Integrand size = 35, antiderivative size = 1641 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^8} \, dx =\text {Too large to display} \] Output:
1/105*b*(8*b^3*c^3*e^2-a*b^2*c^2*e*(28*c*f+5*d*e)-a^2*b*c*(-35*c^2*f^2-28* c*d*e*f+5*d^2*e^2)+a^3*d*(35*c^2*f^2-28*c*d*e*f+8*d^2*e^2))*x*(d*x^2+c)^(1 /2)/a^3/c^3/(b*x^2+a)^(1/2)-1/105*(4*b^3*c^2*d*e^4+3*a^3*c*f^2*(5*c^2*f^2- 16*c*d*e*f+16*d^2*e^2)+a*b^2*c*e^2*(48*c^2*f^2+22*c*d*e*f+13*d^2*e^2)+a^2* b*e*(-48*c^3*f^3-95*c^2*d*e*f^2+22*c*d^2*e^2*f+4*d^3*e^3))*x*(b*x^2+a)^(1/ 2)*(d*x^2+c)^(1/2)/a^3/c^3/e^2-1/105*(12*b^3*c*e^3*(4*c^2*f^2+3*c*d*e*f+d^ 2*e^2)+a^2*b*e*f*(-29*c^3*f^3-134*c^2*d*e*f^2+24*c*d^2*e^2*f+36*d^3*e^3)+2 *a*b^2*e^2*(8*c^3*f^3+12*c^2*d*e*f^2+15*c*d^2*e^2*f+6*d^3*e^3)+a^3*f^2*(10 *c^3*f^3-29*c^2*d*e*f^2+16*c*d^2*e^2*f+48*d^3*e^3))*x^3*(b*x^2+a)^(1/2)*(d *x^2+c)^(1/2)/a^3/c^3/e^3-1/105*f*(4*b^3*c*e^3*(16*c^2*f^2+21*c*d*e*f+9*d^ 2*e^2)+2*a*b^2*e^2*(-10*c^3*f^3-5*c^2*d*e*f^2+6*c*d^2*e^2*f+18*d^3*e^3)+2* a^2*b*e*f*(-c^3*f^3-41*c^2*d*e*f^2-5*c*d^2*e^2*f+42*d^3*e^3)+a^3*f^2*(3*c^ 3*f^3-2*c^2*d*e*f^2-20*c*d^2*e^2*f+64*d^3*e^3))*x^5*(b*x^2+a)^(1/2)*(d*x^2 +c)^(1/2)/a^3/c^3/e^4-1/105*f^2*(3*a^3*d*f^2*(c^2*f^2-4*c*d*e*f+8*d^2*e^2) +4*b^3*c*e^2*(6*c^2*f^2+19*c*d*e*f+9*d^2*e^2)+a*b^2*e*(-12*c^3*f^3-29*c^2* d*e*f^2-14*c*d^2*e^2*f+36*d^3*e^3)+a^2*b*f*(3*c^3*f^3-14*c^2*d*e*f^2-29*c* d^2*e^2*f+76*d^3*e^3))*x^7*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^3/c^3/e^4-1/3 5*b*d*f^3*(4*b^2*c*e^2*(2*c*f+d*e)+a*b*e*(-4*c^2*f^2-3*c*d*e*f+4*d^2*e^2)+ a^2*f*(c^2*f^2-4*c*d*e*f+8*d^2*e^2))*x^9*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a ^3/c^3/e^4-1/7*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)*(f*x^2+e)^3/a/c/e/x^7+1/...
Result contains complex when optimal does not.
