Integrand size = 32, antiderivative size = 1221 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Output:
1/24*(-a*d+b*c)^3*(a*f*(-9*c^2*f^2+36*c*d*e*f+8*d^2*e^2)-b*e*(12*c^2*f^2+1 5*c*d*e*f+8*d^2*e^2))*x*(b*x^2+a)^(1/2)/c/d^2/e^2/(-a*f+b*e)/(-c*f+d*e)^3/ (d*x^2+c)^(3/2)-1/8*b^2*(3*a^2*d*f^2*(-3*c*f+10*d*e)-a*b*f*(-3*c^2*f^2+21* c*d*e*f+17*d^2*e^2)+b^2*e*(4*c^2*f^2+9*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2) /d^2/e^2/f/(-a*f+b*e)/(-c*f+d*e)^2/(d*x^2+c)^(1/2)-1/8*b^3*(a*f*(-3*c*f+10 *d*e)-b*e*(4*c*f+3*d*e))*x^3*(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)^(7/2)/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*(4*c*f+3*d*e))*x*(b*x^2+a)^(7/2)/e^ 2/(-a*f+b*e)/(-c*f+d*e)^2/(d*x^2+c)^(3/2)/(f*x^2+e)-1/24*(5*a*b^2*c^2*d*e^ 2*f*(31*c*f+32*d*e)-b^3*c^2*e^2*(8*c^2*f^2+94*c*d*e*f+3*d^2*e^2)-a^2*b*c*d *e*f*(12*c^2*f^2+271*c*d*e*f+32*d^2*e^2)-a^3*d*f*(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)/d^(1/2)/e^2/f/(-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*b^4*c^2*e^3*(-24*c^2* f^2-12*c*d*e*f+d^2*e^2)-3*a^4*c*d*f^3*(3*c^2*f^2-16*c*d*e*f+48*d^2*e^2)+5* a*b^3*c^2*e^2*f*(16*c^2*f^2+49*c*d*e*f+19*d^2*e^2)-a^2*b^2*c*d*e^2*f*(223* c^2*f^2+319*c*d*e*f+88*d^2*e^2)+a^3*b*d*e*f*(-9*c^3*f^3+221*c^2*d*e*f^2+20 0*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/f/(-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*(b^2*e...
Result contains complex when optimal does not.
Time = 11.78 (sec) , antiderivative size = 761, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\frac {\sqrt {\frac {b}{a}} d e f^2 x \left (a+b x^2\right ) \left (6 c^2 e (b e-a f)^3 (d e-c f) \left (c+d x^2\right )^2+3 c^2 (b e-a f)^2 (a f (-14 d e+3 c f)+b e (d e+10 c f)) \left (c+d x^2\right )^2 \left (e+f x^2\right )+8 c (b c-a d)^3 e^2 (-d e+c f) \left (e+f x^2\right )^2+8 (b c-a d)^2 e^2 (a d (2 d e-11 c f)+b c (8 d e+c f)) \left (c+d x^2\right ) \left (e+f x^2\right )^2\right )-i c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right )^2 \left (-b e f \left (-5 a b^2 c^2 d e^2 f (32 d e+31 c f)+b^3 c^2 e^2 \left (3 d^2 e^2+94 c d e f+8 c^2 f^2\right )+a^2 b c d e f \left (32 d^2 e^2+271 c d e f+12 c^2 f^2\right )+a^3 d f \left (16 d^3 e^3-88 c d^2 e^2 f-42 c^2 d e f^2+9 c^3 f^3\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+b e (d e-c f) \left (-3 a^2 b c d e f^2 (31 d e+4 c f)+15 a b^2 c d e^2 f (2 d e+5 c f)+a^3 d f^2 \left (8 d^2 e^2+36 c d e f-9 c^2 f^2\right )+b^3 c e^2 \left (3 d^2 e^2-30 c d e f-8 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+3 c d (b e-a f)^2 \left (-2 a b e f \left (6 d^2 e^2+33 c d e f-4 c^2 f^2\right )+a^2 f^2 \left (48 d^2 e^2-16 c d e f+3 c^2 f^2\right )+b^2 e^2 \left (-d^2 e^2+12 c d e f+24 c^2 f^2\right )\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{24 \sqrt {\frac {b}{a}} c^2 d e^3 f^2 (d e-c f)^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2} \] Input:
Integrate[(a + b*x^2)^(7/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^3),x]
Output:
(Sqrt[b/a]*d*e*f^2*x*(a + b*x^2)*(6*c^2*e*(b*e - a*f)^3*(d*e - c*f)*(c + d *x^2)^2 + 3*c^2*(b*e - a*f)^2*(a*f*(-14*d*e + 3*c*f) + b*e*(d*e + 10*c*f)) *(c + d*x^2)^2*(e + f*x^2) + 8*c*(b*c - a*d)^3*e^2*(-(d*e) + c*f)*(e + f*x ^2)^2 + 8*(b*c - a*d)^2*e^2*(a*d*(2*d*e - 11*c*f) + b*c*(8*d*e + 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*a*b^2*c^2*d*e^2*f*(32*d*e + 31*c*f) + b ^3*c^2*e^2*(3*d^2*e^2 + 94*c*d*e*f + 8*c^2*f^2) + a^2*b*c*d*e*f*(32*d^2*e^ 2 + 271*c*d*e*f + 12*c^2*f^2) + a^3*d*f*(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)*(-3*a^2*b*c*d*e*f^2*(31*d*e + 4*c*f) + 15*a*b^2*c*d*e^2* f*(2*d*e + 5*c*f) + a^3*d*f^2*(8*d^2*e^2 + 36*c*d*e*f - 9*c^2*f^2) + b^3*c *e^2*(3*d^2*e^2 - 30*c*d*e*f - 8*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[b/a]*x ], (a*d)/(b*c)] + 3*c*d*(b*e - a*f)^2*(-2*a*b*e*f*(6*d^2*e^2 + 33*c*d*e*f - 4*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*(-( d^2*e^2) + 12*c*d*e*f + 24*c^2*f^2))*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqr t[b/a]*x], (a*d)/(b*c)]))/(24*Sqrt[b/a]*c^2*d*e^3*f^2*(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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\left (b x^2+a\right )^{5/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{5/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 419 |
| \(\displaystyle \frac {b \left (\frac {b \left (-\frac {\int -\frac {\sqrt {b x^2+a} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {-\frac {\int -\frac {d \left (b (a d (d e-4 c f)+b c (2 d e+c f)) x^2+a (a d (2 d e-5 c f)+b c (d e+2 c f))\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\int \frac {d \left (b (a d (d e-4 c f)+b c (2 d e+c f)) x^2+a (a d (2 d e-5 c f)+b c (d e+2 c f))\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\int \frac {b (a d (d e-4 c f)+b c (2 d e+c f)) x^2+a (a d (2 d e-5 c f)+b c (d e+2 c f))}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {(a d (2 d e-5 c f)+b c (c f+2 d e)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx-a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(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}\right )}{f}-\frac {(b e-a f) \left (\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}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(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}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(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}\right )}{f}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\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}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\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}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\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 {\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}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\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 {\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}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\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 {\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}\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}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\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 {\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}\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}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\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 {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}\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}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\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 {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}\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}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\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 {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)) \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) \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}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\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 {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)) \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) \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}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\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) \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}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\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) \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}\right )}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (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) \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}\right )}{f}-\frac {(b e-a f) \left (\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) \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}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (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 \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}\right )}{f}-\frac {(b e-a f) \left (\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 \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}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (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}\right )}{f}-\frac {(b e-a f) \left (\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}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (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}\right )}{f}-\frac {(b e-a f) \left (\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}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (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}\right )}{f}-\frac {(b e-a f) \left (\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}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (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}\right )}{f}-\frac {(b e-a f) \left (\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}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (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}\right )}{f}-\frac {(b e-a f) \left (\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}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (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}\right )}{f}-\frac {(b e-a f) \left (\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}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (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}\right )}{f}-\frac {(b e-a f) \left (\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}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {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} (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} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (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}\right )}{f}-\frac {(b e-a f) \left (\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}\right )}{f}\) |
Input:
Int[(a + b*x^2)^(7/2)/((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[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*((b*e - a*f)/(b*c - a*d)^2) Int[(c + d*x^2)^( q + 2)*((e + f*x^2)^(r - 1)/(a + b*x^2)), x], x] - Simp[1/(b*c - a*d)^2 I nt[(c + d*x^2)^q*(e + f*x^2)^(r - 1)*(2*b*c*d*e - a*d^2*e - b*c^2*f + d^2*( b*e - a*f)*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && Gt Q[r, 1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(5212\) vs. \(2(1171)=2342\).
Time = 19.57 (sec) , antiderivative size = 5213, normalized size of antiderivative = 4.27
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(5213\) |
| default | \(\text {Expression too large to display}\) | \(16403\) |
Input:
int((b*x^2+a)^(7/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(7/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(7/2)/(d*x**2+c)**(5/2)/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((b*x^2+a)^(7/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(7/2)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^3), x)
\[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((b*x^2+a)^(7/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(7/2)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^3), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{7/2}}{{\left (d\,x^2+c\right )}^{5/2}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int((a + b*x^2)^(7/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^3),x)
Output:
int((a + b*x^2)^(7/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^3), x)
\[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{\left (d \,x^{2}+c \right )^{\frac {5}{2}} \left (f \,x^{2}+e \right )^{3}}d x \] Input:
int((b*x^2+a)^(7/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,x)
Output:
int((b*x^2+a)^(7/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^3,x)