Integrand size = 32, antiderivative size = 677 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\frac {(b c-a d)^2 (2 d e+3 c f) x \sqrt {a+b x^2}}{6 c d e (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {b^2 x \sqrt {a+b x^2}}{2 d e (d e-c f) \sqrt {c+d x^2}}-\frac {f x \left (a+b x^2\right )^{5/2}}{2 e (d e-c f) \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}-\frac {\left (b^2 c^2 e (13 d e+2 c f)-6 a b c d e (d e+4 c f)-a^2 d \left (4 d^2 e^2-16 c d e f-3 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{6 c^{3/2} \sqrt {d} e (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (3 a^3 c d f^2 (6 d e-c f)-3 b^3 c^2 e^2 (d e+4 c f)+a b^2 c e \left (8 d^2 e^2+23 c d e f+14 c^2 f^2\right )-a^2 b d e \left (2 d^2 e^2+20 c d e f+23 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{6 a \sqrt {c} \sqrt {d} e (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} (b e-a f)^2 (a f (6 d e-c f)-b e (d e+4 c f)) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{2 a \sqrt {d} e^2 (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/6*(-a*d+b*c)^2*(3*c*f+2*d*e)*x*(b*x^2+a)^(1/2)/c/d/e/(-c*f+d*e)^2/(d*x^2 +c)^(3/2)+1/2*b^2*x*(b*x^2+a)^(1/2)/d/e/(-c*f+d*e)/(d*x^2+c)^(1/2)-1/2*f*x *(b*x^2+a)^(5/2)/e/(-c*f+d*e)/(d*x^2+c)^(3/2)/(f*x^2+e)-1/6*(b^2*c^2*e*(2* c*f+13*d*e)-6*a*b*c*d*e*(4*c*f+d*e)-a^2*d*(-3*c^2*f^2-16*c*d*e*f+4*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)/d^(1/2)/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*(3*a^3*c*d*f^2*(-c*f+6*d*e)-3*b^3*c^2*e^2*(4*c*f+d*e)+a *b^2*c*e*(14*c^2*f^2+23*c*d*e*f+8*d^2*e^2)-a^2*b*d*e*(23*c^2*f^2+20*c*d*e* f+2*d^2*e^2))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1 -b*c/a/d)^(1/2))/a/c^(1/2)/d^(1/2)/e/(-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)*(-a*f+b*e)^2*(a*f*(-c*f+6*d*e)-b*e*(4* c*f+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 = 9.10 (sec) , antiderivative size = 481, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\frac {-\sqrt {\frac {b}{a}} d e f x \left (a+b x^2\right ) \left (3 c^2 f (b e-a f)^2 \left (c+d x^2\right )^2+2 c (b c-a d)^2 e (-d e+c f) \left (e+f x^2\right )+2 (b c-a d) e (2 a d (d e-4 c f)+b c (5 d e+c f)) \left (c+d x^2\right ) \left (e+f x^2\right )\right )+i c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \left (-b e f \left (b^2 c^2 e (13 d e+2 c f)-6 a b c d e (d e+4 c f)+a^2 d \left (-4 d^2 e^2+16 c d e f+3 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+b e (-d e+c f) \left (-10 a b c d e f+b^2 c e (3 d e+2 c f)+a^2 d f (2 d e+3 c f)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+3 c d (b e-a f)^2 (a f (-6 d e+c f)+b e (d e+4 c f)) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{6 \sqrt {\frac {b}{a}} c^2 d e^2 f (d e-c f)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \] Input:
Integrate[(a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^2),x]
Output:
(-(Sqrt[b/a]*d*e*f*x*(a + b*x^2)*(3*c^2*f*(b*e - a*f)^2*(c + d*x^2)^2 + 2* c*(b*c - a*d)^2*e*(-(d*e) + c*f)*(e + f*x^2) + 2*(b*c - a*d)*e*(2*a*d*(d*e - 4*c*f) + b*c*(5*d*e + 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*f*(b^2*c^2*e*( 13*d*e + 2*c*f) - 6*a*b*c*d*e*(d*e + 4*c*f) + a^2*d*(-4*d^2*e^2 + 16*c*d*e *f + 3*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]) + b*e*(-( d*e) + c*f)*(-10*a*b*c*d*e*f + b^2*c*e*(3*d*e + 2*c*f) + a^2*d*f*(2*d*e + 3*c*f))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + 3*c*d*(b*e - a*f) ^2*(a*f*(-6*d*e + c*f) + b*e*(d*e + 4*c*f))*EllipticPi[(a*f)/(b*e), I*ArcS inh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(6*Sqrt[b/a]*c^2*d*e^2*f*(d*e - c*f)^3*Sq rt[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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 419 |