Time = 4.16 (sec) , antiderivative size = 520, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^8} \, dx=\frac {-\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (8 b^3 c^3 e^2 x^6-a b^2 c^2 e x^4 \left (5 d e x^2+4 c \left (e+7 f x^2\right )\right )+a^2 b c x^2 \left (-5 d^2 e^2 x^4+2 c d e x^2 \left (e+14 f x^2\right )+c^2 \left (3 e^2+14 e f x^2+35 f^2 x^4\right )\right )+a^3 \left (c+d x^2\right ) \left (8 d^2 e^2 x^4-4 c d e x^2 \left (3 e+7 f x^2\right )+c^2 \left (15 e^2+42 e f x^2+35 f^2 x^4\right )\right )\right )-i b c \left (8 b^3 c^3 e^2-a b^2 c^2 e (5 d e+28 c f)+a^3 d \left (8 d^2 e^2-28 c d e f+35 c^2 f^2\right )+a^2 b c \left (-5 d^2 e^2+28 c d e f+35 c^2 f^2\right )\right ) x^7 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c (b c-a d) \left (8 b^2 c^2 e^2-a b c e (d e+28 c f)+a^2 \left (-4 d^2 e^2+14 c d e f+35 c^2 f^2\right )\right ) x^7 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 a^3 \sqrt {\frac {b}{a}} c^3 x^7 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2)/x^8,x]
Output:
(-(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(8*b^3*c^3*e^2*x^6 - a*b^2*c^2*e*x^4* (5*d*e*x^2 + 4*c*(e + 7*f*x^2)) + a^2*b*c*x^2*(-5*d^2*e^2*x^4 + 2*c*d*e*x^ 2*(e + 14*f*x^2) + c^2*(3*e^2 + 14*e*f*x^2 + 35*f^2*x^4)) + a^3*(c + d*x^2 )*(8*d^2*e^2*x^4 - 4*c*d*e*x^2*(3*e + 7*f*x^2) + c^2*(15*e^2 + 42*e*f*x^2 + 35*f^2*x^4)))) - I*b*c*(8*b^3*c^3*e^2 - a*b^2*c^2*e*(5*d*e + 28*c*f) + a ^3*d*(8*d^2*e^2 - 28*c*d*e*f + 35*c^2*f^2) + a^2*b*c*(-5*d^2*e^2 + 28*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*(b*c - a*d)*(8*b^2*c^2*e^2 - a *b*c*e*(d*e + 28*c*f) + a^2*(-4*d^2*e^2 + 14*c*d*e*f + 35*c^2*f^2))*x^7*Sq rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a *d)/(b*c)])/(105*a^3*Sqrt[b/a]*c^3*x^7*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 1.90 (sec) , antiderivative size = 1012, normalized size of antiderivative = 0.62, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {448, 442, 25, 442, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^8} \, dx\) |
| \(\Big \downarrow \) 448 |
| \(\displaystyle \frac {f \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}{x^6}dx}{e^2}+e \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}{x^8}dx\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {f \left (\frac {\int -\frac {\sqrt {b x^2+a} \left (d (b e-5 a f) x^2+2 b c e-a d e-5 a c f\right )}{x^4 \sqrt {d x^2+c}}dx}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )}{e^2}+e \left (\frac {\int -\frac {\sqrt {b x^2+a} \left (d (3 b e-7 a f) x^2+4 b c e-a d e-7 a c f\right )}{x^6 \sqrt {d x^2+c}}dx}{7 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 a x^7}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f \left (-\frac {\int \frac {\sqrt {b x^2+a} \left (d (b e-5 a f) x^2+2 b c e-a d e-5 a c f\right )}{x^4 \sqrt {d x^2+c}}dx}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )}{e^2}+e \left (-\frac {\int \frac {\sqrt {b x^2+a} \left (d (3 b e-7 a f) x^2+4 b c e-a d e-7 a c f\right )}{x^6 \sqrt {d x^2+c}}dx}{7 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 a x^7}\right )\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle \frac {f \left (-\frac {\frac {\int \frac {d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+b d (b c e+a d e-10 a c f) x^2+2 b^2 c^2 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )}{e^2}+e \left (-\frac {\frac {\int \frac {d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+b d (3 b c e+3 a d e-14 a c f) x^2+4 b^2 c^2 e}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f-a d e+4 b c e)}{5 c x^5}}{7 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 a x^7}\right )\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {f \left (-\frac {\frac {-\frac {\int -\frac {b d \left (\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) x^2+a c (b c e+a d e-10 a c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )}{e^2}+e \left (-\frac {\frac {-\frac {\int \frac {2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+b d \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right ) x^2+8 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {4 b^2 c e}{a}+\frac {4 a d^2 e}{c}-7 a d f-7 b c f-2 b d e\right )}{3 x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f-a d e+4 b c e)}{5 c x^5}}{7 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 