Integrand size = 35, antiderivative size = 608 \[ \int \frac {\left (e+f x^2\right )^2}{x^8 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {2 \left (9 a b c d e (3 b c e+3 a d e-7 a c f)-(b c+a d) \left (8 (b c+a d) e (3 b c e+3 a d e-7 a c f)-5 a c \left (5 b d e^2-7 a c f^2\right )\right )\right ) \sqrt {c+d x^2}}{105 a^3 c^4 x \sqrt {a+b x^2}}-\frac {e^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 a c x^7}+\frac {2 e (3 b c e+3 a d e-7 a c f) \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 a^2 c^2 x^5}-\frac {\left (8 (b c+a d) e (3 b c e+3 a d e-7 a c f)-5 a c \left (5 b d e^2-7 a c f^2\right )\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 a^3 c^3 x^3}+\frac {2 \sqrt {b} \left (24 b^3 c^3 e^2+4 a b^2 c^2 e (5 d e-14 c f)+a^3 d \left (24 d^2 e^2-56 c d e f+35 c^2 f^2\right )+a^2 b c \left (20 d^2 e^2-49 c d e f+35 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 a^{7/2} c^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {b} d \left (24 b^2 c^2 e^2+a b c e (23 d e-56 c f)+a^2 \left (24 d^2 e^2-56 c d e f+35 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 a^{5/2} c^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-2/105*(9*a*b*c*d*e*(-7*a*c*f+3*a*d*e+3*b*c*e)-(a*d+b*c)*(8*(a*d+b*c)*e*(- 7*a*c*f+3*a*d*e+3*b*c*e)-5*a*c*(-7*a*c*f^2+5*b*d*e^2)))*(d*x^2+c)^(1/2)/a^ 3/c^4/x/(b*x^2+a)^(1/2)-1/7*e^2*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/x^7+2/ 35*e*(-7*a*c*f+3*a*d*e+3*b*c*e)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/x^ 5-1/105*(8*(a*d+b*c)*e*(-7*a*c*f+3*a*d*e+3*b*c*e)-5*a*c*(-7*a*c*f^2+5*b*d* e^2))*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^3/c^3/x^3+2/105*b^(1/2)*(24*b^3*c^ 3*e^2+4*a*b^2*c^2*e*(-14*c*f+5*d*e)+a^3*d*(35*c^2*f^2-56*c*d*e*f+24*d^2*e^ 2)+a^2*b*c*(35*c^2*f^2-49*c*d*e*f+20*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))/a^(7/2)/c^4/(b*x^2+a )^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/105*b^(1/2)*d*(24*b^2*c^2*e^2+a* b*c*e*(-56*c*f+23*d*e)+a^2*(35*c^2*f^2-56*c*d*e*f+24*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))/a^(5/2)/ c^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 = 9.66 (sec) , antiderivative size = 575, normalized size of antiderivative = 0.95 \[ \int \frac {\left (e+f x^2\right )^2}{x^8 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (48 b^3 c^3 e^2 x^6-8 a b^2 c^2 e x^4 \left (3 c e-5 d e x^2+14 c f x^2\right )+a^2 b c x^2 \left (40 d^2 e^2 x^4-c d e x^2 \left (23 e+98 f x^2\right )+2 c^2 \left (9 e^2+28 e f x^2+35 f^2 x^4\right )\right )+a^3 \left (48 d^3 e^2 x^6-8 c d^2 e x^4 \left (3 e+14 f x^2\right )+2 c^2 d x^2 \left (9 e^2+28 e f x^2+35 f^2 x^4\right )-c^3 \left (15 e^2+42 e f x^2+35 f^2 x^4\right )\right )\right )+2 i b c \left (24 b^3 c^3 e^2+4 a b^2 c^2 e (5 d e-14 c f)+a^3 d \left (24 d^2 e^2-56 c d e f+35 c^2 f^2\right )+a^2 b c \left (20 d^2 e^2-49 c d e f+35 c^2 f^2\right )\right ) x^7 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c \left (48 b^3 c^3 e^2-16 a b^2 c^2 e (-d e+7 c f)+a^3 d \left (24 d^2 e^2-56 c d e f+35 c^2 f^2\right )+a^2 b c \left (17 d^2 e^2-42 c d e f+70 c^2 f^2\right )\right ) x^7 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 a^4 \sqrt {\frac {b}{a}} c^4 x^7 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(e + f*x^2)^2/(x^8*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
Output:
