Integrand size = 32, antiderivative size = 962 \[ \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 {\left (30 a^3 d^3 e f^3-a^2 b d f^2 \left (749 d^2 e^2-970 c d e f+105 c^2 f^2\right )+2 a b^2 d e f \left (840 d^2 e^2-1358 c d e f+485 c^2 f^2\right )-b^3 e \left (945 d^3 e^3-1680 c d^2 e^2 f+749 c^2 d e f^2-30 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{210 d e f^5 \sqrt {a+b x^2}}+\frac {\left (45 a^2 d^2 f^2-2 a b d f (77 d e-85 c f)+b^2 \left (105 d^2 e^2-154 c d e f+45 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 f^4}-\frac {b d (14 b d e-15 b c f-15 a d f) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 f^3}+\frac {b^2 d^2 x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 f^2}+\frac {(b e-a f)^2 (d e-c f)^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e f^4 \left (e+f x^2\right )}-\frac {\sqrt {a} \left (30 a^3 d^3 e f^3-a^2 b d f^2 \left (749 d^2 e^2-970 c d e f+105 c^2 f^2\right )+2 a b^2 d e f \left (840 d^2 e^2-1358 c d e f+485 c^2 f^2\right )-b^3 e \left (945 d^3 e^3-1680 c d^2 e^2 f+749 c^2 d e f^2-30 c^3 f^3\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 )}{210 \sqrt {b} d e f^5 \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 (60 a^2 d^2 e f^2 (7 d e-9 c f)+b^2 e \left (945 d^3 e^3-1995 c d^2 e^2 f+1267 c^2 d e f^2-225 c^3 f^3\right )-a b f \left (1365 d^3 e^3-2527 c d^2 e^2 f+1235 c^2 d e f^2-105 c^3 f^3\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 )}{210 \sqrt {b} c e f^5 \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)^2 (b e (9 d e-4 c f)-a f (4 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^5 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/210*(30*a^3*d^3*e*f^3-a^2*b*d*f^2*(105*c^2*f^2-970*c*d*e*f+749*d^2*e^2)+ 2*a*b^2*d*e*f*(485*c^2*f^2-1358*c*d*e*f+840*d^2*e^2)-b^3*e*(-30*c^3*f^3+74 9*c^2*d*e*f^2-1680*c*d^2*e^2*f+945*d^3*e^3))*x*(d*x^2+c)^(1/2)/d/e/f^5/(b* x^2+a)^(1/2)+1/105*(45*a^2*d^2*f^2-2*a*b*d*f*(-85*c*f+77*d*e)+b^2*(45*c^2* f^2-154*c*d*e*f+105*d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^4-1/35*b *d*(-15*a*d*f-15*b*c*f+14*b*d*e)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^3+1 /7*b^2*d^2*x^5*(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)^2*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/f^4/(f*x^2+e)-1/210*a^(1/2)*(30 *a^3*d^3*e*f^3-a^2*b*d*f^2*(105*c^2*f^2-970*c*d*e*f+749*d^2*e^2)+2*a*b^2*d *e*f*(485*c^2*f^2-1358*c*d*e*f+840*d^2*e^2)-b^3*e*(-30*c^3*f^3+749*c^2*d*e *f^2-1680*c*d^2*e^2*f+945*d^3*e^3))*(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))/b^(1/2)/d/e/f^5/(b*x^2+a)^(1/2) /(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/210*a^(3/2)*(60*a^2*d^2*e*f^2*(-9*c*f+7 *d*e)+b^2*e*(-225*c^3*f^3+1267*c^2*d*e*f^2-1995*c*d^2*e^2*f+945*d^3*e^3)-a *b*f*(-105*c^3*f^3+1235*c^2*d*e*f^2-2527*c*d^2*e^2*f+1365*d^3*e^3))*(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^5/(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)^2*(b*e*(-4*c*f+9*d*e)-a*f*(c*f+4*d*e))*(d*x^2+c)^(1/2 )*EllipticPi(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^5/(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 = 13.13 (sec) , antiderivative size = 4275, normalized size of antiderivative = 4.44 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^2} \, dx=\text {Result too large to show} \] Input:
Integrate[((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2))/(e + f*x^2)^2,x]
Output:
(315*a*b^2*Sqrt[b/a]*c*d^3*e^5*f^2*x - 518*a*b^2*Sqrt[b/a]*c^2*d^2*e^4*f^3 *x - 518*a^3*(b/a)^(3/2)*c*d^3*e^4*f^3*x + 195*a*b^2*Sqrt[b/a]*c^3*d*e^3*f ^4*x + 760*a^3*(b/a)^(3/2)*c^2*d^2*e^3*f^4*x + 195*a^3*Sqrt[b/a]*c*d^3*e^3 *f^4*x - 210*a^3*(b/a)^(3/2)*c^3*d*e^2*f^5*x - 210*a^3*Sqrt[b/a]*c^2*d^2*e ^2*f^5*x + 105*a^3*Sqrt[b/a]*c^3*d*e*f^6*x + 315*b^3*Sqrt[b/a]*c*d^3*e^5*f ^2*x^3 + 315*a*b^2*Sqrt[b/a]*d^4*e^5*f^2*x^3 - 518*b^3*Sqrt[b/a]*c^2*d^2*e ^4*f^3*x^3 - 910*a*b^2*Sqrt[b/a]*c*d^3*e^4*f^3*x^3 - 518*a^3*(b/a)^(3/2)*d ^4*e^4*f^3*x^3 + 195*b^3*Sqrt[b/a]*c^3*d*e^3*f^4*x^3 + 737*a*b^2*Sqrt[b/a] *c^2*d^2*e^3*f^4*x^3 + 737*a^3*(b/a)^(3/2)*c*d^3*e^3*f^4*x^3 + 195*a^3*Sqr t[b/a]*d^4*e^3*f^4*x^3 - 120*a*b^2*Sqrt[b/a]*c^3*d*e^2*f^5*x^3 - 80*a^3*(b /a)^(3/2)*c^2*d^2*e^2*f^5*x^3 - 120*a^3*Sqrt[b/a]*c*d^3*e^2*f^5*x^3 + 105* a^3*(b/a)^(3/2)*c^3*d*e*f^6*x^3 + 105*a^3*Sqrt[b/a]*c^2*d^2*e*f^6*x^3 + 31 5*b^3*Sqrt[b/a]*d^4*e^5*f^2*x^5 - 392*b^3*Sqrt[b/a]*c*d^3*e^4*f^3*x^5 - 39 2*a*b^2*Sqrt[b/a]*d^4*e^4*f^3*x^5 - 23*b^3*Sqrt[b/a]*c^2*d^2*e^3*f^4*x^5 + 270*a*b^2*Sqrt[b/a]*c*d^3*e^3*f^4*x^5 - 23*a^3*(b/a)^(3/2)*d^4*e^3*f^4*x^ 5 + 90*b^3*Sqrt[b/a]*c^3*d*e^2*f^5*x^5 + 310*a*b^2*Sqrt[b/a]*c^2*d^2*e^2*f ^5*x^5 + 310*a^3*(b/a)^(3/2)*c*d^3*e^2*f^5*x^5 + 90*a^3*Sqrt[b/a]*d^4*e^2* f^5*x^5 + 105*a^3*(b/a)^(3/2)*c^2*d^2*e*f^6*x^5 + 126*b^3*Sqrt[b/a]*d^4*e^ 4*f^3*x^7 - 272*b^3*Sqrt[b/a]*c*d^3*e^3*f^4*x^7 - 272*a*b^2*Sqrt[b/a]*d^4* e^3*f^4*x^7 + 180*b^3*Sqrt[b/a]*c^2*d^2*e^2*f^5*x^7 + 550*a*b^2*Sqrt[b/...
