Integrand size = 32, antiderivative size = 785 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (d+c x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\frac {(a c-b d) \left (b c e^2 (8 c e+27 d f)-a f \left (8 c^2 e^2+36 c d e f-9 d^2 f^2\right )\right ) x \sqrt {a+b x^2}}{24 d e^2 (b e-a f) (c e-d f)^3 \left (d+c x^2\right )^{3/2}}-\frac {f x \left (a+b x^2\right )^{3/2}}{4 e (c e-d f) \left (d+c x^2\right )^{3/2} \left (e+f x^2\right )^2}-\frac {f \left (7 b c e^2-a f (10 c e-3 d f)\right ) x \left (a+b x^2\right )^{3/2}}{8 e^2 (b e-a f) (c e-d f)^2 \left (d+c x^2\right )^{3/2} \left (e+f x^2\right )}+\frac {\sqrt {c} \left (b d e \left (16 c^2 e^2+83 c d e f+6 d^2 f^2\right )+a \left (16 c^3 e^3-88 c^2 d e^2 f-42 c d^2 e f^2+9 d^3 f^3\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{24 d^{3/2} e^2 (c e-d f)^4 \sqrt {\frac {d \left (a+b x^2\right )}{a \left (d+c x^2\right )}} \sqrt {d+c x^2}}+\frac {\sqrt {c} \left (15 b^2 d^2 e^2 f (3 c e+4 d f)+3 a^2 d f^2 \left (48 c^2 e^2-16 c d e f+3 d^2 f^2\right )-a b e \left (8 c^3 e^3+56 c^2 d e^2 f+149 c d^2 e f^2-3 d^3 f^3\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{24 a \sqrt {d} e^2 (c e-d f)^5 \sqrt {\frac {d \left (a+b x^2\right )}{a \left (d+c x^2\right )}} \sqrt {d+c x^2}}+\frac {d^{3/2} f \left (10 a b c e^2 f (6 c e+d f)-5 b^2 c e^3 (3 c e+4 d f)-a^2 f^2 \left (48 c^2 e^2-16 c d e f+3 d^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {d f}{c e},\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{8 a \sqrt {c} e^3 (c e-d f)^5 \sqrt {\frac {d \left (a+b x^2\right )}{a \left (d+c x^2\right )}} \sqrt {d+c x^2}} \] Output:
1/24*(a*c-b*d)*(b*c*e^2*(8*c*e+27*d*f)-a*f*(8*c^2*e^2+36*c*d*e*f-9*d^2*f^2 ))*x*(b*x^2+a)^(1/2)/d/e^2/(-a*f+b*e)/(c*e-d*f)^3/(c*x^2+d)^(3/2)-1/4*f*x* (b*x^2+a)^(3/2)/e/(c*e-d*f)/(c*x^2+d)^(3/2)/(f*x^2+e)^2-1/8*f*(7*b*c*e^2-a *f*(10*c*e-3*d*f))*x*(b*x^2+a)^(3/2)/e^2/(-a*f+b*e)/(c*e-d*f)^2/(c*x^2+d)^ (3/2)/(f*x^2+e)+1/24*c^(1/2)*(b*d*e*(16*c^2*e^2+83*c*d*e*f+6*d^2*f^2)+a*(1 6*c^3*e^3-88*c^2*d*e^2*f-42*c*d^2*e*f^2+9*d^3*f^3))*(b*x^2+a)^(1/2)*Ellipt icE(c^(1/2)*x/d^(1/2)/(1+c*x^2/d)^(1/2),(1-b*d/a/c)^(1/2))/d^(3/2)/e^2/(c* e-d*f)^4/(d*(b*x^2+a)/a/(c*x^2+d))^(1/2)/(c*x^2+d)^(1/2)+1/24*c^(1/2)*(15* b^2*d^2*e^2*f*(3*c*e+4*d*f)+3*a^2*d*f^2*(48*c^2*e^2-16*c*d*e*f+3*d^2*f^2)- a*b*e*(8*c^3*e^3+56*c^2*d*e^2*f+149*c*d^2*e*f^2-3*d^3*f^3))*(b*x^2+a)^(1/2 )*InverseJacobiAM(arctan(c^(1/2)*x/d^(1/2)),(1-b*d/a/c)^(1/2))/a/d^(1/2)/e ^2/(c*e-d*f)^5/(d*(b*x^2+a)/a/(c*x^2+d))^(1/2)/(c*x^2+d)^(1/2)+1/8*d^(3/2) *f*(10*a*b*c*e^2*f*(6*c*e+d*f)-5*b^2*c*e^3*(3*c*e+4*d*f)-a^2*f^2*(48*c^2*e ^2-16*c*d*e*f+3*d^2*f^2))*(b*x^2+a)^(1/2)*EllipticPi(c^(1/2)*x/d^(1/2)/(1+ c*x^2/d)^(1/2),1-d*f/c/e,(1-b*d/a/c)^(1/2))/a/c^(1/2)/e^3/(c*e-d*f)^5/(d*( b*x^2+a)/a/(c*x^2+d))^(1/2)/(c*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 9.28 (sec) , antiderivative size = 575, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (d+c x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\frac {\sqrt {\frac {b}{a}} e x \left (a+b x^2\right ) \left (6 d^2 e f^2 (b e-a f) (c e-d f) \left (d+c x^2\right )^2+3 d^2 f^2 (b e (9 c e+2 d f)+a f (-14 c e+3 d f)) \left (d+c x^2\right )^2 \left (e+f x^2\right )+8 c^2 d (a c-b d) e^2 (c e-d f) \left (e+f x^2\right )^2+8 c^2 e^2 (a c (2 c e-11 d f)+b d (2 c e+7 d f)) \left (d+c x^2\right ) \left (e+f x^2\right )^2\right )-i d \sqrt {1+\frac {b x^2}{a}} \left (d+c x^2\right ) \sqrt {1+\frac {c x^2}{d}} \left (e+f x^2\right )^2 \left (-b e \left (b d e \left (16 c^2 e^2+83 c d e f+6 d^2 f^2\right )+a \left (16 c^3 e^3-88 c^2 d e^2 f-42 c d^2 e f^2+9 d^3 f^3\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a c}{b d}\right )+b e (c e-d f) \left (-b d e (29 c e+6 d f)+a \left (8 c^2 e^2+36 c d e f-9 d^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a c}{b d}\right )+3 d \left (-10 a b c e^2 f (6 c e+d f)+5 b^2 c e^3 (3 c e+4 d f)+a^2 f^2 \left (48 c^2 e^2-16 c d e f+3 d^2 f^2\right )\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a c}{b d}\right )\right )}{24 \sqrt {\frac {b}{a}} d^2 e^3 (c e-d f)^4 \sqrt {a+b x^2} \left (d+c x^2\right )^{3/2} \left (e+f x^2\right )^2} \] Input:
Integrate[(a + b*x^2)^(3/2)/((d + c*x^2)^(5/2)*(e + f*x^2)^3),x]
Output:
(Sqrt[b/a]*e*x*(a + b*x^2)*(6*d^2*e*f^2*(b*e - a*f)*(c*e - d*f)*(d + c*x^2 )^2 + 3*d^2*f^2*(b*e*(9*c*e + 2*d*f) + a*f*(-14*c*e + 3*d*f))*(d + c*x^2)^ 2*(e + f*x^2) + 8*c^2*d*(a*c - b*d)*e^2*(c*e - d*f)*(e + f*x^2)^2 + 8*c^2* e^2*(a*c*(2*c*e - 11*d*f) + b*d*(2*c*e + 7*d*f))*(d + c*x^2)*(e + f*x^2)^2 ) - I*d*Sqrt[1 + (b*x^2)/a]*(d + c*x^2)*Sqrt[1 + (c*x^2)/d]*(e + f*x^2)^2* (-(b*e*(b*d*e*(16*c^2*e^2 + 83*c*d*e*f + 6*d^2*f^2) + a*(16*c^3*e^3 - 88*c ^2*d*e^2*f - 42*c*d^2*e*f^2 + 9*d^3*f^3))*EllipticE[I*ArcSinh[Sqrt[b/a]*x] , (a*c)/(b*d)]) + b*e*(c*e - d*f)*(-(b*d*e*(29*c*e + 6*d*f)) + a*(8*c^2*e^ 2 + 36*c*d*e*f - 9*d^2*f^2))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*c)/(b*d) ] + 3*d*(-10*a*b*c*e^2*f*(6*c*e + d*f) + 5*b^2*c*e^3*(3*c*e + 4*d*f) + a^2 *f^2*(48*c^2*e^2 - 16*c*d*e*f + 3*d^2*f^2))*EllipticPi[(a*f)/(b*e), I*ArcS inh[Sqrt[b/a]*x], (a*c)/(b*d)]))/(24*Sqrt[b/a]*d^2*e^3*(c*e - d*f)^4*Sqrt[ a + b*x^2]*(d + c*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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c x^2+d\right )^{5/2} \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\sqrt {b x^2+a}}{\left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{(c e-d f)^2}-\frac {c \int -\frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{(c e-d f)^2}+\frac {c \int \frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{(c e-d f)^2}+\frac {c \left (\frac {c x \sqrt {a+b x^2} (c e-d f)}{3 d \left (c x^2+d\right )^{3/2} (a c-b d)}-\frac {\int -\frac {b c (c e-d f) x^2+a c (2 c e-5 d f)-3 b d (c e-2 d f)}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{(c e-d f)^2}+\frac {c \left (\frac {\int \frac {b c (c e-d f) x^2+a c (2 c e-5 d f)-3 b d (c e-2 d f)}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{3 d (a c-b d)}+\frac {c x \sqrt {a+b x^2} (c e-d f)}{3 d \left (c x^2+d\right )^{3/2} (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{(c e-d f)^2}+\frac {c \left (\frac {-\frac {c (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \int \frac {\sqrt {b x^2+a}}{\left (c x^2+d\right )^{3/2}}dx}{a c-b d}-\frac {b (a c (c e-4 d f)-3 b d (c e-2 d f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d}}dx}{a c-b d}}{3 d (a c-b d)}+\frac {c x \sqrt {a+b x^2} (c e-d f)}{3 d \left (c x^2+d\right )^{3/2} (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c \left (\frac {-\frac {b (a c (c e-4 d f)-3 b d (c e-2 d f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d}}dx}{a c-b d}-\frac {\sqrt {c} \sqrt {a+b x^2} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}}{3 d (a c-b d)}+\frac {c x \sqrt {a+b x^2} (c e-d f)}{3 d \left (c x^2+d\right )^{3/2} (a c-b d)}\right )}{(c e-d f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{(c e-d f)^2}+\frac {c \left (\frac {-\frac {b \sqrt {d} \sqrt {a+b x^2} (a c (c e-4 d f)-3 b d (c e-2 d f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}}{3 d (a c-b d)}+\frac {c x \sqrt {a+b x^2} (c e-d f)}{3 d \left (c x^2+d\right )^{3/2} (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{\sqrt {a+b x^2} (c e-d f)^2}+\frac {c \left (\frac {-\frac {b \sqrt {d} \sqrt {a+b x^2} (a c (c e-4 d f)-3 b d (c e-2 d f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}}{3 d (a c-b d)}+\frac {c x \sqrt {a+b x^2} (c e-d f)}{3 d \left (c x^2+d\right )^{3/2} (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \left (f x^2+e\right )}dx}{\sqrt {a+b x^2} \sqrt {c x^2+d} (c e-d f)^2}+\frac {c \left (\frac {-\frac {b \sqrt {d} \sqrt {a+b x^2} (a c (c e-4 d f)-3 b d (c e-2 d f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}}{3 d (a c-b d)}+\frac {c x \sqrt {a+b x^2} (c e-d f)}{3 d \left (c x^2+d\right )^{3/2} (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c x^2+d} (c e-d f)^2}+\frac {c \left (\frac {-\frac {b \sqrt {d} \sqrt {a+b x^2} (a c (c e-4 d f)-3 b d (c e-2 d f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}}{3 d (a c-b d)}+\frac {c x \sqrt {a+b x^2} (c e-d f)}{3 d \left (c x^2+d\right )^{3/2} (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^3}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c x^2+d} (c e-d f)^2}+\frac {c \left (\frac {-\frac {b \sqrt {d} \sqrt {a+b x^2} (a c (c e-4 d f)-3 b d (c e-2 d f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} \sqrt {c x^2+d} (a c-b d) \sqrt {\frac {d \left (a+b x^2\right )}{a \left (c x^2+d\right )}}}}{3 d (a c-b d)}+\frac {c x \sqrt {a+b x^2} (c e-d f)}{3 d \left (c x^2+d\right )^{3/2} (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{(c e-d f)^2}-\frac {c \int -\frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{(c e-d f)^2}-\frac {c \int -\frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \int \frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \int \frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) x \sqrt {b x^2+a}}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}-\frac {\int -\frac {b c (c e-d f) x^2+a c (2 c e-5 d f)-3 b d (c e-2 d f)}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) x \sqrt {b x^2+a}}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}-\frac {\int -\frac {b c (c e-d f) x^2+a c (2 c e-5 d f)-3 b d (c e-2 d f)}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {\int \frac {b c (c e-d f) x^2+a c (2 c e-5 d f)-3 b d (c e-2 d f)}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {\int \frac {b c (c e-d f) x^2+a c (2 c e-5 d f)-3 b d (c e-2 d f)}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {c (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \int \frac {\sqrt {b x^2+a}}{\left (c x^2+d\right )^{3/2}}dx}{a c-b d}-\frac {b (a c (c e-4 d f)-3 b d (c e-2 d f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d}}dx}{a c-b d}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {c (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \int \frac {\sqrt {b x^2+a}}{\left (c x^2+d\right )^{3/2}}dx}{a c-b d}-\frac {b (a c (c e-4 d f)-3 b d (c e-2 d f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d}}dx}{a c-b d}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b (a c (c e-4 d f)-3 b d (c e-2 d f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d}}dx}{a c-b d}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b (a c (c e-4 d f)-3 b d (c e-2 d f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d}}dx}{a c-b d}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2 \sqrt {b x^2+a}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2 \sqrt {b x^2+a}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2} \left (f x^2+e\right )}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}-\frac {f \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {f^2 \int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(c e-d f)^2}-\frac {c \int -\frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {f^2 \int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(c e-d f)^2}-\frac {c \int -\frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {c \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx}{(c e-d f)^2}-\frac {c \int -\frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}-\frac {f \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {\int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \int \frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {\int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \int \frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \int \frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}-\frac {f \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {\int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) \int \frac {\sqrt {b x^2+a}}{\left (c x^2+d\right )^{3/2}}dx}{a c-b d}-\frac {(b c e+a c f-2 b d f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d}}dx}{a c-b d}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {\int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) \int \frac {\sqrt {b x^2+a}}{\left (c x^2+d\right )^{3/2}}dx}{a c-b d}-\frac {(b c e+a c f-2 b d f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d}}dx}{a c-b d}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \int \frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}-\frac {f \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {\int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {\sqrt {c} (c e-d f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {(b c e+a c f-2 b d f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d}}dx}{a c-b d}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {\int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {\sqrt {c} (c e-d f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {(b c e+a c f-2 b d f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d}}dx}{a c-b d}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \int \frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}-\frac {f \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {\int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {\sqrt {c} (c e-d f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {\sqrt {d} (b c e+a c f-2 b d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {\int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {\sqrt {c} (c e-d f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {\sqrt {d} (b c e+a c f-2 b d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \int \frac {-c f x^2+c e-2 d f}{\sqrt {b x^2+a} \left (c x^2+d\right )^{5/2}}dx}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}-\frac {f \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {\int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {\sqrt {c} (c e-d f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {\sqrt {d} (b c e+a c f-2 b d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {c \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c x^2}{d}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a c}{b d}\right ) f^2}{\sqrt {b} e (c e-d f)^2 \sqrt {b x^2+a} \sqrt {c x^2+d}}+\frac {c \left (\frac {c (c e-d f) \sqrt {b x^2+a} x}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}+\frac {-\frac {\sqrt {c} (b d (4 c e-7 d f)-a c (2 c e-5 d f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {b \sqrt {d} (a c (c e-4 d f)-3 b d (c e-2 d f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \left (\frac {c \left (\frac {\int \frac {\sqrt {c x^2+d}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {\sqrt {c} (c e-d f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b d}{a c}\right )}{\sqrt {d} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}-\frac {\sqrt {d} (b c e+a c f-2 b d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b d}{a c}\right )}{a \sqrt {c} (a c-b d) \sqrt {\frac {d \left (b x^2+a\right )}{a \left (c x^2+d\right )}} \sqrt {c x^2+d}}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}\right )}{f}-\frac {(b e-a f) \left (\frac {c \left (\frac {c \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )}dx f^2}{(c e-d f)^2}+\frac {c \left (\frac {c (c e-d f) x \sqrt {b x^2+a}}{3 d (a c-b d) \left (c x^2+d\right )^{3/2}}-\frac {\int -\frac {b c (c e-d f) x^2+a c (2 c e-5 d f)-3 b d (c e-2 d f)}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2}}dx}{3 d (a c-b d)}\right )}{(c e-d f)^2}\right )}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}\right )}{c e-d f}-\frac {f \left (\frac {c \int \frac {1}{\sqrt {b x^2+a} \left (c x^2+d\right )^{3/2} \left (f x^2+e\right )^2}dx}{c e-d f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {c x^2+d} \left (f x^2+e\right )^3}dx}{c e-d f}\right )}{c e-d f}\right )}{f}\right )}{f}\) |
Input:
Int[(a + b*x^2)^(3/2)/((d + c*x^2)^(5/2)*(e + f*x^2)^3),x]
Output:
$Aborted
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[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[(((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. \(2690\) vs. \(2(747)=1494\).