(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(48*b^3*c^3*e^2*x^6 - 8*a*b^2*c^2*e*x^4 *(3*c*e - 5*d*e*x^2 + 14*c*f*x^2) + a^2*b*c*x^2*(40*d^2*e^2*x^4 - c*d*e*x^ 2*(23*e + 98*f*x^2) + 2*c^2*(9*e^2 + 28*e*f*x^2 + 35*f^2*x^4)) + a^3*(48*d ^3*e^2*x^6 - 8*c*d^2*e*x^4*(3*e + 14*f*x^2) + 2*c^2*d*x^2*(9*e^2 + 28*e*f* x^2 + 35*f^2*x^4) - c^3*(15*e^2 + 42*e*f*x^2 + 35*f^2*x^4))) + (2*I)*b*c*( 24*b^3*c^3*e^2 + 4*a*b^2*c^2*e*(5*d*e - 14*c*f) + a^3*d*(24*d^2*e^2 - 56*c *d*e*f + 35*c^2*f^2) + a^2*b*c*(20*d^2*e^2 - 49*c*d*e*f + 35*c^2*f^2))*x^7 *Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(48*b^3*c^3*e^2 - 16*a*b^2*c^2*e*(-(d*e) + 7*c*f) + a^3*d*(24*d^2*e^2 - 56*c*d*e*f + 35*c^2*f^2) + a^2*b*c*(17*d^2*e^2 - 42*c* d*e*f + 70*c^2*f^2))*x^7*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF [I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*a^4*Sqrt[b/a]*c^4*x^7*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 1.87 (sec) , antiderivative size = 1038, normalized size of antiderivative = 1.71, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {448, 445, 445, 445, 25, 27, 406, 320, 388, 313, 445, 25, 27, 406, 320, 388, 313}
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.
| \(\displaystyle \int \frac {\left (e+f x^2\right )^2}{x^8 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 448 |
| \(\displaystyle \frac {f \int \frac {f x^2+e}{x^6 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{e^2}+e \int \frac {f x^2+e}{x^8 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {f \left (-\frac {\int \frac {3 b d e x^2+4 b c e+4 a d e-5 a c f}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 a c x^5}\right )}{e^2}+e \left (-\frac {\int \frac {5 b d e x^2+6 b c e+6 a d e-7 a c f}{x^6 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{7 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 a c x^7}\right )\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {f \left (-\frac {-\frac {\int \frac {2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+b d (4 b c e+4 a d e-5 a c f) x^2+8 b^2 c^2 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f+4 a d e+4 b c e)}{3 a c x^3}}{5 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 a c x^5}\right )}{e^2}+e \left (-\frac {-\frac {\int \frac {4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+3 b d (6 b c e+6 a d e-7 a c f) x^2+24 b^2 c^2 e}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f+6 a d e+6 b c e)}{5 a c x^5}}{7 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 a c x^7}\right )\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {f \left (-\frac {-\frac {-\frac {\int -\frac {b d \left (\left (2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+8 b^2 c^2 e\right ) x^2+a c (4 b c e+4 a d e-5 a c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c e}{a}+\frac {8 a d^2 e}{c}-10 a d f-10 b c f+7 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f+4 a d e+4 b c e)}{3 a c x^3}}{5 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 a c x^5}\right )}{e^2}+e \left (-\frac {-\frac {-\frac {\int \frac {8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+b d \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right ) x^2+48 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c e}{a}+\frac {4 a d (6 d e-7 c f)}{c}+b (23 d e-28 c f)\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f+6 a d e+6 b c e)}{5 a c x^5}}{7 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 a c x^7}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f \left (-\frac {-\frac {\frac {\int \frac {b d \left (\left (2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+8 b^2 c^2 e\right ) x^2+a c (4 b c e+4 a d e-5 a c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c e}{a}+\frac {8 a d^2 e}{c}-10 a d f-10 b c f+7 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f+4 a d e+4 b c e)}{3 a c x^3}}{5 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 a c x^5}\right )}{e^2}+e \left (-\frac {-\frac {-\frac {\int \frac {8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+b d \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right ) x^2+48 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c e}{a}+\frac {4 a d (6 d e-7 c f)}{c}+b (23 d e-28 c f)\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f+6 a d e+6 b c e)}{5 a c x^5}}{7 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 a c x^7}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f \left (-\frac {-\frac {\frac {b d \int \frac {\left (2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+8 b^2 c^2 e\right ) x^2+a c (4 b c e+4 a d e-5 a c f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c e}{a}+\frac {8 a d^2 e}{c}-10 a d f-10 b c f+7 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f+4 a d e+4 b c e)}{3 a c x^3}}{5 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 a c x^5}\right )}{e^2}+e \left (-\frac {-\frac {-\frac {\int \frac {8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+b d \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right ) x^2+48 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c e}{a}+\frac {4 a d (6 d e-7 c f)}{c}+b (23 d e-28 c f)\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f+6 a d e+6 b c e)}{5 a c x^5}}{7 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 a c x^7}\right )\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {f \left (-\frac {-\frac {\frac {b d \left (\left (2 a^2 d (4 d e-5 c f)+a b c (7 d e-10 c f)+8 b^2 c^2 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (-5 a c f+4 a d e+4 b c e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c e}{a}+\frac {8 a d^2 e}{c}-10 a d f-10 b c f+7 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f+4 a d e+4 b c e)}{3 a c x^3}}{5 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 a c x^5}\right )}{e^2}+e \left (-\frac {-\frac {-\frac {\int \frac {8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+b d \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right ) x^2+48 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c e}{a}+\frac {4 a d (6 d e-7 c f)}{c}+b (23 d e-28 c f)\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f+6 a d e+6 b c e)}{5 a c x^5}}{7 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 a c x^7}\right )\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f \left (-\frac {-\frac {\frac {b d \left (\left (2 a^2 d (4 d e-5 c f)+a b c (7 d e-10 c f)+8 b^2 c^2 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (-5 a c f+4 a d e+4 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c e}{a}+\frac {8 a d^2 e}{c}-10 a d f-10 b c f+7 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f+4 a d e+4 b c e)}{3 a c x^3}}{5 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 a c x^5}\right )}{e^2}+e \left (-\frac {-\frac {-\frac {\int \frac {8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+b d \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right ) x^2+48 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c e}{a}+\frac {4 a d (6 d e-7 c f)}{c}+b (23 d e-28 c f)\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f+6 a d e+6 b c e)}{5 a c x^5}}{7 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 a c x^7}\right )\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {f \left (-\frac {-\frac {\frac {b d \left (\left (2 a^2 d (4 d e-5 c f)+a b c (7 d e-10 c f)+8 b^2 c^2 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (-5 a c f+4 a d e+4 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c e}{a}+\frac {8 a d^2 e}{c}-10 a d f-10 b c f+7 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f+4 a d e+4 b c e)}{3 a c x^3}}{5 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 a c x^5}\right )}{e^2}+e \left (-\frac {-\frac {-\frac {\int \frac {8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+b d \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right ) x^2+48 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c e}{a}+\frac {4 a d (6 d e-7 c f)}{c}+b (23 d e-28 c f)\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f+6 a d e+6 b c e)}{5 a c x^5}}{7 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 a c x^7}\right )\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle e \left (-\frac {-\frac {-\frac {\int \frac {8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+b d \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right ) x^2+48 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c e}{a}+\frac {4 a d (6 d e-7 c f)}{c}+b (23 d e-28 c f)\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f+6 a d e+6 b c e)}{5 a c x^5}}{7 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 a c x^7}\right )+\frac {f \left (-\frac {-\frac {\frac {b d \left (\left (2 a^2 d (4 d e-5 c f)+a b c (7 d e-10 c f)+8 b^2 c^2 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (-5 a c f+4 a d e+4 b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c e}{a}+\frac {8 a d^2 e}{c}-10 a d f-10 b c f+7 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f+4 a d e+4 b c e)}{3 a c x^3}}{5 a c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 a c x^5}\right )}{e^2}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {f \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (4 b c e+4 a d e-5 a c f)}{3 a c x^3}-\frac {\frac {b d \left (\frac {(4 b c e+4 a d e-5 a c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+8 b^2 c^2 e\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{a c}-\frac {\left (\frac {8 c e b^2}{a}+7 d e b-10 c f b+\frac {8 a d^2 e}{c}-10 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 a c}}{5 a c}\right )}{e^2}+e \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{7 a c x^7}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (6 b c e+6 a d e-7 a c f)}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {24 c e b^2}{a}+(23 d e-28 c f) b+\frac {4 a d (6 d e-7 c f)}{c}\right )}{3 x^3}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right )}{a c x}-\frac {\int -\frac {b d \left (\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) x^2+a c \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}}{3 a c}}{5 a c}}{7 a c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (4 b c e+4 a d e-5 a c f)}{3 a c x^3}-\frac {\frac {b d \left (\frac {(4 b c e+4 a d e-5 a c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+8 b^2 c^2 e\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{a c}-\frac {\left (\frac {8 c e b^2}{a}+7 d e b-10 c f b+\frac {8 a d^2 e}{c}-10 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 a c}}{5 a c}\right )}{e^2}+e \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{7 a c x^7}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (6 b c e+6 a d e-7 a c f)}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {24 c e b^2}{a}+(23 d e-28 c f) b+\frac {4 a d (6 d e-7 c f)}{c}\right )}{3 x^3}-\frac {\frac {\int \frac {b d \left (\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) x^2+a c \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 a c}}{7 a c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (4 b c e+4 a d e-5 a c f)}{3 a c x^3}-\frac {\frac {b d \left (\frac {(4 b c e+4 a d e-5 a c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+8 b^2 c^2 e\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{a c}-\frac {\left (\frac {8 c e b^2}{a}+7 d e b-10 c f b+\frac {8 a d^2 e}{c}-10 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 a c}}{5 a c}\right )}{e^2}+e \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{7 a c x^7}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (6 b c e+6 a d e-7 a c f)}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {24 c e b^2}{a}+(23 d e-28 c f) b+\frac {4 a d (6 d e-7 c f)}{c}\right )}{3 x^3}-\frac {\frac {b d \int \frac {\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) x^2+a c \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 a c}}{7 a c}\right )\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {f \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (4 b c e+4 a d e-5 a c f)}{3 a c x^3}-\frac {\frac {b d \left (\frac {(4 b c e+4 a d e-5 a c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+8 b^2 c^2 e\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{a c}-\frac {\left (\frac {8 c e b^2}{a}+7 d e b-10 c f b+\frac {8 a d^2 e}{c}-10 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 a c}}{5 a c}\right )}{e^2}+e \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{7 a c x^7}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (6 b c e+6 a d e-7 a c f)}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {24 c e b^2}{a}+(23 d e-28 c f) b+\frac {4 a d (6 d e-7 c f)}{c}\right )}{3 x^3}-\frac {\frac {b d \left (a c \left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 a c}}{7 a c}\right )\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (4 b c e+4 a d e-5 a c f)}{3 a c x^3}-\frac {\frac {b d \left (\frac {(4 b c e+4 a d e-5 a c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+8 b^2 c^2 e\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{a c}-\frac {\left (\frac {8 c e b^2}{a}+7 d e b-10 c f b+\frac {8 a d^2 e}{c}-10 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 a c}}{5 a c}\right )}{e^2}+e \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{7 a c x^7}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (6 b c e+6 a d e-7 a c f)}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {24 c e b^2}{a}+(23 d e-28 c f) b+\frac {4 a d (6 d e-7 c f)}{c}\right )}{3 x^3}-\frac {\frac {b d \left (\frac {\left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 a c}}{7 a c}\right )\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {f \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (4 b c e+4 a d e-5 a c f)}{3 a c x^3}-\frac {\frac {b d \left (\frac {(4 b c e+4 a d e-5 a 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 ) 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 (2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+8 b^2 c^2 e\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 )}{a c}-\frac {\left (\frac {8 c e b^2}{a}+7 d e b-10 c f b+\frac {8 a d^2 e}{c}-10 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 a c}}{5 a c}\right )}{e^2}+e \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{7 a c x^7}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (6 b c e+6 a d e-7 a c f)}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {24 c e b^2}{a}+(23 d e-28 c f) b+\frac {4 a d (6 d e-7 c f)}{c}\right )}{3 x^3}-\frac {\frac {b d \left (\frac {\left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\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 (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) \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 )}{a c}-\frac {\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 a c}}{7 a c}\right )\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {f \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (4 b c e+4 a d e-5 a c f)}{3 a c x^3}-\frac {\frac {b d \left (\frac {(4 b c e+4 a d e-5 a 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 ) 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 (2 d (4 d e-5 c f) a^2+b c (7 d e-10 c f) a+8 b^2 c^2 e\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 )}{a c}-\frac {\left (\frac {8 c e b^2}{a}+7 d e b-10 c f b+\frac {8 a d^2 e}{c}-10 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 a c}}{5 a c}\right )}{e^2}+e \left (-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} e}{7 a c x^7}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} (6 