a x^7}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f \left (-\frac {\frac {\frac {\int \frac {b d \left (\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) x^2+a c (b c e+a d e-10 a c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )}{e^2}+e \left (-\frac {\frac {-\frac {\int \frac {2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+b d \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right ) x^2+8 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {4 b^2 c e}{a}+\frac {4 a d^2 e}{c}-7 a d f-7 b c f-2 b d e\right )}{3 x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f-a d e+4 b c e)}{5 c x^5}}{7 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 a x^7}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f \left (-\frac {\frac {\frac {b d \int \frac {\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) x^2+a c (b c e+a d e-10 a c f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )}{e^2}+e \left (-\frac {\frac {-\frac {\int \frac {2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+b d \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right ) x^2+8 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {4 b^2 c e}{a}+\frac {4 a d^2 e}{c}-7 a d f-7 b c f-2 b d e\right )}{3 x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f-a d e+4 b c e)}{5 c x^5}}{7 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 a x^7}\right )\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {f \left (-\frac {\frac {\frac {b d \left (\left (a^2 d (2 d e-5 c f)-a b c (5 c f+2 d e)+2 b^2 c^2 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (-10 a c f+a d e+b c e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )}{e^2}+e \left (-\frac {\frac {-\frac {\int \frac {2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+b d \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right ) x^2+8 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {4 b^2 c e}{a}+\frac {4 a d^2 e}{c}-7 a d f-7 b c f-2 b d e\right )}{3 x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f-a d e+4 b c e)}{5 c x^5}}{7 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 a x^7}\right )\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f \left (-\frac {\frac {\frac {b d \left (\left (a^2 d (2 d e-5 c f)-a b c (5 c f+2 d e)+2 b^2 c^2 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (-10 a c f+a d e+b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )}{e^2}+e \left (-\frac {\frac {-\frac {\int \frac {2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+b d \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right ) x^2+8 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {4 b^2 c e}{a}+\frac {4 a d^2 e}{c}-7 a d f-7 b c f-2 b d e\right )}{3 x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f-a d e+4 b c e)}{5 c x^5}}{7 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 a x^7}\right )\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {f \left (-\frac {\frac {\frac {b d \left (\left (a^2 d (2 d e-5 c f)-a b c (5 c f+2 d e)+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (-10 a c f+a d e+b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )}{e^2}+e \left (-\frac {\frac {-\frac {\int \frac {2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+b d \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right ) x^2+8 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {4 b^2 c e}{a}+\frac {4 a d^2 e}{c}-7 a d f-7 b c f-2 b d e\right )}{3 x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f-a d e+4 b c e)}{5 c x^5}}{7 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 a x^7}\right )\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle e \left (-\frac {\frac {-\frac {\int \frac {2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+b d \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right ) x^2+8 b^3 c^3 e}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {4 b^2 c e}{a}+\frac {4 a d^2 e}{c}-7 a d f-7 b c f-2 b d e\right )}{3 x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-7 a c f-a d e+4 b c e)}{5 c x^5}}{7 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 a x^7}\right )+\frac {f \left (-\frac {\frac {\frac {b d \left (\left (a^2 d (2 d e-5 c f)-a b c (5 c f+2 d e)+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (-10 a c f+a d