| \(\displaystyle \frac {b \left (-\frac {\int -\frac {\sqrt {b x^2+a} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\int \frac {\sqrt {b x^2+a} \left (b f c^2+d^2 (b e-a f) x^2+a d (d e-2 c f)\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {b \left (\frac {-\frac {\int -\frac {d \left (b (a d (d e-4 c f)+b c (2 d e+c f)) x^2+a (a d (2 d e-5 c f)+b c (d e+2 c f))\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {\int \frac {d \left (b (a d (d e-4 c f)+b c (2 d e+c f)) x^2+a (a d (2 d e-5 c f)+b c (d e+2 c f))\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {\frac {\int \frac {b (a d (d e-4 c f)+b c (2 d e+c f)) x^2+a (a d (2 d e-5 c f)+b c (d e+2 c f))}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {\frac {(a d (2 d e-5 c f)+b c (c f+2 d e)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx-a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {f (b e-a f) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}-\frac {\int -\frac {d \left (b (d e-4 c f) x^2+a (2 d e-5 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int \frac {d \left (b (d e-4 c f) x^2+a (2 d e-5 c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int \frac {b (d e-4 c f) x^2+a (2 d e-5 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\frac {a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)+b c (c f+2 d e)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}-\frac {x \sqrt {a+b x^2} (b c-a d) (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f \sqrt {c+d x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {\frac {a b (d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {a+b x^2} (a d (2 d e-5 c f)-b c (d e-4 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c}+\frac {x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2}}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d (d e-c f) x \sqrt {b x^2+a}}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\int \frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {d (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2 \sqrt {b x^2+a}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {\frac {\frac {(a d (2 d e-5 c f)+b c (2 d e+c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}-\frac {(b c-a d) (d e-c f) x \sqrt {b x^2+a}}{3 c \left (d x^2+c\right )^{3/2}}}{(d e-c f)^2}-\frac {a^{3/2} f (b e-a f) \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {d \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{3 c \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \sqrt {c} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(a d (2 d e-5 c f)-b c (d e-4 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c f)) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
Input:
Int[(a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^2),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ [d/c]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b*((b*e - a*f)/(b*c - a*d)^2) Int[(c + d*x^2)^( q + 2)*((e + f*x^2)^(r - 1)/(a + b*x^2)), x], x] - Simp[1/(b*c - a*d)^2 I nt[(c + d*x^2)^q*(e + f*x^2)^(r - 1)*(2*b*c*d*e - a*d^2*e - b*c^2*f + d^2*( b*e - a*f)*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[q, -1] && Gt Q[r, 1]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2 Int[(c + d*x^2)^(q + 2)*((e + f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2 Int[(c + d*x^2)^q*( e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} , x] && LtQ[q, -1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[d/b Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(a + b*x^2)^p*(c + d*x ^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt Q[p, 0] && GtQ[q, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(3433\) vs. \(2(639)=1278\).
