Integrand size = 35, antiderivative size = 1067 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^6} \, dx =\text {Too large to display} \] Output:
-2/15*b*(b^2*c^2*e^2-a*b*c*e*(5*c*f+d*e)+a^2*(-15*c^2*f^2-5*c*d*e*f+d^2*e^ 2))*x*(d*x^2+c)^(1/2)/a^2/c^2/(b*x^2+a)^(1/2)+1/15*(b^2*c*d*e^3+3*a^2*c*f^ 2*(5*c*f+9*d*e)+a*b*e*(27*c^2*f^2+19*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)*( d*x^2+c)^(1/2)/a^2/c^2/e+1/15*(3*b^2*e^2*(4*c^2*f^2+3*c*d*e*f+d^2*e^2)+a^2 *f^2*(10*c^2*f^2+31*c*d*e*f+12*d^2*e^2)+a*b*e*f*(31*c^2*f^2+48*c*d*e*f+9*d ^2*e^2))*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^2+1/15*f*(a^2*f^2*( 3*c^2*f^2+16*c*d*e*f+16*d^2*e^2)+b^2*e^2*(16*c^2*f^2+21*c*d*e*f+9*d^2*e^2) +a*b*e*f*(16*c^2*f^2+50*c*d*e*f+21*d^2*e^2))*x^5*(b*x^2+a)^(1/2)*(d*x^2+c) ^(1/2)/a^2/c^2/e^3+1/15*f^2*(3*a^2*d*f^2*(c*f+2*d*e)+a*b*f*(3*c^2*f^2+22*c *d*e*f+19*d^2*e^2)+b^2*e*(6*c^2*f^2+19*c*d*e*f+9*d^2*e^2))*x^7*(b*x^2+a)^( 1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^3+1/5*b*d*f^3*(a*f*(c*f+2*d*e)+b*e*(2*c*f+d *e))*x^9*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^3-1/5*(b*x^2+a)^(3/2)*( d*x^2+c)^(3/2)*(f*x^2+e)^3/a/c/e/x^5+1/15*(-a*c*f+2*a*d*e+2*b*c*e)*(b*x^2+ a)^(3/2)*(d*x^2+c)^(3/2)*(f*x^2+e)^3/a^2/c^2/e^2/x^3-1/5*(a*f*(c*f+2*d*e)+ b*e*(2*c*f+d*e))*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)*(f*x^2+e)^3/a^2/c^2/e^3/x +2/15*b^(1/2)*(b^2*c^2*e^2-a*b*c*e*(5*c*f+d*e)+a^2*(-15*c^2*f^2-5*c*d*e*f+ 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^(3/2)/c^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/ 2)-1/15*(b^2*c*d*e^2-15*a^2*c*d*f^2+a*b*(-15*c^2*f^2-20*c*d*e*f+d^2*e^2))* (d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(...
Result contains complex when optimal does not.
Time = 2.63 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.36 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^6} \, dx=\frac {-\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 b^2 c^2 e^2 x^4+a b c e x^2 \left (2 d e x^2+c \left (e+10 f x^2\right )\right )+a^2 \left (-2 d^2 e^2 x^4+c d e x^2 \left (e+10 f x^2\right )+c^2 \left (3 e^2+10 e f x^2+15 f^2 x^4\right )\right )\right )-2 i b c \left (-b^2 c^2 e^2+a b c e (d e+5 c f)+a^2 \left (-d^2 e^2+5 c d e f+15 c^2 f^2\right )\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (-2 b^2 c e^2+15 a^2 c f^2+a b e (d e+10 c f)\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \sqrt {\frac {b}{a}} c^2 x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2)/x^6,x]
Output:
(-(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-2*b^2*c^2*e^2*x^4 + a*b*c*e*x^2*(2* d*e*x^2 + c*(e + 10*f*x^2)) + a^2*(-2*d^2*e^2*x^4 + c*d*e*x^2*(e + 10*f*x^ 2) + c^2*(3*e^2 + 10*e*f*x^2 + 15*f^2*x^4)))) - (2*I)*b*c*(-(b^2*c^2*e^2) + a*b*c*e*(d*e + 5*c*f) + a^2*(-(d^2*e^2) + 5*c*d*e*f + 15*c^2*f^2))*x^5*S qrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], ( a*d)/(b*c)] - I*c*(-(b*c) + a*d)*(-2*b^2*c*e^2 + 15*a^2*c*f^2 + a*b*e*(d*e + 10*c*f))*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSin h[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*a^2*Sqrt[b/a]*c^2*x^5*Sqrt[a + b*x^2]*Sq rt[c + d*x^2])
Time = 1.37 (sec) , antiderivative size = 751, normalized size of antiderivative = 0.70, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {448, 442, 25, 442, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^6} \, dx\) |
| \(\Big \downarrow \) 448 |
| \(\displaystyle \frac {f \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}{x^4}dx}{e^2}+e \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}{x^6}dx\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {f \left (\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e+3 a f) x^2+a (d e+3 c f)\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}+e \left (\frac {\int -\frac {\sqrt {b x^2+a} \left (d (b e-5 a f) x^2+2 b c e-a d e-5 a c f\right )}{x^4 \sqrt {d x^2+c}}dx}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f \left (\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e+3 a f) x^2+a (d e+3 c f)\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}+e \left (-\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e-5 a f) x^2+2 b c e-a d e-5 a c f\right )}{x^4 \sqrt {d x^2+c}}dx}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle e \left (-\frac {\frac {\int \frac {d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+b d (b c e+a d e-10 a c f) x^2+2 b^2 c^2 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {\int \frac {b d (b c e+a d e+6 a c f) x^2+a c (2 b d e+3 b c f+3 a d f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle e \left (-\frac {\frac {\int \frac {d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+b d (b c e+a d e-10 a c f) x^2+2 b^2 c^2 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {a c (3 a d f+3 b c f+2 b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+b d (6 a c f+a d e+b c e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle e \left (-\frac {\frac {\int \frac {d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+b d (b c e+a d e-10 a c f) x^2+2 b^2 c^2 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {b d (6 a c f+a d e+b c e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle e \left (-\frac {\frac {\int \frac {d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+b d (b c e+a d e-10 a c f) x^2+2 b^2 c^2 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle e \left (-\frac {\frac {\int \frac {d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+b d (b c e+a d e-10 a c f) x^2+2 b^2 c^2 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle e \left (-\frac {\frac {-\frac {\int -\frac {b d \left (\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) x^2+a c (b c e+a d e-10 a c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle e \left (-\frac {\frac {\frac {\int \frac {b d \left (\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) x^2+a c (b c e+a d e-10 a c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e \left (-\frac {\frac {\frac {b d \int \frac {\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) x^2+a c (b c e+a d e-10 a c f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle e \left (-\frac {\frac {\frac {b d \left (\left (a^2 d (2 d e-5 c f)-a b c (5 c f+2 d e)+2 b^2 c^2 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (-10 a c f+a d e+b c e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle e \left (-\frac {\frac {\frac {b d \left (\left (a^2 d (2 d e-5 c f)-a b c (5 c f+2 d e)+2 b^2 c^2 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (-10 a c f+a d e+b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle e \left (-\frac {\frac {\frac {b d \left (\left (a^2 d (2 d e-5 c f)-a b c (5 c f+2 d e)+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (-10 a c f+a d e+b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle e \left (-\frac {\frac {\frac {b d \left (\left (a^2 d (2 d e-5 c f)-a b c (5 c f+2 d e)+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (-10 a c f+a d e+b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )+\frac {f \left (\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f+3 b c f+2 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b d (6 a c f+a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 c f+d e)}{c x}}{3 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 a x^3}\right )}{e^2}\) |
Input:
Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2)/x^6,x]
Output:
(f*(-1/3*(e*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(a*x^3) + (-((a*(d*e + 3*c* f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x)) + (b*d*(b*c*e + a*d*e + 6*a*c*f )*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*Elli pticE[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)*(2*b*d*e + 3*b*c* f + 3*a*d*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] ))/c)/(3*a)))/e^2 + e*(-1/5*(e*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(a*x^5) - (-1/3*((2*b*c*e - a*d*e - 5*a*c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x ^3) + (-((((2*b^2*c*e)/a - 2*b*d*e + (2*a*d^2*e)/c - 5*b*c*f - 5*a*d*f)*Sq rt[a + b*x^2]*Sqrt[c + d*x^2])/x) + (b*d*((2*b^2*c^2*e + a^2*d*(2*d*e - 5* c*f) - a*b*c*(2*d*e + 5*c*f))*((x*Sqrt[a + b*x^2])/(b*Sqrt[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))]*Sqrt[c + d*x^2])) + (c^(3/2)*(b*c*e + a*d*e - 10*a*c*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sq rt[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*c))/(5*a))
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/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1)) Int[(g*x) ^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 *(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
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 = 7.29 (sec) , antiderivative size = 547, normalized size of antiderivative = 0.51
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 x^{5}}-\frac {e \left (10 a c f +a d e +b c e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{15 a c \,x^{3}}-\frac {\left (15 a^{2} c^{2} f^{2}+10 a^{2} c d e f -2 a^{2} d^{2} e^{2}+10 a b \,c^{2} e f +2 a b c d \,e^{2}-2 b^{2} c^{2} e^{2}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{15 a^{2} c^{2} x}+\frac {\left (a d \,f^{2}+b c \,f^{2}+2 d b e f -\frac {b d e \left (10 a c f +a d e +b c e \right )}{15 a c}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (b d \,f^{2}+\frac {b d \left (15 a^{2} c^{2} f^{2}+10 a^{2} c d e f -2 a^{2} d^{2} e^{2}+10 a b \,c^{2} e f +2 a b c d \,e^{2}-2 b^{2} c^{2} e^{2}\right )}{15 a^{2} c^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(547\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (15 f^{2} x^{4} a^{2} c^{2}+10 a^{2} c d e f \,x^{4}-2 a^{2} d^{2} e^{2} x^{4}+10 a b \,c^{2} e f \,x^{4}+2 a b c d \,e^{2} x^{4}-2 b^{2} c^{2} e^{2} x^{4}+10 a^{2} c^{2} e f \,x^{2}+a^{2} c d \,e^{2} x^{2}+a b \,c^{2} e^{2} x^{2}+3 a^{2} c^{2} e^{2}\right )}{15 x^{5} a^{2} c^{2}}+\frac {\left (-\frac {2 b \left (15 a^{2} c^{2} f^{2}+5 a^{2} c d e f -a^{2} d^{2} e^{2}+5 a b \,c^{2} e f +a b c d \,e^{2}-b^{2} c^{2} e^{2}\right ) c \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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {15 a^{2} b \,c^{3} f^{2} \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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {15 a^{3} c^{2} d \,f^{2} \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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {a \,b^{2} c^{2} d \,e^{2} \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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {a^{2} b c \,d^{2} e^{2} \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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {20 a^{2} b \,c^{2} d e f \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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 a^{2} c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(870\) |
| default | \(\text {Expression too large to display}\) | \(1446\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^6,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)*(-1/5*e^2*(b*d *x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^5-1/15*e*(10*a*c*f+a*d*e+b*c*e)/a/c*(b*d *x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3-1/15/a^2/c^2*(15*a^2*c^2*f^2+10*a^2*c* d*e*f-2*a^2*d^2*e^2+10*a*b*c^2*e*f+2*a*b*c*d*e^2-2*b^2*c^2*e^2)*(b*d*x^4+a *d*x^2+b*c*x^2+a*c)^(1/2)/x+(a*d*f^2+b*c*f^2+2*d*b*e*f-1/15*b*d*e*(10*a*c* f+a*d*e+b*c*e)/a/c)/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d* x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b) ^(1/2))-(b*d*f^2+1/15*b*d*(15*a^2*c^2*f^2+10*a^2*c*d*e*f-2*a^2*d^2*e^2+10* a*b*c^2*e*f+2*a*b*c*d*e^2-2*b^2*c^2*e^2)/a^2/c^2)*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*(Elliptic F(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))))
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^6} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{x^{6}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^6,x, algorithm="fr icas")
Output:
integral((f^2*x^4 + 2*e*f*x^2 + e^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/x^6, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^6} \, dx=\int \frac {\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}{x^{6}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)*(f*x**2+e)**2/x**6,x)
Output:
Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**2/x**6, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^6} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{x^{6}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^6,x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2/x^6, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^6} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{x^{6}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^6,x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2/x^6, x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^6} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2}{x^6} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2)/x^6,x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2)/x^6, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^6} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^6,x)
Output:
( - 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d*f**2*x**2 + 10*sqrt(c + d *x**2)*sqrt(a + b*x**2)*a**2*d**2*f**2*x**4 - 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c**2*f**2*x**2 - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*e** 2 - 10*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*e*f*x**2 + 15*sqrt(c + d* x**2)*sqrt(a + b*x**2)*a*b*c*d*f**2*x**4 + 10*sqrt(c + d*x**2)*sqrt(a + b* x**2)*a*b*d**2*e*f*x**4 + 10*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f **2*x**4 + 10*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*e*f*x**4 - 2*sqrt (c + d*x**2)*sqrt(a + b*x**2)*b**2*d**2*e**2*x**4 - 10*int((sqrt(c + d*x** 2)*sqrt(a + b*x**2)*x**2)/(a**2*c*d + a**2*d**2*x**2 + a*b*c**2 + 2*a*b*c* d*x**2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x**4),x)*a**3*b*d**4*f* *2*x**5 - 20*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c*d + a**2 *d**2*x**2 + a*b*c**2 + 2*a*b*c*d*x**2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x**4),x)*a**2*b**2*c*d**3*f**2*x**5 - 10*int((sqrt(c + d*x**2)*sq rt(a + b*x**2)*x**2)/(a**2*c*d + a**2*d**2*x**2 + a*b*c**2 + 2*a*b*c*d*x** 2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x**4),x)*a**2*b**2*d**4*e*f* x**5 - 20*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c*d + a**2*d* *2*x**2 + a*b*c**2 + 2*a*b*c*d*x**2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b** 2*c*d*x**4),x)*a*b**3*c**2*d**2*f**2*x**5 - 20*int((sqrt(c + d*x**2)*sqrt( a + b*x**2)*x**2)/(a**2*c*d + a**2*d**2*x**2 + a*b*c**2 + 2*a*b*c*d*x**2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x**4),x)*a*b**3*c*d**3*e*f*x...