e+b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}+\frac {2 a d^2 e}{c}-5 a d f-5 b c f-2 b d e\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-5 a c f-a d e+2 b c e)}{3 c x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 a x^5}\right )}{e^2}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {f \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 a x^5}-\frac {\frac {\frac {b d \left (\frac {(b c e+a d e-10 a c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{a c}-\frac {\left (\frac {2 c e b^2}{a}-2 d e b-5 c f b+\frac {2 a d^2 e}{c}-5 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 c}-\frac {(2 b c e-a d e-5 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 c x^3}}{5 a}\right )}{e^2}+e \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{7 a x^7}-\frac {\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {4 c e b^2}{a}-2 d e b-7 c f b+\frac {4 a d^2 e}{c}-7 a d f\right )}{3 x^3}-\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right )}{a c x}-\frac {\int -\frac {b d \left (\left (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) x^2+a c \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}}{3 a c}}{5 c}-\frac {(4 b c e-a d e-7 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 c x^5}}{7 a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 a x^5}-\frac {\frac {\frac {b d \left (\frac {(b c e+a d e-10 a c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{a c}-\frac {\left (\frac {2 c e b^2}{a}-2 d e b-5 c f b+\frac {2 a d^2 e}{c}-5 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 c}-\frac {(2 b c e-a d e-5 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 c x^3}}{5 a}\right )}{e^2}+e \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{7 a x^7}-\frac {\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {4 c e b^2}{a}-2 d e b-7 c f b+\frac {4 a d^2 e}{c}-7 a d f\right )}{3 x^3}-\frac {\frac {\int \frac {b d \left (\left (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) x^2+a c \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\left (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 c}-\frac {(4 b c e-a d e-7 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 c x^5}}{7 a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 a x^5}-\frac {\frac {\frac {b d \left (\frac {(b c e+a d e-10 a c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{a c}-\frac {\left (\frac {2 c e b^2}{a}-2 d e b-5 c f b+\frac {2 a d^2 e}{c}-5 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 c}-\frac {(2 b c e-a d e-5 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 c x^3}}{5 a}\right )}{e^2}+e \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{7 a x^7}-\frac {\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {4 c e b^2}{a}-2 d e b-7 c f b+\frac {4 a d^2 e}{c}-7 a d f\right )}{3 x^3}-\frac {\frac {b d \int \frac {\left (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) x^2+a c \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\left (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 c}-\frac {(4 b c e-a d e-7 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 c x^5}}{7 a}\right )\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {f \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 a x^5}-\frac {\frac {\frac {b d \left (\frac {(b c e+a d e-10 a c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{a c}-\frac {\left (\frac {2 c e b^2}{a}-2 d e b-5 c f b+\frac {2 a d^2 e}{c}-5 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 c}-\frac {(2 b c e-a d e-5 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 c x^3}}{5 a}\right )}{e^2}+e \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{7 a x^7}-\frac {\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {4 c e b^2}{a}-2 d e b-7 c f b+\frac {4 a d^2 e}{c}-7 a d f\right )}{3 x^3}-\frac {\frac {b d \left (a c \left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\left (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 c}-\frac {(4 b c e-a d e-7 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 c x^5}}{7 a}\right )\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 a x^5}-\frac {\frac {\frac {b d \left (\frac {(b c e+a d e-10 a c f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (d (2 d e-5 c f) a^2-b c (2 d e+5 c f) a+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{a