Integrand size = 32, antiderivative size = 995 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=-\frac {(b c-a d)^2 \left (a f \left (8 d^2 e^2+36 c d e f-9 c^2 f^2\right )-b e \left (8 d^2 e^2+21 c d e f+6 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{24 c d e^2 (b e-a f) (d e-c f)^3 \left (c+d x^2\right )^{3/2}}-\frac {b^2 (a f (10 d e-3 c f)-b e (5 d e+2 c f)) x \sqrt {a+b x^2}}{8 d e^2 (b e-a f) (d e-c f)^2 \sqrt {c+d x^2}}-\frac {f x \left (a+b x^2\right )^{5/2}}{4 e (d e-c f) \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}+\frac {f (a f (10 d e-3 c f)-b e (5 d e+2 c f)) x \left (a+b x^2\right )^{5/2}}{8 e^2 (b e-a f) (d e-c f)^2 \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}-\frac {\sqrt {d} \left (5 b^2 c^2 e^2 (11 d e+10 c f)-3 a b c e \left (8 d^2 e^2+59 c d e f+3 c^2 f^2\right )-a^2 \left (16 d^3 e^3-88 c d^2 e^2 f-42 c^2 d e f^2+9 c^3 f^3\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 )}{24 c^{3/2} e^2 (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (3 a^3 c d f^2 \left (48 d^2 e^2-16 c d e f+3 c^2 f^2\right )-3 b^3 c^2 e^2 \left (3 d^2 e^2+24 c d e f+8 c^2 f^2\right )+a b^2 c d e^2 \left (32 d^2 e^2+146 c d e f+137 c^2 f^2\right )-a^2 b d e \left (8 d^3 e^3+128 c d^2 e^2 f+185 c^2 d e f^2-6 c^3 f^3\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 )}{24 a \sqrt {c} \sqrt {d} e^2 (d e-c f)^5 \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} (b e-a f) \left (2 a b e f \left (18 d^2 e^2+19 c d e f-2 c^2 f^2\right )-a^2 f^2 \left (48 d^2 e^2-16 c d e f+3 c^2 f^2\right )-b^2 e^2 \left (3 d^2 e^2+24 c d e f+8 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{8 a \sqrt {d} e^3 (d e-c f)^5 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
-1/24*(-a*d+b*c)^2*(a*f*(-9*c^2*f^2+36*c*d*e*f+8*d^2*e^2)-b*e*(6*c^2*f^2+2 1*c*d*e*f+8*d^2*e^2))*x*(b*x^2+a)^(1/2)/c/d/e^2/(-a*f+b*e)/(-c*f+d*e)^3/(d *x^2+c)^(3/2)-1/8*b^2*(a*f*(-3*c*f+10*d*e)-b*e*(2*c*f+5*d*e))*x*(b*x^2+a)^ (1/2)/d/e^2/(-a*f+b*e)/(-c*f+d*e)^2/(d*x^2+c)^(1/2)-1/4*f*x*(b*x^2+a)^(5/2 )/e/(-c*f+d*e)/(d*x^2+c)^(3/2)/(f*x^2+e)^2+1/8*f*(a*f*(-3*c*f+10*d*e)-b*e* (2*c*f+5*d*e))*x*(b*x^2+a)^(5/2)/e^2/(-a*f+b*e)/(-c*f+d*e)^2/(d*x^2+c)^(3/ 2)/(f*x^2+e)-1/24*d^(1/2)*(5*b^2*c^2*e^2*(10*c*f+11*d*e)-3*a*b*c*e*(3*c^2* f^2+59*c*d*e*f+8*d^2*e^2)-a^2*(9*c^3*f^3-42*c^2*d*e*f^2-88*c*d^2*e^2*f+16* d^3*e^3))*(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))/c^(3/2)/e^2/(-c*f+d*e)^4/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/ (d*x^2+c)^(1/2)+1/24*(3*a^3*c*d*f^2*(3*c^2*f^2-16*c*d*e*f+48*d^2*e^2)-3*b^ 3*c^2*e^2*(8*c^2*f^2+24*c*d*e*f+3*d^2*e^2)+a*b^2*c*d*e^2*(137*c^2*f^2+146* c*d*e*f+32*d^2*e^2)-a^2*b*d*e*(-6*c^3*f^3+185*c^2*d*e*f^2+128*c*d^2*e^2*f+ 8*d^3*e^3))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b *c/a/d)^(1/2))/a/c^(1/2)/d^(1/2)/e^2/(-c*f+d*e)^5/(c*(b*x^2+a)/a/(d*x^2+c) )^(1/2)/(d*x^2+c)^(1/2)-1/8*c^(3/2)*(-a*f+b*e)*(2*a*b*e*f*(-2*c^2*f^2+19*c *d*e*f+18*d^2*e^2)-a^2*f^2*(3*c^2*f^2-16*c*d*e*f+48*d^2*e^2)-b^2*e^2*(8*c^ 2*f^2+24*c*d*e*f+3*d^2*e^2))*(b*x^2+a)^(1/2)*EllipticPi(d^(1/2)*x/c^(1/2)/ (1+d*x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d^(1/2)/e^3/(-c*f+d*e)^5/ (c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 10.78 (sec) , antiderivative size = 670, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\frac {-\sqrt {\frac {b}{a}} e f x \left (a+b x^2\right ) \left (6 c^2 e f (b e-a f)^2 (d e-c f) \left (c+d x^2\right )^2+3 c^2 f (b e-a f) (a f (-14 d e+3 c f)+b e (5 d e+6 c f)) \left (c+d x^2\right )^2 \left (e+f x^2\right )+8 c d (b c-a d)^2 e^2 (-d e+c f) \left (e+f x^2\right )^2+8 d (b c-a d) e^2 (a d (2 d e-11 c f)+b c (5 d e+4 c f)) \left (c+d x^2\right ) \left (e+f x^2\right )^2\right )-i c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right )^2 \left (b e f \left (5 b^2 c^2 