Time = 21.80 (sec) , antiderivative size = 2691, normalized size of antiderivative = 3.43
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2691\) |
| default | \(\text {Expression too large to display}\) | \(8333\) |
Input:
int((b*x^2+a)^(3/2)/(c*x^2+d)^(5/2)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(c*x^2+d))^(1/2)/(b*x^2+a)^(1/2)/(c*x^2+d)^(1/2)*(1/3*(a*c-b*d) /(c*e-d*f)^3/d*x*(b*c*x^4+a*c*x^2+b*d*x^2+a*d)^(1/2)/(x^2+1/c*d)^2+3/2/(-b /a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+c/d*x^2)^(1/2)/(b*c*x^4+a*c*x^2+b*d*x^2+a*d )^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*c+b*d)/d/b)^(1/2))*b*c^2/(c*e-d*f) ^4*a*f+5/8/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+c/d*x^2)^(1/2)/(b*c*x^4+a*c*x ^2+b*d*x^2+a*d)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*c+b*d)/d/b)^(1/2))*b ^2*c/(c*e-d*f)^4*d*f+1/3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+c/d*x^2)^(1/2)/ (b*c*x^4+a*c*x^2+b*d*x^2+a*d)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*c+b*d) /d/b)^(1/2))*b*c^2/(c*e-d*f)^3/d*a+1/3*(b*c*x^2+a*c)*c*(2*a*c^2*e-11*a*c*d *f+2*b*c*d*e+7*b*d^2*f)/(c*e-d*f)^4/d^2*x/((x^2+1/c*d)*(b*c*x^2+a*c))^(1/2 )-5/4/(c*e-d*f)^4/e*f^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+c/d*x^2)^(1/2)/( b*c*x^4+a*c*x^2+b*d*x^2+a*d)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-c/d )^(1/2)/(-b/a)^(1/2))*a*b*c*d-7/8/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+c/d*x^ 2)^(1/2)/(b*c*x^4+a*c*x^2+b*d*x^2+a*d)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+ (a*c+b*d)/d/b)^(1/2))*b^2*c^2/(c*e-d*f)^4*e+15/8/(c*e-d*f)^4*e/(-b/a)^(1/2 )*(1+b*x^2/a)^(1/2)*(1+c/d*x^2)^(1/2)/(b*c*x^4+a*c*x^2+b*d*x^2+a*d)^(1/2)* EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-c/d)^(1/2)/(-b/a)^(1/2))*b^2*c^2-15/8/ (-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+c/d*x^2)^(1/2)/(b*c*x^4+a*c*x^2+b*d*x^2+ a*d)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*c+b*d)/d/b)^(1/2))*b*c/(c*e-d*f )^4/e*a*d*f^2+7/4*d/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+c/d*x^2)^(1/2)/(b...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (d+c x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/2)/(c*x^2+d)^(5/2)/(f*x^2+e)^3,x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (d+c x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(3/2)/(c*x**2+d)**(5/2)/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (d+c x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (c x^{2} + d\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(c*x^2+d)^(5/2)/(f*x^2+e)^3,x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(3/2)/((c*x^2 + d)^(5/2)*(f*x^2 + e)^3), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (d+c x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (c x^{2} + d\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(c*x^2+d)^(5/2)/(f*x^2+e)^3,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)/((c*x^2 + d)^(5/2)*(f*x^2 + e)^3), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (d+c x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (c\,x^2+d\right )}^{5/2}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int((a + b*x^2)^(3/2)/((d + c*x^2)^(5/2)*(e + f*x^2)^3),x)
Output:
int((a + b*x^2)^(3/2)/((d + c*x^2)^(5/2)*(e + f*x^2)^3), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (d+c x^2\right )^{5/2} \left (e+f x^2\right )^3} \, dx=\left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {c \,x^{2}+d}\, x^{2}}{c^{3} f^{3} x^{12}+3 c^{3} e \,f^{2} x^{10}+3 c^{2} d \,f^{3} x^{10}+3 c^{3} e^{2} f \,x^{8}+9 c^{2} d e \,f^{2} x^{8}+3 c \,d^{2} f^{3} x^{8}+c^{3} e^{3} x^{6}+9 c^{2} d \,e^{2} f \,x^{6}+9 c \,d^{2} e \,f^{2} x^{6}+d^{3} f^{3} x^{6}+3 c^{2} d \,e^{3} x^{4}+9 c \,d^{2} e^{2} f \,x^{4}+3 d^{3} e \,f^{2} x^{4}+3 c \,d^{2} e^{3} x^{2}+3 d^{3} e^{2} f \,x^{2}+d^{3} e^{3}}d x \right ) b +\left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {c \,x^{2}+d}}{c^{3} f^{3} x^{12}+3 c^{3} e \,f^{2} x^{10}+3 c^{2} d \,f^{3} x^{10}+3 c^{3} e^{2} f \,x^{8}+9 c^{2} d e \,f^{2} x^{8}+3 c \,d^{2} f^{3} x^{8}+c^{3} e^{3} x^{6}+9 c^{2} d \,e^{2} f \,x^{6}+9 c \,d^{2} e \,f^{2} x^{6}+d^{3} f^{3} x^{6}+3 c^{2} d \,e^{3} x^{4}+9 c \,d^{2} e^{2} f \,x^{4}+3 d^{3} e \,f^{2} x^{4}+3 c \,d^{2} e^{3} x^{2}+3 d^{3} e^{2} f \,x^{2}+d^{3} e^{3}}d x \right ) a \] Input:
int((b*x^2+a)^(3/2)/(c*x^2+d)^(5/2)/(f*x^2+e)^3,x)
Output:
int((sqrt(a + b*x**2)*sqrt(c*x**2 + d)*x**2)/(c**3*e**3*x**6 + 3*c**3*e**2 *f*x**8 + 3*c**3*e*f**2*x**10 + c**3*f**3*x**12 + 3*c**2*d*e**3*x**4 + 9*c **2*d*e**2*f*x**6 + 9*c**2*d*e*f**2*x**8 + 3*c**2*d*f**3*x**10 + 3*c*d**2* e**3*x**2 + 9*c*d**2*e**2*f*x**4 + 9*c*d**2*e*f**2*x**6 + 3*c*d**2*f**3*x* *8 + d**3*e**3 + 3*d**3*e**2*f*x**2 + 3*d**3*e*f**2*x**4 + d**3*f**3*x**6) ,x)*b + int((sqrt(a + b*x**2)*sqrt(c*x**2 + d))/(c**3*e**3*x**6 + 3*c**3*e **2*f*x**8 + 3*c**3*e*f**2*x**10 + c**3*f**3*x**12 + 3*c**2*d*e**3*x**4 + 9*c**2*d*e**2*f*x**6 + 9*c**2*d*e*f**2*x**8 + 3*c**2*d*f**3*x**10 + 3*c*d* *2*e**3*x**2 + 9*c*d**2*e**2*f*x**4 + 9*c*d**2*e*f**2*x**6 + 3*c*d**2*f**3 *x**8 + d**3*e**3 + 3*d**3*e**2*f*x**2 + 3*d**3*e*f**2*x**4 + d**3*f**3*x* *6),x)*a