b c e+6 a d e-7 a c f)}{5 a c x^5}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {24 c e b^2}{a}+(23 d e-28 c f) b+\frac {4 a d (6 d e-7 c f)}{c}\right )}{3 x^3}-\frac {\frac {b d \left (\frac {\left (4 d (6 d e-7 c f) a^2+b c (23 d e-28 c f) a+24 b^2 c^2 e\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 (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\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 )}{a c}-\frac {\left (8 d^2 (6 d e-7 c f) a^3+b c d (40 d e-49 c f) a^2+8 b^2 c^2 (5 d e-7 c f) a+48 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 a c}}{7 a c}\right )\) |
Input:
Int[(e + f*x^2)^2/(x^8*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
Output:
(f*(-1/5*(e*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x^5) - (-1/3*((4*b*c*e + 4*a*d*e - 5*a*c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x^3) - (-((((8*b ^2*c*e)/a + 7*b*d*e + (8*a*d^2*e)/c - 10*b*c*f - 10*a*d*f)*Sqrt[a + b*x^2] *Sqrt[c + d*x^2])/x) + (b*d*((8*b^2*c^2*e + a*b*c*(7*d*e - 10*c*f) + 2*a^2 *d*(4*d*e - 5*c*f))*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sq rt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b* Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2) *(4*b*c*e + 4*a*d*e - 5*a*c*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x )/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2) )]*Sqrt[c + d*x^2])))/(a*c))/(3*a*c))/(5*a*c)))/e^2 + e*(-1/7*(e*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x^7) - (-1/5*((6*b*c*e + 6*a*d*e - 7*a*c*f)*S qrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x^5) - (-1/3*(((24*b^2*c*e)/a + b*(23 *d*e - 28*c*f) + (4*a*d*(6*d*e - 7*c*f))/c)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2 ])/x^3 - (-(((48*b^3*c^3*e + a^2*b*c*d*(40*d*e - 49*c*f) + 8*a*b^2*c^2*(5* d*e - 7*c*f) + 8*a^3*d^2*(6*d*e - 7*c*f))*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) /(a*c*x)) + (b*d*((48*b^3*c^3*e + a^2*b*c*d*(40*d*e - 49*c*f) + 8*a*b^2*c^ 2*(5*d*e - 7*c*f) + 8*a^3*d^2*(6*d*e - 7*c*f))*((x*Sqrt[a + b*x^2])/(b*Sqr t[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt [c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*S qrt[c + d*x^2])) + (c^(3/2)*(24*b^2*c^2*e + a*b*c*(23*d*e - 28*c*f) + 4...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[(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[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[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[e Int[(g*x)^m*(a + b*x ^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] + Simp[f/e^2 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q}, x] && IGtQ[r, 0]
Time = 19.59 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.24
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 a c \,x^{7}}-\frac {2 e \left (7 a c f -3 a d e -3 b c e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{35 a^{2} c^{2} x^{5}}-\frac {\left (35 a^{2} c^{2} f^{2}-56 a^{2} c d e f +24 a^{2} d^{2} e^{2}-56 a b \,c^{2} e f +23 a b c d \,e^{2}+24 b^{2} c^{2} e^{2}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{105 a^{3} c^{3} x^{3}}+\frac {2 \left (35 a^{3} c^{2} d \,f^{2}-56 a^{3} c \,d^{2} e f +24 a^{3} d^{3} e^{2}+35 a^{2} b \,c^{3} f^{2}-49 a^{2} b \,c^{2} d e f +20 a^{2} b c \,d^{2} e^{2}-56 a \,b^{2} c^{3} e f +20 a \,b^{2} c^{2} d \,e^{2}+24 b^{3} c^{3} e^{2}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{105 c^{4} a^{4} x}-\frac {b d \left (35 a^{2} c^{2} f^{2}-56 a^{2} c d e f +24 a^{2} d^{2} e^{2}-56 a b \,c^{2} e f +23 a b c d \,e^{2}+24 b^{2} c^{2} e^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{105 c^{3} a^{3} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {2 b \left (35 a^{3} c^{2} d \,f^{2}-56 a^{3} c \,d^{2} e f +24 a^{3} d^{3} e^{2}+35 a^{2} b \,c^{3} f^{2}-49 a^{2} b \,c^{2} d e f +20 a^{2} b c \,d^{2} e^{2}-56 a \,b^{2} c^{3} e f +20 a \,b^{2} c^{2} d \,e^{2}+24 b^{3} c^{3} e^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{105 c^{3} a^{4} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(755\) |