c}-\frac {\left (\frac {2 c e b^2}{a}-2 d e b-5 c f b+\frac {2 a d^2 e}{c}-5 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 c}-\frac {(2 b c e-a d e-5 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 c x^3}}{5 a}\right )}{e^2}+e \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{7 a x^7}-\frac {\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {4 c e b^2}{a}-2 d e b-7 c f b+\frac {4 a d^2 e}{c}-7 a d f\right )}{3 x^3}-\frac {\frac {b d \left (\frac {\left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 b^2 c^2 e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\left (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\left (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 c}-\frac {(4 b c e-a d e-7 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 c x^5}}{7 a}\right )\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {f \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 a x^5}-\frac {\frac {\frac {b d \left (\frac {(b c e+a d e-10 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 (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 ) \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 {2 c e b^2}{a}-2 d e b-5 c f b+\frac {2 a d^2 e}{c}-5 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 c}-\frac {(2 b c e-a d e-5 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 c x^3}}{5 a}\right )}{e^2}+e \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{7 a x^7}-\frac {\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {4 c e b^2}{a}-2 d e b-7 c f b+\frac {4 a d^2 e}{c}-7 a d f\right )}{3 x^3}-\frac {\frac {b d \left (\frac {\left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 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 (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 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 (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 c}-\frac {(4 b c e-a d e-7 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 c x^5}}{7 a}\right )\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {f \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{5 a x^5}-\frac {\frac {\frac {b d \left (\frac {(b c e+a d e-10 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 (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 ) \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 {2 c e b^2}{a}-2 d e b-5 c f b+\frac {2 a d^2 e}{c}-5 a d f\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{x}}{3 c}-\frac {(2 b c e-a d e-5 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{3 c x^3}}{5 a}\right )}{e^2}+e \left (-\frac {e \sqrt {d x^2+c} \left (b x^2+a\right )^{3/2}}{7 a x^7}-\frac {\frac {-\frac {\sqrt {b x^2+a} \sqrt {d x^2+c} \left (\frac {4 c e b^2}{a}-2 d e b-7 c f b+\frac {4 a d^2 e}{c}-7 a d f\right )}{3 x^3}-\frac {\frac {b d \left (\frac {\left (d (4 d e-7 c f) a^2-b c (2 d e+7 c f) a+4 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 (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 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 (2 d^2 (4 d e-7 c f) a^3-b c d (5 d e-14 c f) a^2-b^2 c^2 (5 d e+14 c f) a+8 b^3 c^3 e\right ) \sqrt {b x^2+a} \sqrt {d x^2+c}}{a c x}}{3 a c}}{5 c}-\frac {(4 b c e-a d e-7 a c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}{5 c x^5}}{7 a}\right )\) |
Input:
Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2)/x^8,x]
Output:
(f*(-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)*Sqrt[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]*S qrt[(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[(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*c))/(5*a)))/e^2 + e*(-1/7*(e*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(a*x^7) - (-1/5*((4*b*c*e - a*d*e - 7*a*c*f)*Sqrt[a + b*x^2]*Sqr t[c + d*x^2])/(c*x^5) + (-1/3*(((4*b^2*c*e)/a - 2*b*d*e + (4*a*d^2*e)/c - 7*b*c*f - 7*a*d*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/x^3 - (-(((8*b^3*c^3*e - a^2*b*c*d*(5*d*e - 14*c*f) + 2*a^3*d^2*(4*d*e - 7*c*f) - a*b^2*c^2*(5*d *e + 14*c*f))*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x)) + (b*d*((8*b^3*c^3 *e - a^2*b*c*d*(5*d*e - 14*c*f) + 2*a^3*d^2*(4*d*e - 7*c*f) - a*b^2*c^2*(5 *d*e + 14*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)*(4* b^2*c^2*e + a^2*d*(4*d*e - 7*c*f) - a*b*c*(2*d*e + 7*c*f))*Sqrt[a + b*x...