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 |
| \(\displaystyle \frac {b \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{f x^2+e}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {b \int \sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\int \sqrt {b x^2+a} \sqrt {d x^2+c} \left (2 d (5 b c-2 a d) x^2+c (7 b c-a d)\right )dx}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\int \frac {\sqrt {b x^2+a} \left (d \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) x^2+c \left (35 b^2 c^2-15 a b d c+4 a^2 d^2\right )\right )}{\sqrt {d x^2+c}}dx}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {\int \frac {d \left (\left (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\right ) x^2+4 a c \left (15 b^2 c^2-4 a b d c+a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} \int \frac {\left (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\right ) x^2+4 a c \left (15 b^2 c^2-4 a b d c+a^2 d^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} \left (4 a c \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} \left (\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} \left (\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )+\frac {1}{3} \left (\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{f x^2+e}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )+\frac {1}{3} \left (\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}dx}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )+\frac {1}{3} \left (\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\int \frac {\sqrt {b x^2+a} \left (2 d (3 b c-a d) x^2+c (5 b c-a d)\right )}{\sqrt {d x^2+c}}dx}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )+\frac {1}{3} \left (\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {\int \frac {d \left (\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) x^2+a c (9 b c-a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )+\frac {1}{3} \left (\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {1}{3} \int \frac {\left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) x^2+a c (9 b c-a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )+\frac {1}{3} \left (\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (9 b c-a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )+\frac {1}{3} \left (\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )+\frac {1}{3} \left (\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \left (\frac {\frac {1}{3} \left (\left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{5 b}+\frac {d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\right )}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c f) \int \frac {\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}{f x^2+e}dx}{f}\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 )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 418 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c f) \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}\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 )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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}\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}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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 {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\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 )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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 {b x^2+a} \sqrt {d x^2+c}\right )}{f^2}\right )}{f}\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 )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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+(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}\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 )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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} (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}\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 )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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} (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}\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 )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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 ((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}\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 {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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}\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 {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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}\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}}{f x^2+e}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}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \int \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}dx}{f}-\frac {(d e-c 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 {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d x \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}{7 b}+\frac {\frac {2 d (5 b c-2 a d) x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\frac {\frac {1}{3} \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} \sqrt {d x^2+c} x+\frac {1}{3} \left (\frac {4 \left (15 b^2 c^2-4 a b d c+a^2 d^2\right ) \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 (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\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}}{7 b}\right )}{f}-\frac {(b e-a f) \left (\frac {d \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 {(d e-c 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}\right )}{f}-\frac {(b e-a f) \left (\frac {b \left (\frac {d \left (\frac {d x \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 b}+\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}\right )}{f}-\frac {(d e-c 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 {\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}{\left (f x^2+e\right )^2}dx}{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[((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. \(2165\) vs. \(2(912)=1824\).
Time = 24.95 (sec) , antiderivative size = 2166, normalized size of antiderivative = 2.25
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2166\) |
| elliptic | \(\text {Expression too large to display}\) | \(4779\) |
| default | \(\text {Expression too large to display}\) | \(7152\) |
Input:
int((b*x^2+a)^(5/2)*(d*x^2+c)^(5/2)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
1/105*x*(15*b^2*d^2*f^2*x^4+45*a*b*d^2*f^2*x^2+45*b^2*c*d*f^2*x^2-42*b^2*d ^2*e*f*x^2+45*a^2*d^2*f^2+170*a*b*c*d*f^2-154*a*b*d^2*e*f+45*b^2*c^2*f^2-1 54*b^2*c*d*e*f+105*b^2*d^2*e^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^4+1/105/ f^4*((270*a^3*c*d^2*f^4-210*a^3*d^3*e*f^3+775*a^2*b*c^2*d*f^4-1736*a^2*b*c *d^2*e*f^3+945*a^2*b*d^3*e^2*f^2+270*a*b^2*c^3*f^4-1736*a*b^2*c^2*d*e*f^3+ 2730*a*b^2*c*d^2*e^2*f^2-1260*a*b^2*d^3*e^3*f-210*b^3*c^3*e*f^3+945*b^3*c^ 2*d*e^2*f^2-1260*b^3*c*d^2*e^3*f+525*b^3*d^3*e^4)/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))-1/f*(15*a^3*d^3*f^3+380*a^2*b*c* d^2*f^3-322*a^2*b*d^3*e*f^2+380*a*b^2*c^2*d*f^3-1148*a*b^2*c*d^2*e*f^2+735 *a*b^2*d^3*e^2*f+15*b^3*c^3*f^3-322*b^3*c^2*d*e*f^2+735*b^3*c*d^2*e^2*f-42 0*b^3*d^3*e^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*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b) ^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))+315/f^2*(a^3*c ^2*d*f^5-2*a^3*c*d^2*e*f^4+a^3*d^3*e^2*f^3+a^2*b*c^3*f^5-6*a^2*b*c^2*d*e*f ^4+9*a^2*b*c*d^2*e^2*f^3-4*a^2*b*d^3*e^3*f^2-2*a*b^2*c^3*e*f^4+9*a*b^2*c^2 *d*e^2*f^3-12*a*b^2*c*d^2*e^3*f^2+5*a*b^2*d^3*e^4*f+b^3*c^3*e^2*f^3-4*b^3* c^2*d*e^3*f^2+5*b^3*c*d^2*e^4*f-2*b^3*d^3*e^5)/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))+105*(a^3*c^3*f^6-3*a^3...
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)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^2} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{\left (f \,x^{2}+e \right )^{2}}d x \] Input:
int((b*x^2+a)^(5/2)*(d*x^2+c)^(5/2)/(f*x^2+e)^2,x)
Output:
int((b*x^2+a)^(5/2)*(d*x^2+c)^(5/2)/(f*x^2+e)^2,x)