Integrand size = 32, antiderivative size = 593 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\frac {d (2 d e+3 c f) x \sqrt {a+b x^2}}{6 c e (d e-c f)^2 \left (c+d x^2\right )^{3/2}}-\frac {f x \sqrt {a+b x^2}}{2 e (d e-c f) \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}-\frac {\sqrt {d} \left (a d \left (4 d^2 e^2-16 c d e f-3 c^2 f^2\right )-b c \left (2 d^2 e^2-14 c d e f-3 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{6 c^{3/2} (b c-a d) e (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {d} \left (15 b^2 c^3 e f^2+3 a^2 c d f^2 (6 d e-c f)-a b \left (2 d^3 e^3-4 c d^2 e^2 f+35 c^2 d e f^2-3 c^3 f^3\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{6 a \sqrt {c} (b c-a d) e (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} f^2 \left (5 b d e^2-a f (6 d e-c f)\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{2 a \sqrt {d} e^2 (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/6*d*(3*c*f+2*d*e)*x*(b*x^2+a)^(1/2)/c/e/(-c*f+d*e)^2/(d*x^2+c)^(3/2)-1/2 *f*x*(b*x^2+a)^(1/2)/e/(-c*f+d*e)/(d*x^2+c)^(3/2)/(f*x^2+e)-1/6*d^(1/2)*(a *d*(-3*c^2*f^2-16*c*d*e*f+4*d^2*e^2)-b*c*(-3*c^2*f^2-14*c*d*e*f+2*d^2*e^2) )*(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)/(-a*d+b*c)/e/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2) /(d*x^2+c)^(1/2)-1/6*d^(1/2)*(15*b^2*c^3*e*f^2+3*a^2*c*d*f^2*(-c*f+6*d*e)- a*b*(-3*c^3*f^3+35*c^2*d*e*f^2-4*c*d^2*e^2*f+2*d^3*e^3))*(b*x^2+a)^(1/2)*I nverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/c^(1/2)/(-a* d+b*c)/e/(-c*f+d*e)^4/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/2* c^(3/2)*f^2*(5*b*d*e^2-a*f*(-c*f+6*d*e))*(b*x^2+a)^(1/2)*EllipticPi(d^(1/2 )*x/c^(1/2)/(1+d*x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d^(1/2)/e^2/( -c*f+d*e)^4/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 6.46 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\frac {-\sqrt {\frac {b}{a}} e x \left (a+b x^2\right ) \left (3 c^2 (b c-a d) f^3 \left (c+d x^2\right )^2+2 c d^2 (b c-a d) e (-d e+c f) \left (e+f x^2\right )+2 d^2 e (2 a d (d e-4 c f)+b c (-d e+7 c f)) \left (c+d x^2\right ) \left (e+f x^2\right )\right )-i c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \left (b e \left (a d \left (4 d^2 e^2-16 c d e f-3 c^2 f^2\right )+b c \left (-2 d^2 e^2+14 c d e f+3 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (-b e \left (2 d^2 e^2+c d e f-3 c^2 f^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+3 c f \left (5 b d e^2+a f (-6 d e+c f)\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{6 \sqrt {\frac {b}{a}} c^2 (b c-a d) e^2 (d e-c f)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \] Input:
Integrate[Sqrt[a + b*x^2]/((c + d*x^2)^(5/2)*(e + f*x^2)^2),x]
Output:
(-(Sqrt[b/a]*e*x*(a + b*x^2)*(3*c^2*(b*c - a*d)*f^3*(c + d*x^2)^2 + 2*c*d^ 2*(b*c - a*d)*e*(-(d*e) + c*f)*(e + f*x^2) + 2*d^2*e*(2*a*d*(d*e - 4*c*f) + b*c*(-(d*e) + 7*c*f))*(c + d*x^2)*(e + f*x^2))) - I*c*Sqrt[1 + (b*x^2)/a ]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*(b*e*(a*d*(4*d^2*e^2 - 16*c* d*e*f - 3*c^2*f^2) + b*c*(-2*d^2*e^2 + 14*c*d*e*f + 3*c^2*f^2))*EllipticE[ I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (b*c - a*d)*(-(b*e*(2*d^2*e^2 + c*d *e*f - 3*c^2*f^2)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]) + 3*c*f* (5*b*d*e^2 + a*f*(-6*d*e + c*f))*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/ a]*x], (a*d)/(b*c)])))/(6*Sqrt[b/a]*c^2*(b*c - a*d)*e^2*(d*e - c*f)^3*Sqrt [a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (-\frac {\int \frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (-\frac {\frac {d (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {d \left (-\frac {\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (-\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (d e-4 c f)-3 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {f^2 \sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{\sqrt {a+b x^2} (d e-c f)^2}+\frac {d \left (-\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (d e-4 c f)-3 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx}{\sqrt {a+b x^2} \sqrt {c+d x^2} (d e-c f)^2}+\frac {d \left (-\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (d e-4 c f)-3 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (d e-c f)^2}+\frac {d \left (-\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (d e-4 c f)-3 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (d e-c f)^2}+\frac {d \left (-\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (d e-4 c f)-3 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (d e-c f)^2}+\frac {d \left (-\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (d e-4 c f)-3 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (d e-c f)^2}+\frac {d \left (-\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (d e-4 c f)-3 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (d e-c f)^2}+\frac {d \left (-\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (d e-4 c f)-3 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (d e-c f)^2}+\frac {d \left (-\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (d e-4 c f)-3 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (-\frac {\int \frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} f^2 \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2} (d e-c f)^2}+\frac {d \left (-\frac {\frac {b \sqrt {c} \sqrt {a+b x^2} (a d (d e-4 c f)-3 