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 = 8.57 (sec) , antiderivative size = 755, normalized size of antiderivative = 0.46
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 x^{7}}-\frac {e \left (14 a c f +a d e +b c e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{35 a c \,x^{5}}-\frac {\left (35 a^{2} c^{2} f^{2}+14 a^{2} c d e f -4 a^{2} d^{2} e^{2}+14 a b \,c^{2} e f +2 a b c d \,e^{2}-4 b^{2} c^{2} e^{2}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{105 a^{2} c^{2} x^{3}}-\frac {\left (35 a^{3} c^{2} d \,f^{2}-28 a^{3} c \,d^{2} e f +8 a^{3} d^{3} e^{2}+35 a^{2} b \,c^{3} f^{2}+28 a^{2} b \,c^{2} d e f -5 a^{2} b c \,d^{2} e^{2}-28 a \,b^{2} c^{3} e f -5 a \,b^{2} c^{2} d \,e^{2}+8 b^{3} c^{3} e^{2}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{105 a^{3} c^{3} x}+\frac {\left (b d \,f^{2}-\frac {b d \left (35 a^{2} c^{2} f^{2}+14 a^{2} c d e f -4 a^{2} d^{2} e^{2}+14 a b \,c^{2} e f +2 a b c d \,e^{2}-4 b^{2} c^{2} e^{2}\right )}{105 a^{2} c^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b \left (35 a^{3} c^{2} d \,f^{2}-28 a^{3} c \,d^{2} e f +8 a^{3} d^{3} e^{2}+35 a^{2} b \,c^{3} f^{2}+28 a^{2} b \,c^{2} d e f -5 a^{2} b c \,d^{2} e^{2}-28 a \,b^{2} c^{3} e f -5 a \,b^{2} c^{2} d \,e^{2}+8 b^{3} c^{3} e^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{105 c^{2} a^{3} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(755\) |
| risch | \(\text {Expression too large to display}\) | \(1163\) |
| default | \(\text {Expression too large to display}\) | \(2169\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^8,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*e^2*(b*d *x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^7-1/35*e*(14*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^5-1/105/a^2/c^2*(35*a^2*c^2*f^2+14*a^2*c *d*e*f-4*a^2*d^2*e^2+14*a*b*c^2*e*f+2*a*b*c*d*e^2-4*b^2*c^2*e^2)*(b*d*x^4+ a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3-1/105/a^3/c^3*(35*a^3*c^2*d*f^2-28*a^3*c*d^ 2*e*f+8*a^3*d^3*e^2+35*a^2*b*c^3*f^2+28*a^2*b*c^2*d*e*f-5*a^2*b*c*d^2*e^2- 28*a*b^2*c^3*e*f-5*a*b^2*c^2*d*e^2+8*b^3*c^3*e^2)*(b*d*x^4+a*d*x^2+b*c*x^2 +a*c)^(1/2)/x+(b*d*f^2-1/105*b*d*(35*a^2*c^2*f^2+14*a^2*c*d*e*f-4*a^2*d^2* e^2+14*a*b*c^2*e*f+2*a*b*c*d*e^2-4*b^2*c^2*e^2)/a^2/c^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)*Ellipt icF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-1/105*b*(35*a^3*c^2*d*f^2-28* a^3*c*d^2*e*f+8*a^3*d^3*e^2+35*a^2*b*c^3*f^2+28*a^2*b*c^2*d*e*f-5*a^2*b*c* d^2*e^2-28*a*b^2*c^3*e*f-5*a*b^2*c^2*d*e^2+8*b^3*c^3*e^2)/c^2/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))-EllipticE(x*(-b/a)^ (1/2),(-1+(a*d+b*c)/c/b)^(1/2))))
Time = 0.10 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.36 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^8} \, dx=\frac {{\left ({\left (8 \, b^{4} c^{3} - 5 \, a b^{3} c^{2} d - 5 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e^{2} - 28 \, {\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} e f + 35 \, {\left (a^{2} b^{2} c^{3} + a^{3} b c^{2} d\right )} f^{2}\right )} \sqrt {a c} x^{7} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (8 \, b^{4} c^{3} + {\left (4 \, a^{2} b^{2} - 5 \, a b^{3}\right )} c^{2} d - {\left (2 \, a^{3} b + 5 \, a^{2} b^{2}\right )} c d^{2} + 4 \, {\left (a^{4} + 2 \, a^{3} b\right )} d^{3}\right )} e^{2} - 14 \, {\left (2 \, a b^{3} c^{3} + {\left (a^{3} b - 2 \, a^{2} b^{2}\right )} c^{2} d + {\left (a^{4} + 2 \, a^{3} b\right )} c d^{2}\right )} e f + 35 \, {\left (a^{2} b^{2} c^{3} + {\left (2 \, a^{4} + a^{3} b\right )} c^{2} d\right )} f^{2}\right )} \sqrt {a c} x^{7} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, a^{4} c^{3} e^{2} + {\left ({\left (8 \, a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d - 5 \, a^{3} b c d^{2} + 8 \, a^{4} d^{3}\right )} e^{2} - 28 \, {\left (a^{2} b^{2} c^{3} - a^{3} b c^{2} d + a^{4} c d^{2}\right )} e f + 35 \, {\left (a^{3} b c^{3} + a^{4} c^{2} d\right )} f^{2}\right )} x^{6} + {\left (35 \, a^{4} c^{3} f^{2} - 2 \, {\left (2 \, a^{2} b^{2} c^{3} - a^{3} b c^{2} d + 2 \, a^{4} c d^{2}\right )} e^{2} + 14 \, {\left (a^{3} b c^{3} + a^{4} c^{2} d\right )} e f\right )} x^{4} + 3 \, {\left (14 \, a^{4} c^{3} e f + {\left (a^{3} b c^{3} + a^{4} c^{2} d\right )} e^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, a^{4} c^{3} x^{7}} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^8,x, algorithm="fr icas")
Output:
1/105*(((8*b^4*c^3 - 5*a*b^3*c^2*d - 5*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*e^2 - 28*(a*b^3*c^3 - a^2*b^2*c^2*d + 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)) - ((8*b^4*c^3 + (4*a^2*b^2 - 5*a*b^3)*c^2*d - (2*a^3*b + 5*a^2*b^2 )*c*d^2 + 4*(a^4 + 2*a^3*b)*d^3)*e^2 - 14*(2*a*b^3*c^3 + (a^3*b - 2*a^2*b^ 2)*c^2*d + (a^4 + 2*a^3*b)*c*d^2)*e*f + 35*(a^2*b^2*c^3 + (2*a^4 + a^3*b)* c^2*d)*f^2)*sqrt(a*c)*x^7*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/ (b*c)) - (15*a^4*c^3*e^2 + ((8*a*b^3*c^3 - 5*a^2*b^2*c^2*d - 5*a^3*b*c*d^2 + 8*a^4*d^3)*e^2 - 28*(a^2*b^2*c^3 - a^3*b*c^2*d + 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 - 2*(2*a^2*b^2*c^3 - a^3* b*c^2*d + 2*a^4*c*d^2)*e^2 + 14*(a^3*b*c^3 + a^4*c^2*d)*e*f)*x^4 + 3*(14*a ^4*c^3*e*f + (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^4*c^3*x^7)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^8} \, dx=\int \frac {\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}{x^{8}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)*(f*x**2+e)**2/x**8,x)
Output:
Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**2/x**8, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^8} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{x^{8}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^8,x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2/x^8, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^8} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{x^{8}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^8,x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2/x^8, x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^8} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2}{x^8} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2)/x^8,x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2)/x^8, x)
\[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2}{x^8} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2/x^8,x)
Output:
(7*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f**2*x**2 + 7*sqrt(c + d*x**2)*sq rt(a + b*x**2)*b*c*f**2*x**2 - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e** 2 - 14*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*f*x**2 - 21*sqrt(c + d*x**2 )*sqrt(a + b*x**2)*b*d*f**2*x**4 + 35*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**3*c*d**2*f**2*x**7 + 70*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*b*c**2*d*f**2*x**7 - 28*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*b*c*d**2*e*f* x**7 + 3*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*b*d**3*e**2*x**7 + 35*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**2*c**3*f* *2*x**7 - 28*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**2*c**2*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...