e^2 (11 d e+10 c f)-3 a b c e \left (8 d^2 e^2+59 c d e f+3 c^2 f^2\right )+a^2 \left (-16 d^3 e^3+88 c d^2 e^2 f+42 c^2 d e f^2-9 c^3 f^3\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+b e (d e-c f) \left (-a b c e f (61 d e+9 c f)+b^2 c e^2 (9 d e+26 c f)+a^2 f \left (8 d^2 e^2+36 c d e f-9 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 c (b e-a f) \left (-2 a b e f \left (18 d^2 e^2+19 c d e f-2 c^2 f^2\right )+a^2 f^2 \left (48 d^2 e^2-16 c d e f+3 c^2 f^2\right )+b^2 e^2 \left (3 d^2 e^2+24 c d e f+8 c^2 f^2\right )\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{24 \sqrt {\frac {b}{a}} c^2 e^3 f (d e-c f)^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \] Input:
Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^3),x]
Output:
(-(Sqrt[b/a]*e*f*x*(a + b*x^2)*(6*c^2*e*f*(b*e - a*f)^2*(d*e - c*f)*(c + d *x^2)^2 + 3*c^2*f*(b*e - a*f)*(a*f*(-14*d*e + 3*c*f) + b*e*(5*d*e + 6*c*f) )*(c + d*x^2)^2*(e + f*x^2) + 8*c*d*(b*c - a*d)^2*e^2*(-(d*e) + c*f)*(e + f*x^2)^2 + 8*d*(b*c - a*d)*e^2*(a*d*(2*d*e - 11*c*f) + b*c*(5*d*e + 4*c*f) )*(c + d*x^2)*(e + f*x^2)^2)) - I*c*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)^2*(b*e*f*(5*b^2*c^2*e^2*(11*d*e + 10*c*f) - 3*a* b*c*e*(8*d^2*e^2 + 59*c*d*e*f + 3*c^2*f^2) + a^2*(-16*d^3*e^3 + 88*c*d^2*e ^2*f + 42*c^2*d*e*f^2 - 9*c^3*f^3))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d )/(b*c)] + b*e*(d*e - c*f)*(-(a*b*c*e*f*(61*d*e + 9*c*f)) + b^2*c*e^2*(9*d *e + 26*c*f) + a^2*f*(8*d^2*e^2 + 36*c*d*e*f - 9*c^2*f^2))*EllipticF[I*Arc Sinh[Sqrt[b/a]*x], (a*d)/(b*c)] - 3*c*(b*e - a*f)*(-2*a*b*e*f*(18*d^2*e^2 + 19*c*d*e*f - 2*c^2*f^2) + a^2*f^2*(48*d^2*e^2 - 16*c*d*e*f + 3*c^2*f^2) + b^2*e^2*(3*d^2*e^2 + 24*c*d*e*f + 8*c^2*f^2))*EllipticPi[(a*f)/(b*e), I* ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(24*Sqrt[b/a]*c^2*e^3*f*(d*e - c*f)^4 *Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2)^2)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}-\frac {\int -\frac {d \left (b (d e-4 c f) x^2+a (2 d e-5 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int \frac {d \left (b (d e-4 c f) x^2+a (2 d e-5 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int \frac {b (d e-4 c f) x^2+a (2 d e-5 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\frac {a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d \left (\frac {\frac {a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)-b c (d e-4 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)-b c (d e-4 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {d \left (\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)-b c (d e-4 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {d \left (\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)-b c (d e-4 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\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}{(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 )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\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}{(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 )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \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 )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \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 )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d (d e-c f) x \sqrt {b x^2+a}}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d (d e-c f) x \sqrt {b x^2+a}}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\int \frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\int \frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {d (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {d (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\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 f^2}{(d