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (-\frac {\frac {d (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\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 )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d \left (\frac {\int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d \left (\frac {\int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(b d e-2 b c f+a d f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {d (d e-c f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d \left (\frac {\int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(b d e-2 b c f+a d f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d \left (\frac {\int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d \left (\frac {c^{3/2} \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}\right )}{f}\) |
| \(\Big \downarrow \) 424 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {d \left (\frac {c^{3/2} \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}-\frac {b d \int \frac {f x^2+e}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{2 e (b e-a f) (d e-c f)}\right )}{d e-c f}\right )}{d e-c f}\right )}{f}\) |
Input:
Int[Sqrt[a + b*x^2]/((c + d*x^2)^(5/2)*(e + f*x^2)^2),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[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ [d/c]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2 Int[(c + d*x^2)^(q + 2)*((e + f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2 Int[(c + d*x^2)^q*( e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} , x] && LtQ[q, -1]
Int[1/(((a_) + (b_.)*(x_)^2)^2*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)* (x_)^2]), x_Symbol] :> Simp[b^2*x*Sqrt[c + d*x^2]*(Sqrt[e + f*x^2]/(2*a*(b* c - a*d)*(b*e - a*f)*(a + b*x^2))), x] + (Simp[(b^2*c*e + 3*a^2*d*f - 2*a*b *(d*e + c*f))/(2*a*(b*c - a*d)*(b*e - a*f)) Int[1/((a + b*x^2)*Sqrt[c + d *x^2]*Sqrt[e + f*x^2]), x], x] - Simp[d*(f/(2*a*(b*c - a*d)*(b*e - a*f))) Int[(a + b*x^2)/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[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. \(2932\) vs. \(2(561)=1122\).
Time = 18.63 (sec) , antiderivative size = 2933, normalized size of antiderivative = 4.95
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2933\) |
| default | \(\text {Expression too large to display}\) | \(5111\) |
Input:
int((b*x^2+a)^(1/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/2*f^3*d/e/( c^2*f^2-2*c*d*e*f+d^2*e^2)/(c*f-d*e)*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2) /(d*f*x^2+d*e)+1/3/(c*f-d*e)^2/c*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^ 2+c/d)^2+1/3*(b*d*x^2+a*d)/c^2*d/(a*d-b*c)*x*(8*a*c*d*f-2*a*d^2*e-7*b*c^2* f+b*c*d*e)/(c*f-d*e)/(c^2*f^2-2*c*d*e*f+d^2*e^2)/((x^2+c/d)*(b*d*x^2+a*d)) ^(1/2)+8/3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x ^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*d ^2/c/(c*f-d*e)/(c^2*f^2-2*c*d*e*f+d^2*e^2)*a*f-2/3/(-b/a)^(1/2)*(1+b*x^2/a )^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x* (-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*d^3/c^2/(c*f-d*e)/(c^2*f^2-2*c*d*e* f+d^2*e^2)*a*e+1/3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x ^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^ (1/2))*d^2/c/(c*f-d*e)/(c^2*f^2-2*c*d*e*f+d^2*e^2)*b*e+1/2/e^2/(c^2*f^2-2* c*d*e*f+d^2*e^2)/(c*f-d*e)*f^3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^ (1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/ e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*c-3/(c^2*f^2-2*c*d*e*f+d^2*e^2)/(c*f-d*e) *f^2/e/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b *c*x^2+a*c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^ (1/2))*a*d-17/6/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+ a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^...
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^2,x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {a + b x^{2}}}{\left (c + d x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)/(d*x**2+c)**(5/2)/(f*x**2+e)**2,x)
Output:
Integral(sqrt(a + b*x**2)/((c + d*x**2)**(5/2)*(e + f*x**2)**2), x)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^2,x, algorithm="maxima ")
Output:
integrate(sqrt(b*x^2 + a)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^2), x)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^2,x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^2), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b\,x^2+a}}{{\left (d\,x^2+c\right )}^{5/2}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int((a + b*x^2)^(1/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^2),x)
Output:
int((a + b*x^2)^(1/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^2), x)
\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d^{3} f^{2} x^{10}+3 c \,d^{2} f^{2} x^{8}+2 d^{3} e f \,x^{8}+3 c^{2} d \,f^{2} x^{6}+6 c \,d^{2} e f \,x^{6}+d^{3} e^{2} x^{6}+c^{3} f^{2} x^{4}+6 c^{2} d e f \,x^{4}+3 c \,d^{2} e^{2} x^{4}+2 c^{3} e f \,x^{2}+3 c^{2} d \,e^{2} x^{2}+c^{3} e^{2}}d x \] Input:
int((b*x^2+a)^(1/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^2,x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c**3*e**2 + 2*c**3*e*f*x**2 + c** 3*f**2*x**4 + 3*c**2*d*e**2*x**2 + 6*c**2*d*e*f*x**4 + 3*c**2*d*f**2*x**6 + 3*c*d**2*e**2*x**4 + 6*c*d**2*e*f*x**6 + 3*c*d**2*f**2*x**8 + d**3*e**2* x**6 + 2*d**3*e*f*x**8 + d**3*f**2*x**10),x)