Time = 18.74 (sec) , antiderivative size = 3434, normalized size of antiderivative = 5.07
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(3434\) |
| default | \(\text {Expression too large to display}\) | \(6707\) |
Input:
int((b*x^2+a)^(5/2)/(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)*(f^2/(c^2*f^2- 2*c*d*e*f+d^2*e^2)/(c*f-d*e)/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^2*b*c+4*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)* (1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*b^2/(c*f-d*e)/(c^2*f ^2-2*c*d*e*f+d^2*e^2)*a*f*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(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)*d*b/(c*f-d*e)/(c^2*f^2-2*c*d*e*f+d^2*e^2)*a^2*f*Elliptic E(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-2/3/(-b/a)^(1/2)*(1+b*x^2/a)^(1 /2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/ a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*b^2/(c^2*f^2-2*c*d*e*f+d^2*e^2)*a+1/3*( a^2*d^2-2*a*b*c*d+b^2*c^2)/(c*f-d*e)^2/c/d^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a* c)^(1/2)/(x^2+c/d)^2+1/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b *d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*d*b^2/(c*f-d*e)/(c^2*f^2-2*c*d*e*f+d^2*e ^2)*a*e*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+2/3/c/(-b/a)^(1 /2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2 )*d^2*b/(c*f-d*e)/(c^2*f^2-2*c*d*e*f+d^2*e^2)*a^2*e*EllipticE(x*(-b/a)^(1/ 2),(-1+(a*d+b*c)/c/b)^(1/2))+1/3*(b*d*x^2+a*d)*(8*a^2*c*d^2*f-2*a^2*d^3*e- 9*a*b*c^2*d*f-3*a*b*c*d^2*e+b^2*c^3*f+5*b^2*c^2*d*e)/d/(c*f-d*e)/(c^2*f^2- 2*c*d*e*f+d^2*e^2)/c^2*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+1/3/(-b/a)^(1/...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^2,x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(5/2)/(d*x**2+c)**(5/2)/(f*x**2+e)**2,x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^2,x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^2), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^2,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(5/2)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^{5/2}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^2),x)
Output:
int((a + b*x^2)^(5/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^2), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^2,x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*x - 20*int((sqrt(c + d*x**2 )*sqrt(a + b*x**2)*x**6)/(4*a**2*c**3*e**2*f + 8*a**2*c**3*e*f**2*x**2 + 4 *a**2*c**3*f**3*x**4 + 12*a**2*c**2*d*e**2*f*x**2 + 24*a**2*c**2*d*e*f**2* x**4 + 12*a**2*c**2*d*f**3*x**6 + 12*a**2*c*d**2*e**2*f*x**4 + 24*a**2*c*d **2*e*f**2*x**6 + 12*a**2*c*d**2*f**3*x**8 + 4*a**2*d**3*e**2*f*x**6 + 8*a **2*d**3*e*f**2*x**8 + 4*a**2*d**3*f**3*x**10 + a*b*c**3*e**3 + 6*a*b*c**3 *e**2*f*x**2 + 9*a*b*c**3*e*f**2*x**4 + 4*a*b*c**3*f**3*x**6 + 3*a*b*c**2* d*e**3*x**2 + 18*a*b*c**2*d*e**2*f*x**4 + 27*a*b*c**2*d*e*f**2*x**6 + 12*a *b*c**2*d*f**3*x**8 + 3*a*b*c*d**2*e**3*x**4 + 18*a*b*c*d**2*e**2*f*x**6 + 27*a*b*c*d**2*e*f**2*x**8 + 12*a*b*c*d**2*f**3*x**10 + a*b*d**3*e**3*x**6 + 6*a*b*d**3*e**2*f*x**8 + 9*a*b*d**3*e*f**2*x**10 + 4*a*b*d**3*f**3*x**1 2 + b**2*c**3*e**3*x**2 + 2*b**2*c**3*e**2*f*x**4 + b**2*c**3*e*f**2*x**6 + 3*b**2*c**2*d*e**3*x**4 + 6*b**2*c**2*d*e**2*f*x**6 + 3*b**2*c**2*d*e*f* *2*x**8 + 3*b**2*c*d**2*e**3*x**6 + 6*b**2*c*d**2*e**2*f*x**8 + 3*b**2*c*d **2*e*f**2*x**10 + b**2*d**3*e**3*x**8 + 2*b**2*d**3*e**2*f*x**10 + b**2*d **3*e*f**2*x**12),x)*a**2*b**3*c**2*d*e*f**2 - 20*int((sqrt(c + d*x**2)*sq rt(a + b*x**2)*x**6)/(4*a**2*c**3*e**2*f + 8*a**2*c**3*e*f**2*x**2 + 4*a** 2*c**3*f**3*x**4 + 12*a**2*c**2*d*e**2*f*x**2 + 24*a**2*c**2*d*e*f**2*x**4 + 12*a**2*c**2*d*f**3*x**6 + 12*a**2*c*d**2*e**2*f*x**4 + 24*a**2*c*d**2* e*f**2*x**6 + 12*a**2*c*d**2*f**3*x**8 + 4*a**2*d**3*e**2*f*x**6 + 8*a*...