e-c f)^2 \sqrt {b x^2+a}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\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 f^2}{(d e-c f)^2 \sqrt {b x^2+a}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\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 f^2}{(d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\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 f^2}{(d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{d e-c 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}{d e-c f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{d e-c 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 )^3}dx}{d e-c f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \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 )^{5/2}}dx}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \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 )^{5/2}}dx}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \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 )^{5/2}}dx}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{d e-c 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 )^3}dx}{d e-c f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \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 )^{5/2}}dx}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \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 )^{5/2}}dx}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \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 )^{5/2}}dx}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{d e-c 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 )^3}dx}{d e-c f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d (d e-c f) x \sqrt {b x^2+a}}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d (d e-c f) x \sqrt {b x^2+a}}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d (d e-c f) x \sqrt {b x^2+a}}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{d e-c 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 )^3}dx}{d e-c f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\int \frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\int \frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\int \frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{d e-c 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 )^3}dx}{d e-c f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {d (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {d (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {d (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{d e-c 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 )^3}dx}{d e-c f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{d e-c 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 )^3}dx}{d e-c f}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c 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 {(a d (2 d e-5 c f)-b c (d e-4 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} \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}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\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 (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c 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}{d e-c f}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{d e-c 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 )^3}dx}{d e-c f}\right )}{f}\right )}{f}\right )}{f}\) |
Input:
Int[(a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^3),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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[d/b Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(a + b*x^2)^p*(c + d*x ^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt Q[p, 0] && GtQ[q, 0]
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. \(4003\) vs. \(2(951)=1902\).