| risch | \(\text {Expression too large to display}\) | \(1163\) |
| default | \(\text {Expression too large to display}\) | \(2173\) |
Input:
int((f*x^2+e)^2/x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOS E)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-1/7/a/c*e^2* (b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^7-2/35*e*(7*a*c*f-3*a*d*e-3*b*c*e)/a ^2/c^2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^5-1/105/a^3/c^3*(35*a^2*c^2*f ^2-56*a^2*c*d*e*f+24*a^2*d^2*e^2-56*a*b*c^2*e*f+23*a*b*c*d*e^2+24*b^2*c^2* e^2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3+2/105*(35*a^3*c^2*d*f^2-56*a^ 3*c*d^2*e*f+24*a^3*d^3*e^2+35*a^2*b*c^3*f^2-49*a^2*b*c^2*d*e*f+20*a^2*b*c* d^2*e^2-56*a*b^2*c^3*e*f+20*a*b^2*c^2*d*e^2+24*b^3*c^3*e^2)/c^4/a^4*(b*d*x ^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x-1/105*b*d*(35*a^2*c^2*f^2-56*a^2*c*d*e*f+2 4*a^2*d^2*e^2-56*a*b*c^2*e*f+23*a*b*c*d*e^2+24*b^2*c^2*e^2)/c^3/a^3/(-b/a) ^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^( 1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+2/105*b*(35*a^3*c^ 2*d*f^2-56*a^3*c*d^2*e*f+24*a^3*d^3*e^2+35*a^2*b*c^3*f^2-49*a^2*b*c^2*d*e* f+20*a^2*b*c*d^2*e^2-56*a*b^2*c^3*e*f+20*a*b^2*c^2*d*e^2+24*b^3*c^3*e^2)/c ^3/a^4/(-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))-Elli pticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))
Time = 0.09 (sec) , antiderivative size = 598, normalized size of antiderivative = 0.98 \[ \int \frac {\left (e+f x^2\right )^2}{x^8 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {2 \, {\left (4 \, {\left (6 \, b^{4} c^{3} + 5 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2} + 6 \, a^{3} b d^{3}\right )} e^{2} - 7 \, {\left (8 \, a b^{3} c^{3} + 7 \, a^{2} b^{2} c^{2} d + 8 \, a^{3} b c d^{2}\right )} e f + 35 \, {\left (a^{2} b^{2} c^{3} + a^{3} b c^{2} d\right )} f^{2}\right )} \sqrt {a c} x^{7} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (48 \, b^{4} c^{3} + 8 \, {\left (3 \, a^{2} b^{2} + 5 \, a b^{3}\right )} c^{2} d + {\left (23 \, a^{3} b + 40 \, a^{2} b^{2}\right )} c d^{2} + 24 \, {\left (a^{4} + 2 \, a^{3} b\right )} d^{3}\right )} e^{2} - 14 \, {\left (8 \, a b^{3} c^{3} + {\left (4 \, a^{3} b + 7 \, a^{2} b^{2}\right )} c^{2} d + 4 \, {\left (a^{4} + 2 \, a^{3} b\right )} c d^{2}\right )} e f + 35 \, {\left (2 \, a^{2} b^{2} c^{3} + {\left (a^{4} + 2 \, a^{3} b\right )} c^{2} d\right )} f^{2}\right )} \sqrt {a c} x^{7} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + {\left (15 \, a^{4} c^{3} e^{2} - 2 \, {\left (4 \, {\left (6 \, a b^{3} c^{3} + 5 \, a^{2} b^{2} c^{2} d + 5 \, a^{3} b c d^{2} + 6 \, a^{4} d^{3}\right )} e^{2} - 7 \, {\left (8 \, a^{2} b^{2} c^{3} + 7 \, a^{3} b c^{2} d + 8 \, a^{4} c d^{2}\right )} e f + 35 \, {\left (a^{3} b c^{3} + a^{4} c^{2} d\right )} f^{2}\right )} x^{6} + {\left (35 \, a^{4} c^{3} f^{2} + {\left (24 \, a^{2} b^{2} c^{3} + 23 \, a^{3} b c^{2} d + 24 \, a^{4} c d^{2}\right )} e^{2} - 56 \, {\left (a^{3} b c^{3} + a^{4} c^{2} d\right )} e f\right )} x^{4} + 6 \, {\left (7 \, a^{4} c^{3} e f - 3 \, {\left (a^{3} b c^{3} + a^{4} c^{2} d\right )} e^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, a^{5} c^{4} x^{7}} \] Input:
integrate((f*x^2+e)^2/x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fr icas")
Output:
