e*(b*e - a*f)*(d*e - c*f)*(e + f*x^ 2)) - (b*d*(f*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d ]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*e*Sqr t[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*S qrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/(2*e*(b*e - a*f)*(d*e - c*f)) + (Sqrt[-a]*(b*e*(3*d*e - 2*c*f) - a*f*(2*d*e - c*f))* Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), ArcSin[(Sq rt[b]*x)/Sqrt[-a]], (a*d)/(b*c)])/(2*Sqrt[b]*e^2*(b*e - a*f)*(d*e - c*f)*S qrt[a + b*x^2]*Sqrt[c + d*x^2])))/(b*e - a*f)) + (b*((b*((Sqrt[b]*(b*e ...
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[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !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[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
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[(((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[1/(((a_) + (b_.)*(x_)^2)^2*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)* (x_)^2]), x_Symbol] :> Simp[b^2*x*Sqrt[c + d*x^2]*(Sqrt[e + f*x^2]/(2*a*(b* c - a*d)*(b*e - a*f)*(a + b*x^2))), x] + (Simp[(b^2*c*e + 3*a^2*d*f - 2*a*b *(d*e + c*f))/(2*a*(b*c - a*d)*(b*e - a*f)) Int[1/((a + b*x^2)*Sqrt[c + d *x^2]*Sqrt[e + f*x^2]), x], x] - Simp[d*(f/(2*a*(b*c - a*d)*(b*e - a*f))) Int[(a + b*x^2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x]
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. \(2508\) vs. \(2(715)=1430\).
Time = 20.53 (sec) , antiderivative size = 2509, normalized size of antiderivative = 3.36
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2509\) |
| default | \(\text {Expression too large to display}\) | \(9310\) |
Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(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)*(1/2*f^4*b/(a* f-b*e)/(a^2*f^2-2*a*b*e*f+b^2*e^2)/e/(c*f-d*e)*x*(b*d*x^4+a*d*x^2+b*c*x^2+ a*c)^(1/2)/(b*f*x^2+b*e)-1/3*b/(a*f-b*e)^2/(a*d-b*c)/a*x*(b*d*x^4+a*d*x^2+ b*c*x^2+a*c)^(1/2)/(x^2+a/b)^2-2/3*(b*d*x^2+b*c)*b^2/a^2/(a*d-b*c)^2*x*(5* a^2*d*f-4*a*b*c*f-2*a*b*d*e+b^2*c*e)/(a^2*f^2-2*a*b*e*f+b^2*e^2)/(a*f-b*e) /((x^2+a/b)*(b*d*x^2+b*c))^(1/2)-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))*d*b^2/(a*f-b*e)^2/(a*d-b*c)/a-8/3*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^4/(a*d-b*c)^2/a/(a^2*f^2-2*a*b*e*f+b^2*e^2)/(a*f-b*e)*f*EllipticE(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*b^3/(a*d-b*c)^2/(a^ 2*f^2-2*a*b*e*f+b^2*e^2)/(a*f-b*e)*f*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c )/c/b)^(1/2))-4/3/(-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)/a/(a^2*f^2-2*a*b*e*f+b^2*e^2)/(a*f-b*e)*d*e+2/3/(-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^4/(a*d-b*c)/a^2/ (a^2*f^2-2*a*b*e*f+b^2*e^2)/(a*f-b*e)*c*e-4/3*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^4/(a*d-b...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**2,x)
Output:
Integral(1/((a + b*x**2)**(5/2)*sqrt(c + d*x**2)*(e + f*x**2)**2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="maxi ma")
Output:
integrate(1/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="giac ")
Output:
integrate(1/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2),x)
Output:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{2}}d x \] Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)
Output:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)