Integrand size = 32, antiderivative size = 719 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=-\frac {b \left (2 a b d e f (85 d e-62 c f)-a^2 d f^2 (61 d e-15 c f)-b^2 e \left (105 d^2 e^2-95 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{30 d e f^4 \sqrt {a+b x^2}}-\frac {b (10 b d e-6 b c f-11 a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 f^3}+\frac {b^2 d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 f^2}-\frac {(b e-a f)^2 (d e-c f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e f^3 \left (e+f x^2\right )}+\frac {\sqrt {a} \sqrt {b} \left (2 a b d e f (85 d e-62 c f)-a^2 d f^2 (61 d e-15 c f)-b^2 e \left (105 d^2 e^2-95 c d e f+6 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{30 d e f^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (30 a^2 d^2 e f^2-a b f \left (135 d^2 e^2-128 c d e f+15 c^2 f^2\right )+b^2 e \left (105 d^2 e^2-130 c d e f+33 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{30 \sqrt {b} c e f^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (b e-a f) (d e-c f) (b e (7 d e-4 c f)-a f (2 d e+c f)) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c e^2 f^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/30*b*(2*a*b*d*e*f*(-62*c*f+85*d*e)-a^2*d*f^2*(-15*c*f+61*d*e)-b^2*e*(6* c^2*f^2-95*c*d*e*f+105*d^2*e^2))*x*(d*x^2+c)^(1/2)/d/e/f^4/(b*x^2+a)^(1/2) -1/15*b*(-11*a*d*f-6*b*c*f+10*b*d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^3 +1/5*b^2*d*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^2-1/2*(-a*f+b*e)^2*(-c*f+ d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/f^3/(f*x^2+e)+1/30*a^(1/2)*b^(1/2 )*(2*a*b*d*e*f*(-62*c*f+85*d*e)-a^2*d*f^2*(-15*c*f+61*d*e)-b^2*e*(6*c^2*f^ 2-95*c*d*e*f+105*d^2*e^2))*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+ b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/d/e/f^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/( b*x^2+a))^(1/2)+1/30*a^(3/2)*(30*a^2*d^2*e*f^2-a*b*f*(15*c^2*f^2-128*c*d*e *f+135*d^2*e^2)+b^2*e*(33*c^2*f^2-130*c*d*e*f+105*d^2*e^2))*(d*x^2+c)^(1/2 )*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/e /f^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/2*a^(3/2)*(-a*f+b*e )*(-c*f+d*e)*(b*e*(-4*c*f+7*d*e)-a*f*(c*f+2*d*e))*(d*x^2+c)^(1/2)*Elliptic Pi(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),1-a*f/b/e,(1-a*d/b/c)^(1/2))/b^(1/2 )/c/e^2/f^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 11.64 (sec) , antiderivative size = 600, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\frac {i b c e f \left (2 a b d e f (85 d e-62 c f)+a^2 d f^2 (-61 d e+15 c f)+b^2 e \left (-105 d^2 e^2+95 c d e f-6 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i e \left (30 a^3 d^3 e f^3+2 a b^2 d e f \left (120 d^2 e^2-70 c d e f-23 c^2 f^2\right )+a^2 b d f^2 \left (-165 d^2 e^2+82 c d e f+15 c^2 f^2\right )+b^3 e \left (-105 d^3 e^3+60 c d^2 e^2 f+35 c^2 d e f^2-6 c^3 f^3\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d \left (\sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (15 a^2 f^2 (-d e+c f)+2 a b e f \left (26 d e-15 c f+11 d f x^2\right )+b^2 e \left (3 c f \left (9 e+4 f x^2\right )+d \left (-35 e^2-14 e f x^2+6 f^2 x^4\right )\right )\right )-15 i (b e-a f)^2 (-d e+c f) (a f (2 d e+c f)+b e (-7 d e+4 c f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{30 \sqrt {\frac {b}{a}} d e^2 f^5 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2))/(e + f*x^2)^2,x]
Output:
(I*b*c*e*f*(2*a*b*d*e*f*(85*d*e - 62*c*f) + a^2*d*f^2*(-61*d*e + 15*c*f) + b^2*e*(-105*d^2*e^2 + 95*c*d*e*f - 6*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*e*(30*a^3*d^3*e*f^3 + 2*a*b^2*d*e*f*(120*d^2*e^2 - 70*c*d*e*f - 23*c^2* f^2) + a^2*b*d*f^2*(-165*d^2*e^2 + 82*c*d*e*f + 15*c^2*f^2) + b^3*e*(-105* d^3*e^3 + 60*c*d^2*e^2*f + 35*c^2*d*e*f^2 - 6*c^3*f^3))*Sqrt[1 + (b*x^2)/a ]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/ (b*c)] + d*(Sqrt[b/a]*e*f^2*x*(a + b*x^2)*(c + d*x^2)*(15*a^2*f^2*(-(d*e) + c*f) + 2*a*b*e*f*(26*d*e - 15*c*f + 11*d*f*x^2) + b^2*e*(3*c*f*(9*e + 4* f*x^2) + d*(-35*e^2 - 14*e*f*x^2 + 6*f^2*x^4))) - (15*I)*(b*e - a*f)^2*(-( d*e) + c*f)*(a*f*(2*d*e + c*f) + b*e*(-7*d*e + 4*c*f))*Sqrt[1 + (b*x^2)/a] *Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/ a]*x], (a*d)/(b*c)]))/(30*Sqrt[b/a]*d*e^2*f^5*Sqrt[a + b*x^2]*Sqrt[c + d*x ^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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {b \int \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a} \left (2 d (3 b c-a d) x^2+c (5 b c-a d)\right )}{\sqrt {d x^2+c}}dx}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\int \frac {d \left (\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) x^2+a c (9 b c-a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \int \frac {\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) x^2+a c (9 b c-a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (9 b c-a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 418 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}+\frac {d \int -\frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\sqrt {d x^2+c}}dx}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\sqrt {d x^2+c}}dx}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {\int \frac {d \left ((3 b d e-4 b c f-a d f) x^2+a (3 d e-5 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \int \frac {(3 b d e-4 b c f-a d f) x^2+a (3 d e-5 c f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left (a (3 d e-5 c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+(-a d f-4 