-1/105*(2*(4*(6*b^4*c^3 + 5*a*b^3*c^2*d + 5*a^2*b^2*c*d^2 + 6*a^3*b*d^3)*e ^2 - 7*(8*a*b^3*c^3 + 7*a^2*b^2*c^2*d + 8*a^3*b*c*d^2)*e*f + 35*(a^2*b^2*c ^3 + a^3*b*c^2*d)*f^2)*sqrt(a*c)*x^7*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(- b/a)), a*d/(b*c)) - ((48*b^4*c^3 + 8*(3*a^2*b^2 + 5*a*b^3)*c^2*d + (23*a^3 *b + 40*a^2*b^2)*c*d^2 + 24*(a^4 + 2*a^3*b)*d^3)*e^2 - 14*(8*a*b^3*c^3 + ( 4*a^3*b + 7*a^2*b^2)*c^2*d + 4*(a^4 + 2*a^3*b)*c*d^2)*e*f + 35*(2*a^2*b^2* c^3 + (a^4 + 2*a^3*b)*c^2*d)*f^2)*sqrt(a*c)*x^7*sqrt(-b/a)*elliptic_f(arcs in(x*sqrt(-b/a)), a*d/(b*c)) + (15*a^4*c^3*e^2 - 2*(4*(6*a*b^3*c^3 + 5*a^2 *b^2*c^2*d + 5*a^3*b*c*d^2 + 6*a^4*d^3)*e^2 - 7*(8*a^2*b^2*c^3 + 7*a^3*b*c ^2*d + 8*a^4*c*d^2)*e*f + 35*(a^3*b*c^3 + a^4*c^2*d)*f^2)*x^6 + (35*a^4*c^ 3*f^2 + (24*a^2*b^2*c^3 + 23*a^3*b*c^2*d + 24*a^4*c*d^2)*e^2 - 56*(a^3*b*c ^3 + a^4*c^2*d)*e*f)*x^4 + 6*(7*a^4*c^3*e*f - 3*(a^3*b*c^3 + a^4*c^2*d)*e^ 2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a^5*c^4*x^7)
\[ \int \frac {\left (e+f x^2\right )^2}{x^8 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {\left (e + f x^{2}\right )^{2}}{x^{8} \sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate((f*x**2+e)**2/x**8/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
Output:
Integral((e + f*x**2)**2/(x**8*sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{x^8 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{8}} \,d x } \] Input:
integrate((f*x^2+e)^2/x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="ma xima")
Output:
integrate((f*x^2 + e)^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^8), x)
\[ \int \frac {\left (e+f x^2\right )^2}{x^8 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{8}} \,d x } \] Input:
integrate((f*x^2+e)^2/x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="gi ac")
Output:
integrate((f*x^2 + e)^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^8), x)
Timed out. \[ \int \frac {\left (e+f x^2\right )^2}{x^8 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^2}{x^8\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int((e + f*x^2)^2/(x^8*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
Output:
int((e + f*x^2)^2/(x^8*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{x^8 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:
int((f*x^2+e)^2/x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
Output:
( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*e**2 + 14*int((sqrt(c + d*x**2)*sqrt (a + b*x**2))/(a**2*c*d*x**6 + a**2*d**2*x**8 + a*b*c**2*x**6 + 2*a*b*c*d* x**8 + a*b*d**2*x**10 + b**2*c**2*x**8 + b**2*c*d*x**10),x)*a**2*c*d*e*f*x **7 - 6*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*d*x**6 + a**2*d**2 *x**8 + a*b*c**2*x**6 + 2*a*b*c*d*x**8 + a*b*d**2*x**10 + b**2*c**2*x**8 + b**2*c*d*x**10),x)*a**2*d**2*e**2*x**7 + 14*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*d*x**6 + a**2*d**2*x**8 + a*b*c**2*x**6 + 2*a*b*c*d*x** 8 + a*b*d**2*x**10 + b**2*c**2*x**8 + b**2*c*d*x**10),x)*a*b*c**2*e*f*x**7 - 12*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*d*x**6 + a**2*d**2*x **8 + a*b*c**2*x**6 + 2*a*b*c*d*x**8 + a*b*d**2*x**10 + b**2*c**2*x**8 + b **2*c*d*x**10),x)*a*b*c*d*e**2*x**7 - 6*int((sqrt(c + d*x**2)*sqrt(a + b*x **2))/(a**2*c*d*x**6 + a**2*d**2*x**8 + a*b*c**2*x**6 + 2*a*b*c*d*x**8 + a *b*d**2*x**10 + b**2*c**2*x**8 + b**2*c*d*x**10),x)*b**2*c**2*e**2*x**7 + 7*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*d*x**4 + a**2*d**2*x**6 + a*b*c**2*x**4 + 2*a*b*c*d*x**6 + a*b*d**2*x**8 + b**2*c**2*x**6 + b**2*c *d*x**8),x)*a**2*c*d*f**2*x**7 + 7*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)) /(a**2*c*d*x**4 + a**2*d**2*x**6 + a*b*c**2*x**4 + 2*a*b*c*d*x**6 + a*b*d* *2*x**8 + b**2*c**2*x**6 + b**2*c*d*x**8),x)*a*b*c**2*f**2*x**7 - 5*int((s qrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*d*x**4 + a**2*d**2*x**6 + a*b*c* *2*x**4 + 2*a*b*c*d*x**6 + a*b*d**2*x**8 + b**2*c**2*x**6 + b**2*c*d*x*...