Time = 20.40 (sec) , antiderivative size = 4004, normalized size of antiderivative = 4.02
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(4004\) |
| default | \(\text {Expression too large to display}\) | \(12398\) |
Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,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)*(1/8*f*(3*a^2* c*f^3-14*a^2*d*e*f^2+3*a*b*c*e*f^2+19*a*b*d*e^2*f-6*b^2*c*e^2*f-5*b^2*d*e^ 3)/(c*f-d*e)^4/e^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+e)-1/3*(a^ 2*d^2-2*a*b*c*d+b^2*c^2)/d/(c*f-d*e)^3/c*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^( 1/2)/(x^2+c/d)^2+1/4*(a^2*f^2-2*a*b*e*f+b^2*e^2)*f/(c*f-d*e)^3/e*x*(b*d*x^ 4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+e)^2-1/3*(b*d*x^2+a*d)*(11*a^2*c*d^2*f -2*a^2*d^3*e-15*a*b*c^2*d*f-3*a*b*c*d^2*e+4*b^2*c^3*f+5*b^2*c^2*d*e)/(c*f- d*e)^4/c^2*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+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^2/(c*f-d*e)^3*a*d-3/4/(-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)*El lipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))/(c*f-d*e)^4*c^2*b^3*f-15/ 8/(-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*d/(c*f-d *e)^4/e*f^2*a^2*c-1/(c*f-d*e)^4*f/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/ c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f /b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b^3*c^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/(c*f-d*e)^3/c*a^2*d^2+3/2/(-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...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(5/2)/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^3), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^3), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{5/2}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^3),x)
Output:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^3), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*x - 54*int((sqrt(c + d*x**2 )*sqrt(a + b*x**2)*x**6)/(6*a**2*c**3*d*e**3*f + 18*a**2*c**3*d*e**2*f**2* x**2 + 18*a**2*c**3*d*e*f**3*x**4 + 6*a**2*c**3*d*f**4*x**6 + 18*a**2*c**2 *d**2*e**3*f*x**2 + 54*a**2*c**2*d**2*e**2*f**2*x**4 + 54*a**2*c**2*d**2*e *f**3*x**6 + 18*a**2*c**2*d**2*f**4*x**8 + 18*a**2*c*d**3*e**3*f*x**4 + 54 *a**2*c*d**3*e**2*f**2*x**6 + 54*a**2*c*d**3*e*f**3*x**8 + 18*a**2*c*d**3* f**4*x**10 + 6*a**2*d**4*e**3*f*x**6 + 18*a**2*d**4*e**2*f**2*x**8 + 18*a* *2*d**4*e*f**3*x**10 + 6*a**2*d**4*f**4*x**12 + 2*a*b*c**4*e**3*f + 6*a*b* c**4*e**2*f**2*x**2 + 6*a*b*c**4*e*f**3*x**4 + 2*a*b*c**4*f**4*x**6 + a*b* c**3*d*e**4 + 15*a*b*c**3*d*e**3*f*x**2 + 39*a*b*c**3*d*e**2*f**2*x**4 + 3 7*a*b*c**3*d*e*f**3*x**6 + 12*a*b*c**3*d*f**4*x**8 + 3*a*b*c**2*d**2*e**4* x**2 + 33*a*b*c**2*d**2*e**3*f*x**4 + 81*a*b*c**2*d**2*e**2*f**2*x**6 + 75 *a*b*c**2*d**2*e*f**3*x**8 + 24*a*b*c**2*d**2*f**4*x**10 + 3*a*b*c*d**3*e* *4*x**4 + 29*a*b*c*d**3*e**3*f*x**6 + 69*a*b*c*d**3*e**2*f**2*x**8 + 63*a* b*c*d**3*e*f**3*x**10 + 20*a*b*c*d**3*f**4*x**12 + a*b*d**4*e**4*x**6 + 9* a*b*d**4*e**3*f*x**8 + 21*a*b*d**4*e**2*f**2*x**10 + 19*a*b*d**4*e*f**3*x* *12 + 6*a*b*d**4*f**4*x**14 + 2*b**2*c**4*e**3*f*x**2 + 6*b**2*c**4*e**2*f **2*x**4 + 6*b**2*c**4*e*f**3*x**6 + 2*b**2*c**4*f**4*x**8 + b**2*c**3*d*e **4*x**2 + 9*b**2*c**3*d*e**3*f*x**4 + 21*b**2*c**3*d*e**2*f**2*x**6 + 19* b**2*c**3*d*e*f**3*x**8 + 6*b**2*c**3*d*f**4*x**10 + 3*b**2*c**2*d**2*e...