b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left ((-a d f-4 b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left ((-a d f-4 b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(-a d f-4 b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} \sqrt {c+d x^2} (d e-c f)^2 \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d \left (\frac {1}{3} \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(-a d f-4 b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 418 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\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 (d e-c f)^2}{f^2}+\frac {d \int -\frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\sqrt {d x^2+c}}dx}{f^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\sqrt {d x^2+c}}dx}{f^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {\int \frac {d \left ((3 b d e-4 b c f-a d f) x^2+a (3 d e-5 c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \int \frac {(3 b d e-4 b c f-a d f) x^2+a (3 d e-5 c f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left (a (3 d e-5 c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+(3 b d e-4 b c f-a d f) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left (\frac {\sqrt {c} (3 d e-5 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 b d e-4 b c f-a d f) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left (\frac {\sqrt {c} (3 d e-5 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 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {2}{3} (3 b c-a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {(9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{5 b}\right )}{f}-\frac {(b e-a f) \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {a^{3/2} (d e-c f)^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 )}{\sqrt {b} c e 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 {1}{3} \left ((3 b d e-4 b c f-a d f) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} (3 d e-5 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}}\right )-\frac {1}{3} f x \sqrt {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \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)^(3/2))/(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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
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[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
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)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[(b*c - a*d)^2/b^2 Int[Sqrt[e + f*x^2]/((a + b*x ^2)*Sqrt[c + d*x^2]), x], x] + Simp[d/b^2 Int[(2*b*c - a*d + b*d*x^2)*(Sq rt[e + f*x^2]/Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && P osQ[d/c] && PosQ[f/e]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[d/b Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*x^2 )), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[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. \(1810\) vs. \(2(675)=1350\).
Time = 21.96 (sec) , antiderivative size = 1811, normalized size of antiderivative = 2.52
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1811\) |
| elliptic | \(\text {Expression too large to display}\) | \(3298\) |
| default | \(\text {Expression too large to display}\) | \(4969\) |
Input:
int((b*x^2+a)^(5/2)*(d*x^2+c)^(3/2)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
1/15*b*x*(3*b*d*f*x^2+11*a*d*f+6*b*c*f-10*b*d*e)*(b*x^2+a)^(1/2)*(d*x^2+c) ^(1/2)/f^3+1/15/f^3*((15*a^3*d^2*f^3+79*a^2*b*c*d*f^3-90*a^2*b*d^2*e*f^2+3 9*a*b^2*c^2*f^3-170*a*b^2*c*d*e*f^2+135*a*b^2*d^2*e^2*f-30*b^3*c^2*e*f^2+9 0*b^3*c*d*e^2*f-60*b^3*d^2*e^3)/f^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^ 2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(- 1+(a*d+b*c)/c/b)^(1/2))+15/f^2*(2*a^3*c*d*f^4-2*a^3*d^2*e*f^3+3*a^2*b*c^2* f^4-12*a^2*b*c*d*e*f^3+9*a^2*b*d^2*e^2*f^2-6*a*b^2*c^2*e*f^3+18*a*b^2*c*d* e^2*f^2-12*a*b^2*d^2*e^3*f+3*b^3*c^2*e^2*f^2-8*b^3*c*d*e^3*f+5*b^3*d^2*e^4 )/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))+15*(a^3*c^2*f^5-2*a^3*c*d*e*f^4+a^3*d^2*e^2*f^3-3*a^2*b*c^2*e*f^4+6*a^ 2*b*c*d*e^2*f^3-3*a^2*b*d^2*e^3*f^2+3*a*b^2*c^2*e^2*f^3-6*a*b^2*c*d*e^3*f^ 2+3*a*b^2*d^2*e^4*f-b^3*c^2*e^3*f^2+2*b^3*c*d*e^4*f-b^3*d^2*e^5)/f^2*(1/2* f^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1 /2)/(f*x^2+e)-1/2*d*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/(-b/a)^(1/2)*(1+b* x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*Ellipti cF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+1/2*f*b/(a*c*f^2-a*d*e*f-b*c*e *f+b*d*e^2)/e*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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/ 2))-1/2*f*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e*c/(-b/a)^(1/2)*(1+b*x^2...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(5/2)*(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(5/2)*(d*x**2+c)**(3/2)/(f*x**2+e)**2,x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)*(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)/(f*x^2 + e)^2, x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)*(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)/(f*x^2 + e)^2, x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{3/2}}{{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2))/(e + f*x^2)^2,x)
Output:
int(((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2))/(e + f*x^2)^2, x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{\left (f \,x^{2}+e \right )^{2}}d x \] Input:
int((b*x^2+a)^(5/2)*(d*x^2+c)^(3/2)/(f*x^2+e)^2,x)
Output:
int((b*x^2+a)^(5/2)*(d*x^2+c)^(3/2)/(f*x^2+e)^2,x)